I think I understand what you’re getting at.
Currently Badges are grouped into Bronze, Silver, Gold and Custom.
Though this choice of grouping could be debated IMHO it works and is a concept that is easily understood,
But I don’ think anything similar would work for Groups, that is, it would be difficult to come up with a one size fits all scheme. At least nothing that I can think of at the moment.
Maybe a parent child approach similar to categories would be better than grouping groups?
Maybe a way to order them would be good enough?
As for separate pages, those already exist unless I’m not understanding what you’re saying