Don't forget Presidential nominations
All members of congress, senators, and the VP can have up to 5 appointees at a service academy at any time. And each of them can nominate up to 10 candidates for each of the academies to consider for appointment (the academy can choose from the slate of 10, or the MOC can specify a principal nominee (uncommon), or provide a ranking for the academy to appoint from (also uncommon)). Additionally, there is another pool of nominations that can be made from Presidential nomination pool of which there can be quite a few, but the number of these who can be appointed is limited.