Number of rotation subgroups of S_n

[N. J. A. Sloane]

A friend suggested the following set of sequences

Let S_n be the symmetric group of all perms of n letters,
of order n!

A transposition is a permutation like (3,5).
A reflection is any element of order 2, that is, a product
   of disjoint transpositions
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?

How many subgroups of S_n have the property that they are
generated by reflections?  (Is this the Bell numbers, A000110 ?)
How many non-isomorphic ones?  (Again, this may be a well-known sequence)

Same two questions for groups generated by rotations AND reflections?

NJAS