  A $\vee$-complement of a subgroup $H \leqslant \mathbb{F}_n$ is a subgroup $K
\leqslant \mathbb{F}_n$ such that $H \vee K = \mathbb{F}_n$. If we also ask $K$
to have trivial intersection with $H$, then we say that $K$ is a
$\oplus$-complement of $H$. The minimum possible rank of a $\vee$-complement
(resp. $\oplus$-complement) of $H$ is called the $\vee$-corank (resp.
$\oplus$-corank) of $H$. We use Stallings automata to study these notions and
the relations between them. In particular, we characterize when complements
exist, compute the $\vee$-corank, and provide language-theoretical descriptions
of the sets of cyclic complements. Finally, we prove that the two notions of
corank coincide on subgroups that admit cyclic complements of both kinds.
