Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega(M)$, is an undirected simple graph whose vertex set $V(\Omega)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\neq M$. We study the connectivity, the core and the clique number of $\Omega(M)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega(M)$ is infinite and $\Omega(M)$ is a planar graph. Moreover, we give several examples for which $n$ the graph $\Omega(\mathbb{Z}_{n})$ is connected, bipartite and planar.
