Is it possible to find all subgroup of $Alt_4$ from all subgroup of the permutation group $S_4$?

Question

Is it possible to find all subgroup of $Alt_4$ from all subgroup of the permutation group $S_4$?

I think the answer is yes from Lagrange's theorem and Sylow's theorem. Is anyone is able to give me a hint?

Answer 1

It shouldn't be too hard, the elements of $A_4$ are:

$e,(2,3,4),(2,4,3),(1,3,4),(1,4,3),(1,2,4),(1,4,2),(1,2,3),(1,3,2),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)$

If the order is $1$: There is only one.

If the order is $2$: There are three, they are generated by the double transpositions.

If the order is $3$: They are four, generated by the $3$-cycles.

If the order is $4$: The subgroup can't contain $3$-cycles. So the only options is the subgroup consisting of the identity and all the double-transpositions, this in fact is a subgroup. So add one more to the list.

If the order is $6$: Because of Cauchy's theorem we must have a $3$- cycle $c$ and a double transposition $t$. so we already have three elements, $e,c,c^2,t$. Now prove $ct$ and $tc$ give us a $3$-cycle and a transposition $c'\neq c,c^2 $ and $t'\neq t$ so we have at least $7$ elements. $e,c,c^2,t,t',c',c'^2$. There are no subgroups of order $6$.

If the order is $12$: There is clearly only one.