1) Faction 2) Sect 1) Group 2) Grouping 3) Set within a set 
In mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually represented notationally by H ≤ G, read as `H is a subgroup of G`. A proper .....
• (n.) A subdivision of a group, as of animals.
A subset of a group that is a group under the same operation.
The next level of mineral classification after categorization in groups.
Sub'group` noun (Biol.) A subdivision of a group, as of animals. Darwin. 
A subpart of the study population distinguished by a particular characteristic or set of characteristics (e.g., males under age 45 at entry).
[n] - a distinct and often subordinate group within a group 2. [n] - (mathematics) a subset (that is not empty) of a mathematical group
noun (mathematics) a subset (that is not empty) of a mathematical group
a subordinate group; a division of a group. · a division of a group in the periodic table. · a subset of a group that is closed under the group operation and in which every element has an inverse in the subset.