This plugin helped me address the issue in one of my projects: http://community.elgg.org/pg/plugins/project/594083/developer/madbohem/supergroups

"SuperGroups" ;-) original author MIA for quite a long time ;)
will let you clone and rename groups --
eg. Groups, Clubs, Schools --
all will still be Elgg groups, but called different names.
The extra fields, different menu addressing, disparatedness,
special searching.. will need custom code inside
the relevant modules within Super Groups.

Definitely recommend the subclass/subtype approach, although cloning the groups plugin is a significant amount of work no matter how you slice it. It just wasn't written to be extensible, and the fact that the default groups plugin doesn't use a subtype just makes things that much messier.

That said, it's definitely possible to do this. I did so with https://courseware.stanford.edu