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A subgroup functor τ\tau is said Φ\Phi -regular if for all primitive groups GG, whenever H∈τ(G)H\in \tau \left(G) is a pp-subgroup and NN is a minimal normal subgroup of GG, then ∣G:NG(H∩N)∣=pd| G:{N}_{G}\left(H\cap N)| ={p}^{d} for some integer dd. In this article, we investigate groups in which some primary subgroups are τ\tau -subgroups for a Φ\Phi -regular subgroup functor τ\tau , and we obtain new criteria for the supersolubility or pp-nilpotency of a group.