Groups of order 168

This article gives information about, and links to more details on, groups of order 168
See pages on algebraic structures of order 168 | See pages on groups of a particular order

Statistics at a glance

The prime factorization of 168 is:

$\!169=2^{3}\cdot 3\cdot 7=8\cdot 3\cdot 7$

Quantity	Value	List/comment
Total number of groups	57
Total number of abelian groups	3	((number of abelian groups of order 8) = 3) times (number of abelian groups of order 3) = 1) times (number of abelian groups of order 7) = 1). See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Total number of nilpotent groups	5	((number of groups of order 8) = 5) times ((number of groups of order 3) = 1) times ((number of groups of order 5) = 1). See equivalence of definitions of finite nilpotent group
Total number of solvable groups	56	the only non-solvable group is the simple non-abelian group projective special linear group:PSL(3,2), which is also isomorphic to $PSL(2,7)$.
Total number of simple groups	1	the simple non-abelian group projective special linear group:PSL(3,2), which is also isomorphic to $PSL(2,7)$.