  In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial
characterization of the isomorphism types of embeddings of a cyclic subgroup in
a finite abelian group. In this paper we first use partial maps on
Littlewood-Richardson tableaux to generalize this result to finite direct sums
of such embeddings. We then focus on an application to invariant subspaces of
nilpotent linear operators. We develop a criterion to decide if two irreducible
components in the representation space are in the boundary partial order.
