Show that the composition factors of a group $G$ of order $p^2q$ where $p$ and $q$ are prime and $p \neq q$ are $C_p$, $C_p$ and $C_q$.
So I know that $|G|=p^2q$ is not a simple group. Not really sure where to go from here?
Any tips would be much appreciated.
My attempt at a solution (this feels pretty incomplete):
$e \triangleleft G_2 \triangleleft G_1 \triangleleft G$
Where $|G_1|=pq$ and $|G_2|=p$. Since $\frac{G_1}{G}$ and $\frac{G_1}{G_2}$ and $G_1$ are all of prime order they are cyclic and simple so this is a valid decomposition.
My question is how do I know that $G_1$ and $G_2$ are subgroups of $G$?
Thank you for your time.