The reasons behind the gauge symmetry of the Standard Model, $U(1)\times SU(2)\times SU(3)$, are still unsettled. One obvious feature is the low dimensionality of all its subgroups.
Under certain conditions, a negative answer to the question "why not larger groups like SU(15), or for that matter, SP(26) or E7?" is possible.

We have recently observed that fermions charged under large groups acquire much bigger dynamical masses, all things being equal at a high e.g. GUT scale, than ordinary quarks. Should such multicharged fermions exist, they are just too heavy to be observed today
(and have either decayed early on if coupled to the rest of the Standard Model, or become reliquial dark matter if uncoupled).
Their mass scale is dictated by strong antiscreening of the running coupling for those larger groups (with an appropriately small number of flavors) together with scaling properties of the Dyson-Schwinger equation for the fermion mass.

The generated fermion mass (assuming only few flavors, to avoid spoiling antiscreening) grows exponentially with the number of colors as $M(N_c) \propto e^{N_c} \times \theta(N_f^{\rm critical} -N_f)$
for scales much below the GUT scale. Large groups would be strongly coupled already near the GUT scale and fermions charged thereunder have correspondingly large masses.