  Let G be a transitive group of permutations of a finite set X, and suppose
that some element of G has at most two orbits on X. We prove that any two
maximal chains of groups between G and a point-stabilizer of G have the same
length, and the same sequence of relative indices between consecutive groups
(up to permutation). We also deduce the same conclusion when G has a transitive
quasi-Hamiltonian subgroup.
