It is known that the following three computational problems for subgroups $G$ of $S_n$ are polynomially equivalent:

1. Computing (generators of) the centralizer $C_G(g)$ of an element $g \in S_n$ (and also testing $g,h \in S_n$ for conjugacy in $G$).

2. Computing (generators of) the setwise stabilizer of a subset of the set of size $n$ on which $S_n$ acts (and also testing two such subsets for being in the same orbit under $S_n$).

3. Computing (generators of) the intersection of $G$ with another subgroup $H \le S_n$.

As Mark says, these are all at least as difficult as graph isomorphism.

The proofs are clever but basically elementary and interesting, so I recommend them! One reference is:

E.M. Luks, ``Permutation groups and polynomial-time computation'', in L. Finkelstein and W.M. Kantor (eds.), Groups and Computation, Dimacs Series in Discrete Mathematics and Theoretical Computer Science vol. 11, American Math. Soc., 139-176, 1993.

I have just noticed that Luks has a recently published book with the same title, which I have not seen yet.

It is known that the following three computational problems for subgroups $G$ of $S_n$ are polynomially equivalent:

1. Computing (generators of) the centralizer $C_G(g)$ of an element $g \in S_n$ (and also testing $g,h \in S_n$ for conjugacy in $G$).

2. Computing (generators of) the setwise stabilizer of a subset of the set of size $n$ on which $S_n$ acts (and also testing two such subsets for being in the same orbit under $S_n$).

3. Computing (generators of) the intersection of $G$ with another subgroup $H \le S_n$.

As Mark says, these are all at least as difficult as graph isomorphism.

The proofs are clever but basically elementary and interesting, so I recommend them! One reference is:

E.M. Luks, ``Permutation groups and polynomial-time computation'', in L. Finkelstein and W.M. Kantor (eds.), Groups and Computation, Dimacs Series in Discrete Mathematics and Theoretical Computer Science vol. 11, American Math. Soc., 139-176, 1993.

I have just noticed that Luks has a recently published book with the same title, which I have not seen yet.

Added later: It should also be mentioned that the implementations of the above algorithms in GAP and Magma involve backtrack searches, and so are potentially exponential, but in practice they run fast for most examples of moderate degree.