There is a question in Artin's algebra whereby you are given the canonical homomophism $\varphi : S_4 \rightarrow S_3 $ and asked to find the 6 subgroups of $S_4$ that contain the kernel.
My question is, why are there only 6? Since the kernel is a normal subgroup of $S_4$ why can't I take every subgroup of $S_4$ (say $H$) and create a subgroup $HK$? Will this not contain the kernel? Thanks.
It might happen that $HK=GK$ for different subgroups $H,G$, and in fact this is the case.