The most general form of Lagrange's group theorem, also known as Lagrange's lemma, states that for a group , a subgroup of , and a subgroup of , , where the products are taken as cardinalities
(thus the theorem holds even for infinite groups)
and
denotes the subgroup index for the subgroup of . A frequently stated corollary (which follows from taking
,
where
is the identity element) is that the order of
is equal to the product of the order of and the subgroup index of
.

The corollary is easily proven in the case of being a finite group, in which
case the left cosets of form a partition of , thus giving the order of as the number of blocks in the partition (which is ) multiplied by the number of elements in each partition
(which is just the order of ).

For a finite group , this corollary gives that the order of must divide the order of . Then, because the order of an element of is the order of the cyclic subgroup generated by , we must have that the order of any element of divides the order of .

The converse of Lagrange's theorem is not, in general, true (Gallian 1993, 1994).