The Reductive Subgroups of F_4

Let G=G(K) be a simple algebraic group defined over an algebraically closed field K of characteristic p0. A subgroup X of G is said to be G-completely reducible if, whenever it is contained in a parabolic subgroup of G, it is contained in a Levi subgroup of that parabolic. A subgroup X of G is said to be G-irreducible if X is in no proper parabolic subgroup of G; and G-reducible if it is in some proper parabolic of G. In this paper, the author considers the case that G=F4(K). The author finds all conjugacy classes of closed, connected, semisimple G-reducible subgroups X of G. Thus he also finds all non-G-completely reducible closed, connected, semisimple subgroups of G. When X is closed, connected and simple of rank at least two, he finds all conjugacy classes of G-irreducible subgroups X of G. Together with the work of Amende classifying irreducible subgroups of type A1 this gives a complete classification of the simple subgroups of G. The author also uses this classification to find all subgroups of G=F4 which are generated by short root elements of G, by utilising and extending the results of Liebeck and Seitz.