First of all yes, your answer for $S_7$ is correct. And indeed listing all cycle types quickly becomes unmanageable as $n$ grows.
It is quite helpful and not difficult to prove that the maximum is always assumed for a permutation that is the product of disjoint cycles of prime power lengths. For increasing values of $n$ this eliminates an increasing proportion of permutations; for $n=7$ and $n=10$ it does not eliminate much.
It is also quite helpful and not difficult to prove that the maximum is always assumed for a permutation that has at most one fixed point. This eliminates a large proportion of permutations for any $n$.
With these two facts in mind, for $n=7$ we are left with permutations of cycle types
$$(7),\ (5,2),\ (4,3),\ (4,2,1),\ (3,3,1),\ (3,2,2)\quad
\text{and}\quad (2,2,2,1).$$
For $n=10$ we are left with permutations of cycle types
$$(9,1),\ (8,2),\ (7,3),\ (7,2,1),\ (5,5),\ (5,4,1),\ (5,3,2),\ (4,4,2),\ (4,3,3),\ (4,3,2,1),\ (3,3,3,1),\ (3,3,2,2),\quad\text{and}\quad (2,2,2,2,2).$$
Of course one can already see some recursion in this problem; given that the maxima for $n=3$ and $n=5$ are assumed for cycle types $(3)$ and $(3,2)$, respectively, for $n=10$ we can eliminate the cycle types
$$(7,2,1),\ (5,5)\quad\text{and}\quad(5,4,1).$$
There are a lot of simple facts to note that each reduce the number of cycle types to consider by a little bit. For your problem there aren't that many to begin with, and the facts above are enough to make the problem manageable by hand. But it can just as well be done by hand for $n=100$ and $n=1000$, though this requires more careful analysis of the problem which I won't go into here. I'll leave you with this vague intuition though:
Heuristically, for large $n$ the maximum is assumed when writing $n$ as a sum of prime powers close to $\sqrt{n}$, and padding the remainder with small primes. It is not easy to compute the maximum precisely for large values of $n$, though asymptotically it is $\exp((1+o(1))\sqrt{n\ln{n}})$.
Best Answer
You will have to break it up by the partitions of n
In your case with $S_{7}$ you can break it up by, where (n) denotes a cycle of length n,
For a general approach to generating partitions of n, see this geeksforgeeks post: https://www.geeksforgeeks.org/generate-unique-partitions-of-an-integer/.
It can be proven that the order of each permutation that is a product of disjoint cycles is the least common multiple of all the orders of the disjoint cycles.