Quantum Subgroups of the Haagerup Fusion Categories

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the quantum subgroups in the sense of Ocneanu), we find all irreducible subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.