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

Let G be a finite group and C(G) denote the set of all non-normal non-cyclic subgroups of G. In this paper, the function δc(G) =1/|G|∑H(G)|H| is introduced. In fact, we prove that, if δc(G)≤ 10/3, then either G A₅, or G is solvable. We also find some examples of finite groups G with δc(G)≤ 10/3.