  We determine the structure of automorphism groups of finite graphs of bounded
Hadwiger number. This in particular settles three of Babai's conjectures from
the 1980s. The first one states that the order of non-alternating, non-abelian
composition factors for automorphism groups of graphs of bounded Hadwiger
number is bounded. The second one, the subcontraction conjecture, states that a
non-trivial minor-closed graph class represents only finitely many non-abelian
simple groups. And the third one states that if the order of such a group does
not have small prime factors, then the group is obtained by iterated wreath and
direct products from abelian groups. Our proof includes a structural analysis
of finite edge-transitive graphs.
