  We systematically analyze the large-$N$ limit of the superconformal index of
$\mathcal{N}=1$ superconformal theories having a quiver description. The index
of these theories is known in terms of unitary matrix integrals, which we
calculate using the recently-developed technique of elliptic extension. This
technique allows us to easily evaluate the integral as a sum over saddle points
of an effective action in the limit where the rank of the gauge group is
infinite. For a generic quiver theory under consideration, we find a special
family of saddles whose effective action takes a universal form controlled by
the anomaly coefficients of the theory. This family includes the known
supersymmetric black hole solution in the holographically dual AdS$_5$
theories. We then analyze the index refined by turning on flavor chemical
potentials. We show that, for a certain range of chemical potentials, the
effective action again takes a universal cubic form that is controlled by the
anomaly coefficients of the theory. Finally, we present a large class of
solutions to the saddle-point equations which are labelled by group
homomorphisms of finite abelian groups of order $N$ into the torus.
