  A few years ago a connection between the elliptic genus of the K3 manifold
and the largest Mathieu group M$_{24}$ was proposed. We study the elliptic
genera for Calabi-Yau manifolds of larger dimensions and discuss potential
connections between the expansion coefficients of these elliptic genera and
sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the
elliptic genera of certain Calabi-Yau $d$-folds for $d>3$ have expansions that
could potentially arise from underlying sporadic symmetry groups. We explore
such potential connections by calculating twined elliptic genera for a large
number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected
spaces, for a toroidal orbifold and two Gepner models.
