Not entirely sure if this should be in the discrete math section or the algebra section, as it deals with both, however I feel its really a combinatorics question so I will post it here.
In the Polya Enumeration Theorem (PET) as I understand it (which I must admit I hardly do) one uses the number of orbits of an assigned group to count the kinds of combinatorial structures. I was wondering if there were other approaches to counting using groups, in particular, whether there has been any research done using the number of conjugate subgroups of an assigned group to count the number of combinatorial structures? Any info would be great. Thanks in advance.