What's a "good" way of describing $S_4$ subgroups by hand? The following post gives a pretty detailed stategy but i can't figure out some tools that are used:
I think I'm not familiar with the use of the "Normalizer" that is used in this post, it may be useful to understand the strategy.
For subgroups of orders less than $8$, I can describe them mentally with intuition, but for subgroups of order $8$ and $12$, I realize I may not be using a good method as I can't describe them by hand.
More precisely, for the example of $A_4$ , the order $12$ subgroup of $S_4$, I understand the following reasoning: Prime factorization of $12$ is $2^2*3$, so, we know that this subgroup of order $12$ should (if it exists, what I can't prove except by exposing $A_4$) contain a $2$-sylow or a $3$-sylow, more precisely a subgroup of order $2^2$ or $3$. Since this point, I can't really figure out how to move forward.
I also tried to find a proper post that explains it but I never found explanation about the use of the normalizer,