Let G be a finite group. We say that a subgroup H of G is Uc-normal in G if G has a subnormal subgroup T such that TH ＝ G and (H ∩ T)HG/HG is contained in the U-hypercenter Zu∞ (G/HG) of G/HG, where U is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are Uc-normal in G.