On the complement of the intersection graph of subgroups of a group

The complement of the intersection graph of subgroups of a group G, denoted by I c (G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and its two distinct vertices H and K are adjacent if and only if H ∩ K = 1, where 1 denotes the trivial subgroup of G. In this paper, we classify all finite groups whose complement of the intersection graph of subgroups is one of totally disconnected, bipartite, complete bipartite, tree, star graph or C3-free. Also we characterize all the finite groups whose complement of the intersection graph of subgroups is planar.