  In this paper we consider unitary highest weight irreducible representations
of the `Large' $\mathcal{N}=4$ superconformal algebra $A_\gamma$ in the Ramond
sector as infinite-dimensional graded modules of its zero mode subalgebra,
$\mathfrak{su}(2|2)$. We describe how representations of $\mathfrak{su}(2|2)$
may be classified using Young supertableaux, and use the decomposition of
$A_\gamma$ as an $\mathfrak{su}(2|2)$ module to discuss the states which
contribute to the supersymmetric index $I_1$, previously proposed in the
literature by Gukov, Martinec, Moore and Strominger.
