Let $G$ be a finite group and $H$ a subgroup of $G$. In 2012, David F. Anderson et al. introduced the subgroup graph of $H$ in $G$ as a simple graph with vertex set consisting all elements of $G$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy \in H$. They denoted this graph by $\Gamma_H(G)$. In this paper, we consider the complement of ${\Gamma}_H(G)$, denoted by $\overline{\Gamma_H(G)}$ and will give some graph properties of this graph, for instance diameter, girth, independent and dominating sets, regularity. Moreover, the structure of this graph, planerity and 1-planerity are also investigated in the paper.

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