  We exhibit an infinite family of supersymmetric phases in the
three-dimensional ABJM superconformal field theory and the dual asymptotically
AdS$_4$ gravity. They are interpreted as partially deconfined phases which
generalize the confined/pure AdS phase and deconfined/supersymmetric black hole
phase. Our analysis involves finding a family of saddle-points of the
superconformal index labelled by rational points (equivalently, roots of
unity), separately in the bulk and boundary theories. In the ABJM theory we
calculate the free energy of each saddle by the large-$N$ asymptotic expansion
of the superconformal index to all orders in perturbation theory near the
saddle-point. We find that this expansion terminates at finite order. In the
gravitational theory we show that there is a corresponding family of solutions,
constructed by orbifolding the eleven-dimensional uplift of the supersymmetric
black hole. The on-shell gravitational action of each orbifold agrees with the
free energy of the corresponding saddle in the SCFT. We find that there are two
saddles in the ABJM theory with the same entropy as the supersymmetric black
hole, corresponding to the two primitive fourth-roots of unity, which causes
macroscopic oscillations in the microcanonical index.
