The fourth group isomorphism theorem, also called the lattice group isomorphism theorem, lets 
be a group and let 
, where 
indicates that 
is a normal subgroup of
.
Then there is a bijection from the set of subgroups 
of 
that contain 
onto the set of subgroups 
of 
. In particular, every subgroup 
is of the form 
for some subgroup 
of 
containing 
(namely, its preimage in 
under the natural projection homomorphism from 
to 
.) This bijection has the following properties: for all 
with 
and 
,
1. 
iff 
2. If 
,
then 
3. 
,
where 
denotes the subgroup generated by 
and 
4. 
5. 
iff 
.
