  We show how one may classify all semisimple algebras containing the
$\mathfrak{su}(3)\oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$ symmetry of
the Standard Model and acting on some given matter sector, enabling theories
beyond the Standard Model with unification (partial or total) of symmetries
(gauge or global) to be catalogued. With just a single generation of Standard
Model fermions plus a singlet neutrino, the only {gauge} symmetries correspond
to the well-known algebras $\mathfrak{su}(5),\mathfrak{so}(10),$ and
$\mathfrak{su}(4)\oplus \mathfrak{su}(2) \oplus \mathfrak{su}(2)$, but with two
or more generations a limited number of exotic symmetries mixing flavour,
colour, and electroweak degrees of freedom become possible. We provide a
complete catalogue in the case of 3 generations or fewer and outline how our
method generalizes to cases with additional matter.
