Finite groups in which the order of each non-normal subgroup has at most \(3\) prime factors (Q6656709)

Let \(G\) be a finite group of order \(\Pi_{i=1}^{n} p_{i}^{k_{i}}\). Define \(\pi(G)\) as the set of primes dividing \(|G| \) and \(w(G)=\Sigma_{i=1}^{n} k_{i}\).\N\NIt is well-known that every finite group of order \(pqr\) where \(p, q\) and \(r\) are prime numbers, is solvable. The authors somehow extend this result in a neat way by describing the structure of finite groups whose all non-normal subgroups \(H\) have orders with \(w(H) \leq 3\). The proofs depend on a very detailed analysis of the solvable and non-solvable cases.