Determine all the generators and subgroups of $H = ⟨(123456)⟩ \subset S_6$.
I calculated all of the elements, so, I got
generator : (135)(246), (14)(25)(36), (153)(264), and (165432) and subgroups : <(135)(246)>, <(14)(25)(36)>, <(153)(264), and <(165432)>
am I missing subgroups or generators?
I believe $\langle (135)(246)\rangle=\langle (153)(264)\rangle$, and $(165432)$ generates the whole group. Thus you've found two proper subgroups. Add the trivial subgroup and you're done with the subgroups. A cyclic group of order $6$ has $3$ proper subgroups.