  A braided subfactor determines a coupling matrix Z which commutes with the S-
and T-matrices arising from the braiding. Such a coupling matrix is not
necessarily of "type I", i.e. in general it does not have a block-diagonal
structure which can be reinterpreted as the diagonal coupling matrix with
respect to a suitable extension. We show that there are always two intermediate
subfactors which correspond to left and right maximal extensions and which
determine "parent" coupling matrices Z^\pm of type I. Moreover it is shown that
if the intermediate subfactors coincide, so that Z^+=Z^-, then Z is related to
Z^+ by an automorphism of the extended fusion rules. The intertwining relations
of chiral branching coefficients between original and extended S- and
T-matrices are also clarified. None of our results depends on non-degeneracy of
the braiding, i.e. the S- and T-matrices need not be modular. Examples from
SO(n) current algebra models illustrate that the parents can be different,
Z^+\neq Z^-, and that Z need not be related to a type I invariant by such an
automorphism.
