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In this article we generalize the theory of subgroup graphs of subgroups offree groups to finite index subgroups $H$ of finitely generated groups $G$. Westudy and prove various properties of $H$ in relation to its subgroup graph$\Gamma(H)$. For a finitely generated group $G$ we consider the poset$P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of$G$. We use the theory of subgroup graphs to prove that for many finitelygenerated infinite groups the order complex $\Delta P_{\text{fi}} (G)$ and thecorresponding nerve complex are contractible.

Subgroup graph methods for presentations of finitely generated groups and the connectivity of associated simplicial complexes