How do nations maintain sovereignty over conquered nations?

William did not have any claim to the throne, except by conquest after winning. The kings of Denmark and Norway had better claims.

King Edward the confessor did not have the right to promise the throne to either Harold or William (except as a cynical political maneuver to keep them from making trouble during his reign). King Edward did not have the right to sincerely promise either of them the throne or anything more than an important place in the regency for his rightful heir, Edgar the Atheling, who in fact was proclaimed king in London after the battle of Hastings.

Since Harold Godwinson was not the rightful heir to England, any promise that William forced him to make could not give William any rightful claim to the throne. Harold would have been equally traitorous if he pushed for the throne for himself or for William. As for punishing Harold for breaking his oath to do something unlawful, it would have been far more ethical for William to send hordes of hired assassins after Harold than to punish all 1.7 million English people with William's cruel tyranny.

I think the general answer is that successful conquerers:

• take the top positions of power (ideally decapitating the entire top eschelon) but leave the administration/bureaucratic system below them largely in place initially.

• exploit divisions among those who might unite against them

• use religion

See all three of these techniques in use by William, Cortez, Pizarro and Caesar.

This answer is more political that purely historical; it should have been a comment but I needed the extra space.

I believe you are thinking in modern terms: the state-nation, where the people is sovereign and can elect its own form of government. This concept, stablished as it seems, is relatively new (Age of Enlightment, American Declaration of Independence and French Revolution). Before that, the people sheldom was a political subject¹ and was considered under the control of a higher, usually military caste. The idea that english people should revolt because they liked Harold better than Edward was like political-fiction at the time (and it is worth noting that it isn't that such people were never asked if they liked Harold in the first place).

Even the Roman Empire worked this way, with a tiny nucleus of citizens dominating a mass of non-citizens. In the conquered territories, if the chieftain was amenable to submit and pay taxes they would not interfere with him or they would even support him in dominating his subjects; if the local power caused trouble then they would send in the legions.

Later this Roman citizen/non-citizen relationship was changed to a military/civil relationship, with the emperors being so due to its control of the Army. They still used the civilian structures (v.g. Senate) but whenever it became unruly the army would step in to "restore order"

When the Germanic tribes took over they did pretty much the same, they simply destroyed / took over the Roman military and replaced it with its own; what was left was already used to obey whoever was in charge. Those power structures left by the Romans (vg. Church, or local chieftains) simply had to negotiate with the new lords which would be the balance of power; some people could not like it and rebel, but those were not "popular" uprisings but rebellions by local leaders².

The conquest of England was more or less the same: take out the top dog, destroy organized opposition and then rule; the main difficulty being that you have to reward your allies with lands so you would need to disposses the previous local nobles (at the risk of the occasional local rebellion).

¹: There were rebellions in the Middle Ages or before, but mostly due to lack of food and they sheldom tried to replace the stablished order but to lower taxes and similar situations.

²: The most notorious exception would be the case of religious wars, when the invader was of a different religion. For example, many Germanic tribes found it expedient to convert from Arrianism to Catholicism in order to avoid alienating too much their subjects.

A more "normal" ratio of military to population might be something like 1%. That ratio would imply 17,000 men for the Norman conquerers instead of 7,000.

There was one other factor in the Normans' favor. In modern times, guns are a great "equalizer." Not "everyone," but a large part of the population can be taught to use a gun in a short period of time. During World War II, the Russians (and later the Germans) sent teenaged boys into battle with only a few days' training in basic weapons management.

The "hand" weapons of the time took years, not days, to master. One trained man with armor and skill with a "regulation" weapon was easily the match of five or even ten untrained men with crude weapons and no armor. This inequality meant that the Normans could maintain control with even less than 1% of the population. And it was also true that the Normans were better and more heavily armed than Harold's "Anglo-Saxons" that they had just defeated.

There is a difference between "control" and sovereignty. After the Battle of Hastings it was clear William had the most powerful force, so he became sovereign. When he marched to London there was noone to oppose him, so the town capitulated to him. He took hostages in London and then went around demonstrating his power. This activity which took place for about two months between the middle of October and December 1066 is not well documented, but apparently he went in a circle around London killing, burning and pillaging as he went while various small local forces made sporadic resistance.

As this happened Edgar Ætheling was declared king by a Saxon Witenaġemot and so it became a sort of race against time as William tried to get local lords to pledge fealty to him before any kind of serious resistance could be formed against him. During this period of chaos his army made many massacres to discourage resistance. Eventually, late in December, the Archbishop and the nobles in the London area decided to crown William King.

Even after he was king in Westminster, there were many petty revolts and insubordination all throughout England to his rule and William spent years marching around subduing different people and places and building castles to enforce his rule. In many cases the forces in these castles were tiny compared to the local population, but their armor and weapons made it no easy matter to kill them.

In cases where a garrison was wiped out or there was serious resistance from a lord, William would march with an army to defeat it. Such expeditions were always extremely punitive. The army would kill everyone and burn anything they could to punish the local people for resisting. Such actions often created large numbers of refugees, fleeing for their lives. During this time Scotland actually gained a significant amount of population, just from refugees who were fleeing the conflict in England.

To answer your question in general: a military force acts as a coordinated team and coupled with powerful weapons it is difficult for lightly armed, unorganized people to contest. If all of England had gathered together, financed an army and organized themselves into a fighting unit they might have been able to defeat William, but they failed to do this. Do not underestimate the power of a military unit. For example, Cortez, with only about 120 men, conquered a civilization in which he battled against thousands of men all gathered against him. If you read the account of Cortez given by Bernal Diaz, one of his lieutenants, you will get an idea of how a tiny band of men who are well armed and coordinated can defeat much larger forces.