I don't understand how using group-theory language to describe number-theoretic properties provides extra insight in this case (e.g. conjecture: all perfect numbers are even is more concise than the group-theoretic description given in the page). Can you expand on why you believe the tools of group theory have something to say about this? (e.g. for polynomial roots, the connection with symmetry groups comes from symmetries of factorized polynomials, while there's no obvious-to-me connection here as there is no unique-up-to-symmetry integer factorization)

This "classification" is pretty much a tautology and provides no information. The enumeration of all subgroups of a cyclic group and their orders is one of the most basic facts in all of group theory and known by pretty much everyone who has studied the subject at all. Thus, I'm completely failing to see the point in any of this.

Some independent math research just for fun.