Subgroup structure of symmetric group:S8
From Groupprops
This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S8.
View subgroup structure of particular groups | View other specific information about symmetric group:S8
This article discusses the subgroup structure of symmetric group:S8, which is the symmetric group on the set . The group has order 40320.
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate
Quick summary
Item | Value |
---|---|
Number of subgroups | 151221 Compared with : 1,2,6,30,156,1455,11300,151221 |
Number of conjugacy classes of subgroups | 296 Compared with : 1,2,4,11,19,56,96,296,554,1593,... |
Number of automorphism classes of subgroups | 96 Compared with : 1,2,4,11,19,37,96,296,554,1593,... |