Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups.

- Is there any classification of such groups?
- In what conditions, we can say that $G$ is solvable?
- In what conditions, we can say that $G$ has a non-trivial normal abelian subgroup?

My main question is about arbitrary primes p,q and r. If the answer of the above questions are not known in general, one can answer my questions for centerless $\{2,3,5\}$-groups.