This question may be too naive, in which case I apologise in advance. Anyway, it is a well-known fact (see e.g. Milne's notes) that any abelian variety A has only finitely many direct factors up to automorphisms of A. (Here a direct factor of A is an abelian subvariety B for which there exists another abelian subvariety C of A such that $A \cong B \times C$.)
My question is: how much is known about the corresponding question for arbitrary abelian subvarieties, rather than direct factors? That is, is it known whether every abelian variety A has finitely many abelian subvarieties, up to automorphisms of A? If not, what's the best known result in this direction?
I've asked a couple of people about this, and their opinion seems to be that it's "more or less" known. But I would like something a little more concrete, if possible. Any relevant references would be appreciated!