I am trying to prove the title. I have shown that there exists a non-trivial normal subgroup of $G$:
$$|G|=200=5^2\cdot8$$
$\implies$the amount $s_5$ of 5-Sylow subgroups of $G$ divides $8$ and $s_5\equiv1\mod{5}$
$\Longleftrightarrow$ $s_5\in\{1,2,4,8\}\cap\{1+5k|k\in\mathbb{Z}\}=\{1\}$
$\implies$the 5-Sylow subgroup of $G$ is a (non-trivial) normal subgroup of $G$
My question: How can I show that there exists an abelian non-trivial normal subgroup of $G$?
Also please let me know if you see any mistakes in my proof
