On the planarity of a graph related to the join of subgroups of a finite group

‎Let $G$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎We define an undirected simple graph $\Delta(G)$ whose‎ ‎vertices are the proper subgroups of $G$, which are not contained in the‎ ‎Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge‎ ‎if and only if $G=\langle H‎ , ‎K\rangle$‎. ‎In this paper we classify finite groups with planar graph‎. ‎%For this‎, ‎by Kuratowskiʹs Theorem‎, ‎we have to study subdivisions‎ ‎%of the Kuratowski graphs $K_{3‎ , ‎3}$ and $K_5$ in the graph $\Delta(G)$‎. ‎Our result shows that only few groups have planar graphs‎.