Let $G$ be a finite group with the identity $e$. The subgroup intersection graph $\Gamma_{SI}(G)$ of $G$ is the graph with vertex set $V(\Gamma_{SI}(G)) = G-e$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma_{SI}(G)$ if and only if $|\left\langle x\right\rangle \cap\left\langle y\right\rangle|>1$, where $\left\langle x\right\rangle $ is the cyclic subgroup of $G$ generated by $x\in G$. In this paper, we obtain a lower bound for the independence number of subgroup intersection graph. We characterize certain classes of subgroup intersection graphs corresponding to finite abelian groups. Finally, we characterize groups whose automorphism group is the same as that of its subgroup intersection graph.

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