Order of subgroup of symmetric group within an interval

Question

Consider symmetric group $S_n$. I want to find a subgroup of $S_n$ whose order is around a number $m$. For example I want to find a subgroup whose order is within 10000 to 12000 of $S_{17}$. What is the procedure? Please help

Answer 1

The group $S_{17}$ has order $$ 2^{15}\cdot 3^6\cdot 5^3\cdot 7^2\cdot 11\cdot 13\cdot 17. $$ Hence the largest nilpotent subgroup has order $2^{15}$. It is a Sylow $2$-subgroup. The subgroups of it have orders $2^k$ with $k\le 15$, which are not between $10000$ and $12000$. What about a direct product of Sylow subgroups?

Concerning large solvable subgroups, see

What are the solvable subgroups of $S_n$?