Fixed point spaces in actions of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p, and let H be a maximal closed non-subspace subgroup of G. Given such a pair (G,H), we obtain a close to best possible upper bound for the ratio dim (x^G n H) / dim (x^G), where x is a semisimple or unipotent element in G of prime order. We apply this result to the study of fixed point spaces.