Are the Groups having all subgroups of same order conjugate classified?
I haven't read it, but a classification is claimed in this article:
Robert W. van der Waall (September 2012), "The classification of the finite groups whose subgroups of equal order are conjugate", Indagationes Mathematicae 23 (3) 448–478 http://dx.doi.org/10.1016/j.indag.2012.02.009