  We provide a unified presentation of extensions of the Minimal Dark Matter
framework in which new fermionic electroweak multiplets are coupled to each
other via the Standard Model Higgs doublet. We study systematically the generic
features of all the possibilities, starting with a singlet and two doublets
(akin to Bino-Higgsino dark matter) up to a Majorana quintuplet coupled to two
Weyl quadruplets. We pay special attention to this last case, since it has not
yet been discussed in the literature. We estimate the parameter space for
viable dark matter candidates. This includes an estimate for the mass of a
quasi-pure quadruplet dark matter candidate taking into account the Sommerfeld
effect. We also argue how the coupling to the Higgs can bring the Minimal Dark
Matter scenario within the reach of present and future direct detection
experiments.
