Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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 | #560 |
| 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. #19128828 | C2 | semidirect |
| 2. #19128827 | C2 | semidirect |
| 3. #19128814 | C2 | semidirect |
| 4. #19128807 | C2 | semidirect |
| 5. #19128806 | C2 | semidirect |
| 6. #19128805 | C2 | semidirect |
| 7. #19128802 | C2 | semidirect |
| 8. #19128800 | C2 | semidirect |
| 9. #123×C167 | C2 | semidirect |
| 13. C2×D668 | C22 | semidirect |
| 16. C2×C22⋊D167 | C22 | semidirect |
| 17. C2×C4×D167 | C22 | semidirect |
| 18. C4⋊Dic167 | C22 | semidirect |
| 19. Dic167⋊C4 | C22 | semidirect |
| 24. (C2×D167)⋊C4 | C22 | semidirect |
| 25. C4×Dic167 | C22 | semidirect |
| 27. C22×C4×C167 | C22 | semidirect |
| 28. C4⋊C4×C167 | C22 | semidirect |
| 30. C22⋊C4×C167 | C22 | semidirect |
| 74. C167 | #560 | semidirect |
| 79. #123 | D167 | semidirect |
| 94. C22×C4 | C2×D167 | semidirect |
| 97. C4⋊C4 | C2×D167 | semidirect |
| 104. C22⋊C4 | C2×D167 | semidirect |