The classification of the finite groups whose subgroups of equal order are conjugate

All groups in this paper will be finite. Notations and conventions follow the same scheme as in Bensaid and van der Waall (1990) [1], Sezer and van der Waall (2006) [7], and van der Waall (1993) [9]. An outline of the details of the classification in the title, will be given in the Introduction.