On groups whose conjugacy class sizes are not divisible by each other

Let G be a finite group and $N(G)$ be the set of its conjugacy class sizes excluding~$1$. Let us define a directed graph Γ(G), the set of vertices of this graph is $N(G)$ and the vertices x and y are connected by a directed edge from x to y if x divides y and $N(G)$ does not contain a number z different from x and y such that x divides z and z divides y. We will call the graph Γ(G) the conjugate graph of the group G. In this work, we will study finite groups whose conjugate graph is a set of points.