As seen in this answer, a group of order 144 is not simple. Now, I understand the main part of the answer, i.e. where it is concluded that, upon deducing that $n_3 = 16$, it is forced that $n_2 = 1$, and hence the group is not simple.
Why does $n_3=4$ imply that $G$ maps nontrivially to $S_4$? And why does this imply that $G$ is not simple?
In the part "so by Sylow's theorem it has at least 4 Sylow 3-subgroups, and hence has order at least 36, so $|G:N_G(T)|≤4$ and G cannot be simple.", what exactly is the Sylow Theorem being applied to, and how is it used to deduce (using the fact that $|G:N_G(T)|≤4$) that $G$ is not simple?
Are there other techniques besides the counting argument and the non-trivial mapping argument of $G$ to $S_n$ to show that $G$ of a certain order is not simple (using the Sylow theorems, that is)?
Any help would be much appreciated.