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LetGbe a connected reductive linear algebraic group defined over an algebraically closed field of characteristicp. Assume thatpis good forG. In this note we consider particular classes of connected reductive subgroupsHofGand show that the cocharacters ofHthat are associated to a given nilpotent element e in the Lie algebra ofHare precisely the cocharacters ofGassociated to e that take values inH. In particular, we show that this is the case providedHis a connected reductive subgroup ofGofdoi:10.1017/s0027763000009582 fatcat:rxzsa5bvqbdcff4p7lpwmbfwla