Number of rotation subgroups of S_n

[Jens Voss]

 > > A "rotation" is a product of two transpositions, not necessarily 
disjoint,
 > >    so either (1,2,3) or (1,2)(3,4), etc.
 >
 > > Questions:
 > > How many subgroups of S_n have the property that they are
 > > generated by rotations?
 > > How many non-isomorphic ones?
 >
 > [...]
 >
 > I counted by hand cases up to n=5
 >
 > Unlabeled: 1,1,2,5,8
 > Labeled:   1,1,2,10,53

Hmm, I did the same and got

Unlabeled: 1, 1, 2, 4, 7
Labeled:   1, 1, 2, 7, 48

As for the case n=4, I can't see how you get 5/10: The groups I
see are {1}, the 1 characteristic V_4, the 4 C_3s and the 1 A_4.

Similarly for the case n=5: We get the 5 * (7 - 1) + 1 = 31 groups
of the already counted isomorphism classes living in one the 5 A_4s
plus the 10 conjugates of <(1 2 3), (1 2)(4 5)> (isomorphic to S_3)
plus the 6 conjugates of <(1 2)(3 4), (1 3)(2 5)> (isom. to D_10)
plus the 1 A_5 itself, giving a total of 31 + 10 + 6 + 1 = 48 groups
in 4 + 3 = 7 isomorphism classes.

Did I miss any?

Jens