TITLE: Are the Groups having all subgroups of same order conjugate classified?
QUESTION [8 upvotes]: As we know, Alternating Groups $A_n$ of degree $n\leq 5$, Dihedral Groups $D_{2n}$ of order $2n$ (for odd natural number $n$) and Cyclic Groups $C_n$ of order $n$ (for positive integer $n$) have subgroups of same order conjugate. Are such (finite) groups classified? Is there some more groups of this type?

REPLY [5 votes]: 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