If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple.
By the Sylow theorem, we have that the number of $2$-sylow subgroups of $G$ $n_2$ satisfy that $$ n_2 \equiv1\mod2\mbox{ and } n_2\mid5^6 $$ Similarly for $n_5$ we have, $$ n_5 \equiv1\mod5\mbox{ and } n_5\mid2^4 $$ Hence, $$ n_2 \in \{1,5,5^2,5^3,5^4,5^5,5^6\}, \mbox{ and } n_5 \in \{1,16\} $$
But assuming that none of the $n_p'$s are one and using sylow theorems, I can't surpass the order of $G$ as the professor show us in class with one example. Now I am pretty sure I will need another approach but nothing comes to my mind. Any help would be appreciated.