Let G be a group. The Cardinality Prime Graph of intersection of two cyclic subgroups of G, denoted by Г_CP (G) is the graph whose vertex set is the set of all the non-trivial elements of G, and two vertices a and b are adjacent if and only if |〈a〉∩〈b〉|=p. In this paper, we classify all the finite groups whose graph is one of complete, totally disconnected, bipartite, planar.