On the sum of orders of non-cyclic and non-normal subgroups in a finite group (Q6595664)

Let \(G\) be a finite group.~In this paper, the authors considered the set of all non-normal non-cyclic subgroups of \(G\), which is denoted by \(\mathcal{C}(G)\).~Let \(\delta_{c}(G)=\frac{1}{|G|}\sum\limits_{H\in \mathcal{C}(G)}|H|\).~The authors proved that if \(\delta_{c}(G)\leq \frac{10}{3}\), then either \(G\cong \mathrm{A}_{5}\), or \(G\) is solvable.