  We classify all different composite Higgs models (CHMs) characterised by the
coset space $\mathcal{G}/\cal{H}$ of compact semi-simple Lie groups
$\mathcal{G}$ and $\mathcal{H}$ involving up to 13 Nambu-Goldstone bosons
(NGBs), together with mild phenomenological constraints. As a byproduct of this
work, we prove several simple yet, to the best of our knowledge, mostly unknown
results: (1) Under certain conditions, a given set of massless scalars can be
UV completed into a CHM in which they arise as NGBs; (2) The set of all CHMs
with a fixed number of NGBs is finite, and in particular there are 642 of them
with up to 13 massless scalars (factoring out models that differ by extra
$U(1)$'s); (3) Any scalar representation of the Standard Model group can be
realised within a CHM; (4) Certain symmetries of the scalar sector allowed from
the IR perspective are never realised within CHMs. On top of this, we make a
simple statistical analysis of the landscape of CHMs, determining the frequency
of models with scalar singlets, doublets, triplets and other multiplets of the
custodial group as well as their multiplicity. We also count the number of
models with a symmetric coset.
