How many undead can a single Lich keep under his control permanently?

Question

Suppose your average Lich wants to collect some more souls, but doesn't want to leave his lair for whatever reason. So given he has basically unlimited amounts of magic and access to spells like Finger of Death and Animate Dead, how many undead can he control (can be both direct and indirect) for an unlimited amount of time, given he has as much time to set up as needed? Assume he rolls average on his dice to regain a spell slot if relevant for your solution.

I know he could get an unlimited number via creating a zombie via Animate Dead and then killing it with Finger of Death, but ignore this exploit of sorts for the time being.

Answer 1

Undead (Zombies) created using only Finger of Death - unlimited.

1802 (7201 with max rolls) per 24 hours

Since "unlimited" is not a satisfying answer, I will calculate how many Zombies can be created within 24 hours / one week. This also requires thousands of "willing" humanoids to come to the Lichs lair, however that might be accomplished.

Starting out, the Lich can cast Finger of Death 3 times using his 9th, 8th and 7th level spell slot: +3 Zombies in 3 rounds (18 seconds)

The following rounds he has a 25% chance to regain a 7th or 8th level spell slot whenever he takes a lair action, needed to cast Finger of Death. Unfortunately, his lair actions description states the following (emphasis mine):

Lair Actions

On initiative count 20 (losing initiative ties), the Lich can take a lair action to cause one of the following magical effects; the Lich can’t use the same effect two rounds in a row:

This means, on average, he regains a 7th or 8th level slot every 8 rounds, which can then be used in a subsequent casting of Finger of Death.

With the remaining 23 hours, 59 minutes and 42 seconds (= 14,397 rounds) of the day, assuming average dice rolls, the Lich can cast Finger of Death 1799 times (rounded down from 1799.625): 1799 Zombies in 14,397 rounds

This brings us to a total of 1802 Zombies withing 24 hours (assuming enough humanoids around)

If he were to always roll max on his dice to regain spent spell slots, this number increases to a total of 7201 Zombies.

If he continues this for a full week, he will end up with 12,614 (50,407) zombies in one week.