The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data DHg fits into a family DωHg2n+1, where n≥0 and ω∈ ₂n+1. We show D0Hg2n+1
Nearby in the stack
is related to the subfactors Izumi hypothetically associates to the cyclic groups
Z2n+1
. Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type
Z7
,
Z9
and
Z32
, and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type
Z11,Z13,Z15,Z17,Z19
(previously, Izumi had shown uniqueness for
Z3
and
Z5
), and we identify their modular data. We explain how DHg (more generally
DωHg2n+1
) is a graft of the quantum double DSym(3) (resp. the twisted double
DωD2n+1
) by affine so(13) (resp. so(4n²+4n+5)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data
DωHg2n+1
. For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c=8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.