$S_4\times S_3$ is a group of order $144$.
$2$--Sylow subgroup is of order 16 and $3$--Sylow subgroup is of order $9$.
Is there any way to find whether these subgroups are normal or not ?
In general, If $H$ is a a subgroup of $G$, to prove $H$ is not normal, we need to find an element $a$ $\in$ $G$ such that $aH$ $\neq$ $Ha$. This is the only way I know to prove it is not normal.
We know that by Sylow's $3^\text{rd}$ theorem, the number of subgroups of order $16$ can be $1$,$3$ or $9$. I came across the statement $\textbf{“$2$-Sylow subgroup is isomorphic to $D_4 \times Z_2$}$ $\textbf{and hence number of $2$--Sylow subgroups are 9}$ $\textbf{and hence it is not normal.''}$ I am clueless how that isomorphism can be found and about the conclusion of number of $2$-Sylow subgroups are $9$!! If i am sure about that, then by "A Sylow $p$-subgroup of a finite group $G$ is a normal subgroup of $G$ if and only if it is the only Sylow $p$-subgroup of $G$". I can say all those subgroups are not normal.
Please let me know what topic's knowledge in group theory I am lacking to understand that. Please suggest me if any source available to learn that topic. Thank you “