  We classify the four-dimensional purely fermionic gauge theories that give a
UV completion of composite Higgs models. Our analysis is at the group
theoretical level, addressing the necessary (but not sufficient) conditions for
the viability of these models, such as the existence of top partners and
custodial symmetry. The minimal cosets arising are those of type SU(5)/SO(5)
and SU(4)/Sp(4). We list all the possible "hyper-color" groups allowed and
point out the simplest and most promising ones.
