Controlling composition factors of a finite group by its character degree ratio

For a finite nonabelian group G   let rat(G)rat(G) be the largest ratio of degrees of two nonlinear irreducible characters of G. We show that nonabelian composition factors of G   are controlled by rat(G)rat(G) in some sense. Specifically, if S   different from the simple linear groups PSL2(q)PSL2(q) is a nonabelian composition factor of G, then the order of S and the number of composition factors of G isomorphic to S   are both bounded in terms of rat(G)rat(G). Furthermore, when the groups PSL2(q)PSL2(q) are not composition factors of G  , we prove that |G:O∞(G)|⩽rat(G)21|G:O∞(G)|⩽rat(G)21 where O∞(G)O∞(G) denotes the solvable radical of G.