Showing that G is solvable

Question

Let $|G|=200$. Show that G is solvable.

My beginning of the proof:

1. $|G|=200=2^3*5^2$
2. Let $n_5$ be the number of Sylow p-subgroups. Then $n_5|8$ and $n_5\equiv1 mod5$. And it implies that $n_5=1$.
3. From 2. we know that there exists $N$, such that $N\unlhd G$ and $|N|=25$.

What should be my next steps?

Answer 1

You're almost there:

1. $\;N\;$ is solvable (why?)

2. $\;G/N\;$ is also solvable (again, why?)

and thus $\;G\;$ is solvable.