We investigate the finite primitive permutation groups G which have a trans
itive subgroup containing no nontrivial subnormal subgroup of G. The conclu
sion is that such primitive groups are rather rare, and that their existenc
e is intimately connected with factorisations of almost simple groups. A co
rollary is obtained on primitive groups which contain a regular subgroup. H
eavily involved in our proofs are some new results on subgroups of simple g
roups which have orders divisible by various primes. For example, another c
orollary implies that for every simple group T apart from L-3(3), U-3(3), a
nd L-2(p) with p a Mersenne prime, there is a collection IT consisting of t
wo 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 pri
me divisors of \T\, and we obtain a classification of such subgroups M. (C)
2000 Academic Press.