# List the cyclic subgroups of S4? [duplicate]

I am trying to list all the cyclic subgroups of $$S_4$$, and also list two examples of proper non-cyclic subgroups of $$S_4$$.

Here's the general idea of what I have for the list: $$S_4$$, $$A_4$$, $$\langle e\rangle$$,

$$\langle(1 2)\rangle, \langle(2 3)\rangle$$, and so on...

$$\langle(134)\rangle, \langle(234)\rangle$$, and so on...

$$\langle(12)(34)\rangle, \langle(13)(24)\rangle$$, and so on...

$$\langle(234)(23)\rangle, \langle(123)(12)\rangle$$ and so on...

You get the idea. There are thirty in total by my count.

Is this $$i.$$ proper notation for a subgroup of $$S_4$$ and $$ii.$$ how would I find a proper non-cyclic subgroup?

• $S_4$ has twenty-four elements. How can it have more than twenty-four cyclic subgroups? – Angina Seng Nov 9 '19 at 19:54
• Use $\langle X\rangle$ for $\langle X\rangle$. – Shaun Nov 9 '19 at 19:58
• @LordSharktheUnknown, you are correct, there are 30 total subgroups but not all of them are cyclic, that is my mistake. There are only 17 cyclic subgroups, – o's1234 Nov 9 '19 at 20:26