The question is fairly dry: Is there any semigroup analogue of Lagrange's theorem for groups (counting as a generalization of the latter)? Let me guess the answer: Obviously yes. So the real question is: Any reference? Thank you in advance.

P.S.: I've notice of a Lagrange's theorem for Smarandache semigroups, but I would like to hear of different extensions, if possible (I don't think this is quite standard, but somebody defines a Smarandache semigroup to be any semigroup $(A, \star)$ for which there exist a proper subset $G$ of $A$, a unary operation $u: G \to G$ and a distinguished element $e \in G$ such that $(G, \star, u, e)$ is a group).

Edit. This is basically a comment to the subsequent answer of Vladimir Dotsenko. Let me highlight that I'm not asking for (possible) extensions to arbitrary semigroups. And I don't expect that, if any non-trivial extension is possible, it looks exactly like Lagrange's theorem for groups.

I'm just asking for any possible non-trivial extension that is already there, in the literature. Say, for instance, an extension to some interesting classes of semigroups.

Apart from the ones where the theorem sounds true by definition (e.g. Smarandache lagrangian semigroups). I know, non-trivial and interesting are not well-defined terms. But I have faith in your common sense.