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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.

So in the algebraic structure of groups, is a subquotient of if there exists a subgroup of and a normal subgroup of so that is isomorphic to .

In the literature about sporadic groups wordings like „ is involved in “^{[1]} can be found with the apparent meaning of „ is a subquotient of “.

As in the context of subgroups, in the context of subquotients the term *trivial* may be used for the two subquotients and which are present in every group .^{[citation needed]}

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.^{[2]}

There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to article Sporadic group, *Fi*_{22} has a double cover which is a subgroup of *Fi*_{23}, so it is a subquotient of *Fi*_{23} without being a subgroup or quotient of it.

The relation *subquotient of* is an order relation – which shall be denoted by . It shall be proved for groups.

- Notation
- For group , subgroup of and normal subgroup of the quotient group is a subquotient of , i. e. .

- Reflexivity: , i. e. every element is related to itself. Indeed, is isomorphic to the subquotient of .
- Antisymmetry: if and then , i. e. no two distinct elements precede each other. Indeed, a comparison of the group orders of and then yields from which .
- Transitivity: if and then .

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

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

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.

**^**Griess, Robert L. (1982), "The Friendly Giant",*Inventiones Mathematicae*,**69**: 1−102, Bibcode:1982InMat..69....1G, doi:10.1007/BF01389186, hdl:2027.42/46608, S2CID 123597150**^**Dixmier, Jacques (1996) [1974],*Enveloping algebras*, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740 p. 310