Prove that every group $G$ whose order is the form $|G|=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$ is not solvable

Question

Prove that every group $G$, whose order is the form $|G|=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$, where $p,q,r$ are distinct prime numbers and $\alpha_i >1$, is not solvable.

Any hint or guidance will be great or any reference to study will be great. thanks a lot.

Answer 1

This is not true: counterexample $S_3 \times C_5$, which is a non- abelian solvable group of order $2\cdot 3\cdot 5$.