  The intersection graph of a group $G$ is an undirected graph without loops
and multiple edges defined as follows: the vertex set is the set of all proper
non-trivial subgroups of $G$, and there is an edge between two distinct
vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the
trivial subgroup of $G$. In this paper, we classify finite solvable groups
whose intersection graphs are not $2$-connected and finite nilpotent groups
whose intersection graphs are not $3$-connected. Our methods are elementary.
