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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. Our main interest is 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.