Authors’ abstract: We show that the number of the conjugacy classes of the AFD type \(\text{II}_ 1\) subfactors with the principal graph \(D_ n^{(1)}\) is \(n-2\). This gives the last missing number in the complete classification list of subfactors with index 4 by S. Popa. This also disproves an announcement of A. Ocneanu that such a subfactor is unique for each \(n\).
We give two different proofs. One is by an application of an idea of an orbifold model in solvable lattice model theory to Ocneanu’s paragroup theory and the other is by reduction to classification of dihedral group actions. The latter also shows that the AFD type \(\text{III}_ 1\) subfactors with the principal graph \(D_ n^{(1)}\) split as type \(\text{II}_ 1\) subfactors tensored with the common AFD type \(\text{III}_ 1\) factor. We also discuss a relation between these proofs and a construction of subfactors using Cuntz algebra endomorphisms.