Why is $\Bbb{Z}_2 \times A_5$ not solvable?

Question

We can write $\Bbb{Z}_2 \times A_5 > 1 \times A_5 > 1$ as a composition series, but why is $\Bbb{Z}_2 \times A_5$ not solvable? I know that $A_5$ is not solvable, but it is enough? I'm really having a bad time with this.

Thanks in advance!

Answer 1

For a finite solvable group $G$, the composition factors of $G$ must be cyclic groups of prime order.

The composition factors of $\Bbb{Z}_2\times A_5$ are $\Bbb{Z}_2$ and $A_5$. Since $A_5$ is not cyclic group of prime order, we conclude that $\Bbb{Z}_2\times A_5$ is not solvable.

Answer 2

Every subgroup of a solvable group is solvable. But $\mathbb Z_2×A_5$ has a subgroup $\{1\}×A_5\cong A_5$, which is not solvable.