  The Georgi-Machacek model adds two SU(2)_L-triplet scalars to the Standard
Model in such a way as to preserve custodial SU(2) symmetry. We study the
generalizations of the Georgi-Machacek model to SU(2)_L representations larger
than triplets. Perturbative unitarity considerations limit the possibilities to
models containing only SU(2)_L quartets, quintets, or sextets. These models are
phenomenologically interesting because they allow the couplings of the 125 GeV
Higgs boson to WW and ZZ to be larger than their values in the Standard Model.
We write down the most general custodial SU(2)-preserving scalar potentials for
these models and outline their phenomenology. We find that experimental and
theoretical constraints on the fermiophobic custodial-fiveplet states present
in each of the models lead to absolute upper bounds on the 125 GeV Higgs boson
coupling strength to WW and ZZ.
