  In this paper we present a conjecture on intermediate subfactors which is a
generalization of Wall's conjecture from the theory of finite groups. Motivated
by this conjecture, we determine all intermediate subfactors of
Goodman-Harpe-Jones subfactors, and as a result we verify that
Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a
negative answer to a question motivated by a conjecture of
Aschbacher-Guralnick.
