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Let  G  be a finite group and   u   be the class of all finite supersoluble groups. A supersoluble subgroup  U  of  G  is called   u  -maximal in  G  if for any supersoluble subgroup  V  of  G  containing  U ,    V  =  U   . Moreover,Int    uGis the intersection of all   u  -maximal subgroups of  G . This paper obtains some new criteria onInt    uG, by assuming that some subgroups of  G  are either   Φ  - I -supplemented or   Φ  - I -embedded in  G . Here, a subgroup  H  of  G  is called   Φ  - I -supplemented in  G  if there exists a subnormal subgroup  T  of  G  such that    G  =  H  T    andH  ∩  TH    G/H    G≤  Φ    H  /    H    GInt    uGand   Φ  - I -embedded in  G  if there exists a  S -quasinormal subgroup  T  of  G  such that    H  T    is  S -quasinormal in  G  andH  ∩  TH    G/H    G≤  Φ    H  /    H    GInt    uG.

Theoretical Researches about u-Maximal Subgroups and Its Applications in Charactering IntuG