On the solvability of a finite group by the sum of subgroup orders

Let $G$ be a finite group and ${\sigma }_1(G)=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{H\le G}\mathrm{\mid }H\mathrm{\mid }$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if ${\sigma }_1(G)<\frac{117}{20}$, then $G$ is solvable. This partially solves an open problem posed in \cite{9}.