  In this paper we construct two new fusion categories and many new subfactors
related to the exceptional Extended Haagerup subfactor.
  The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion
categories are mysterious and are the only known fusion categories which appear
to be unrelated to finite groups, quantum groups, or Izumi quadratic
categories. One key technique which has previously revealed hidden structure in
fusion categories is to study all other fusion categories in the Morita
equivalence class, and hope that one of the others is easier to understand. In
this paper we show that there are exactly four categories (EH1, EH2, EH3, EH4)
in the Morita equivalence class of Extended Haagerup, and that there is a
unique Morita equivalence between each pair. The existence of EH3 and EH4 gives
a number of interesting new subfactors. Neither EH3 nor EH4 appears to be
easier to understand than the Extended Haaerup subfactor, providing further
evidence that Extended Haagerup does not come from known constructions. We also
find several interesting intermediate subfactor lattices related to Extended
Haagerup.
  The method we use to construct EH3 and EH4 is interesting in its own right
and gives a general computational recipe for constructing fusion categories in
the Morita equivalence class of a subfactor. We show that pivotal module $\rm
C^*$ categories over a given subfactor correspond exactly to realizations of
that subfactor planar algebra as a planar subalgebra of a graph planar algebra.
This allows us to construct EH3 and EH4 by realizing the Extended Haagerup
subfactor planar algebra inside the graph planar algebras of two new graphs.
This technique also answers a long-standing question of Jones: which graph
planar algebras contain a given subfactor planar algebra?
