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

Studying superconformal symmetry enhancement through indices