In an attempt to extend the property of being supercompact but not hod-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not hod-supercompact holds for the least supercompact cardinal κ 0 , κ 0 is indestructibly supercompact, the strongly

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... t and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above κ 0 but fails below κ 0 . Additionally, we get the property of being supercompact but not hod-supercompact at the least supercompact cardinal, in a model where level by level equivalence between strong compactness and supercompactness holds.