Subquotient
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory. In the literature about sporadic groups wordings like " is involved in " can be found with the apparent meaning of " is a subquotient of ." A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem. Examples
Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 are called "pariah groups." Order relation
The relation subquotient of is an order relation. Proof of transitivity for groups
Notation
For group , subgroup  of   and normal subgroup  of   the quotient group  is a subquotient of 

Let  be subquotient of , furthermore  be subquotient of  and  be the canonical homomorphism. Then all vertical () maps 

with suitable  are surjective for the respective pairs

The preimages  and  are both subgroups of  containing  and it is  and , because every  has a preimage  with  Moreover, the  subgroup  is normal in 

As a consequence, the subquotient  of  is a subquotient of  in the form

Relation to cardinal order
In constructive set theory, where the law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual order relation(s) on cardinals. When one has the law of the excluded middle, then a subquotient  of  is either the empty set or there is an onto function . This order relation is traditionally denoted 
If additionally the axiom of choice holds, then  has a one-to-one function to  and this order relation is the usual  on corresponding cardinals. See also
 Homological algebra
 Subcountable

References

Category theory
Abstract algebra