Transitive subgroups of primitive permutation groups

We investigate the finite primitive permutation groups G which have a transitive subgroup containing no nontrivial subnormal subgroup of G. The conclusion is that such primitive groups are rather rare, and that their existence is intimately connected with factorisations of almost simple groups. A corollary is obtained on primitive groups which contain a regular subgroup. Heavily involved in our proofs are some new results on subgroups of simple groups which have orders divisible by various primes. For example, another corollary implies that for every simple group T apart from L-3(3), U-3(3), and L-2(p) with p a Mersenne prime, there is a collection IT consisting of two or three odd prime divisors of \T\, such that if M is a subgroup of T of order divisible by every prime in Pi, then \M\ is divisible by all the prime divisors of \T\, and we obtain a classification of such subgroups M. (C) 2000 Academic Press.