  Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we
prove Lagrange's theorem for Hopf monoids in the category of connected species.
As a corollary, we obtain necessary conditions for a given subspecies K of a
Hopf monoid H to be a Hopf submonoid: the quotient of any one of the generating
series of H by the corresponding generating series of K must have nonnegative
coefficients. Other corollaries include a necessary condition for a sequence of
nonnegative integers to be the sequence of dimensions of a Hopf monoid in the
form of certain polynomial inequalities, and of a set-theoretic Hopf monoid in
the form of certain linear inequalities. The latter express that the binomial
transform of the sequence must be nonnegative.
