Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D167 |
| commutator, socle, cosocle | C22×C167 | C24 |
| Fitting | #123×C167 | C2 |
| hypercenter | #123 | D167 |
| hypocenter | C167 | #562 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128823 | C2 | semidirect |
| 2. #19128818 | C2 | semidirect |
| 3. #19128818 | C2 | semidirect |
| 4. #19128817 | C2 | semidirect |
| 5. #19128816 | C2 | semidirect |
| 6. #19128814 | C2 | semidirect |
| 7. #19128812 | C2 | semidirect |
| 8. #19128811 | C2 | semidirect |
| 9. #19128810 | C2 | semidirect |
| 10. #19128809 | C2 | semidirect |
| 11. #19128806 | C2 | semidirect |
| 12. #19128805 | C2 | semidirect |
| 13. #19128804 | C2 | semidirect |
| 14. #19128803 | C2 | semidirect |
| 15. #123×C167 | C2 | semidirect |
| 109. C167 | #562 | semidirect |
| 117. #123 | D167 | semidirect |