Order Equal to 400

If the order of a group is 400, then the following holds:

$$\begin{tabular}{\vert c\vert c\vert l\vert} \hline prime & Sy... ...2 & 16 & 1, 5 \\ \hline 5 & 25 & 1, 16 \\ \hline \end{tabular}$$

Claim that n₂ =1. This follows from the Index Theorem. If n₂ = 5 then the index in G of N_G(Syl₂) = 5 and so G must be isomorphic to a subgroup of S₅. But $\mid G \mid$ does not divide $\mid S_{5} \mid$ and this is a contradiction. Therefore G is not simple.