Complexity of computing the order of a permutation group Given two permutations $g$ and $h$ over $n$ elements (i.e., members of $S_n$), what is the complexity of computing the order of the subgroup generated by $g,h$? Or just of deciding whether the subgroup is of order $n!$ (i.e., all of $S_n$)?
The order of permutation groups can be computed in polynomial-time. In fact, I believe even in $\mathsf{NC}$ and also nearly linear Las Vegas time. See, e.g., the book by Seress. 

For reference, subgroups of $S_n$ (and algorithms related thereto) are typically called "permutation groups" rather than merely "subgroups (of $S_n$)". So you can google "permutation group algorithms" etc.