  In this note we classify the necessary and the sufficient conditions that an
index of a superconformal theory in $3\leq d \leq 6$ must obey for the theory
to have enhanced supersymmetry. We do that by noting that the index
distinguishes a superconformal multiplet contribution to the index only up to a
certain equivalence class it lies in. We classify the equivalence classes in
$d=4$ and build a correspondence between ${\cal N} = 1$ and ${\cal N}>1$
equivalence classes. Using this correspondence, we find a set of necessary
conditions and a sufficient condition on the $d=4$ ${\cal N} = 1$ index for the
theory to have ${\cal N}>1$ SUSY. We also find a necessary and sufficient
condition on a $d=4$ ${\cal N}>1$ index to correspond to a theory with ${\cal
N} > 2$. We then use our results to study some of the $d=4$ theories described
by Agarwal, Maruyoshi and Song, and find that the theories in question have
only ${\cal N} = 1$ SUSY despite having rational central charges. In $d=3$ we
classify the equivalence classes, and build a correspondence between ${\cal N}
= 2$ and ${\cal N}>2$ equivalence classes. Using this correspondence, we
classify all necessary or sufficient conditions on an ${\cal N}=1-3$
superconformal index in $d=3$ to correspond to a theory with higher SUSY, and
find a necessary and sufficient condition on an ${\cal N} = 4$ index to
correspond to an ${\cal N} > 4$ theory. Finally, in $d=6$ we find a necessary
and sufficient condition for an ${\cal N} = 1$ index to correspond to an ${\cal
N}=2$ theory.
