Let $G$ be a group of order $p^3q^5$ where $p$ and $q$ are two distinct prime numbers. Is $G$ solvable? If it is, how to justify it (without using the Burnside's Theorem [wiki])?
Basically, I am trying to find a non-trivial normal subgroup $H$ of $G$ and to show that both $H$ and $G/H$ are solvable.
To find an $H \triangleleft G$, I think the Sylow theorems will be helpful. However, for both $n_p$ (the number of Sylow $p$-subgroups) and $n_q$, there are too many cases to consider, due to the "high powers" of $p$ and $q$.