  In this article we generalize the theory of subgroup graphs of subgroups of
free groups to finite index subgroups $H$ of finitely generated groups $G$. We
study 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 finitely
generated infinite groups the order complex $\Delta P_{\text{fi}} (G)$ and the
corresponding nerve complex are contractible.
