Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C4×D4×C167 | C2 |
| hypercenter | C4×D4 | D167 |
| hypocenter | C167 | D42 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D668 | C2 | semidirect |
| 2. C2×D4×D167 | C2 | semidirect |
| 3. #19128829 | C2 | semidirect |
| 4. #19128807 | C2 | semidirect |
| 5. #19128806 | C2 | semidirect |
| 6. #19128802 | C2 | semidirect |
| 7. C4×D668 | C2 | semidirect |
| 8. C4×D4×C167 | C2 | semidirect |
| 9. C23×D167 | C22 | semidirect |
| 12. C2×D668 | C22 | semidirect |
| 13. C2×D668 | C22 | semidirect |
| 14. C2×D668 | C22 | semidirect |
| 15. C2×D668 | C22 | semidirect |
| 16. C2×C22⋊D167 | C22 | semidirect |
| 18. C2×C4×D167 | C22 | semidirect |
| 19. C4⋊Dic167 | C22 | semidirect |
| 20. (C2×D167)⋊C4 | C22 | semidirect |
| 21. (C2×D167)⋊C4 | C22 | semidirect |
| 23. C22×C4×C167 | C22 | semidirect |
| 24. C4⋊C4×C167 | C22 | semidirect |
| 25. C22⋊C4×C167 | C22 | semidirect |
| 26. C42×C167 | C22 | semidirect |
| 36. D668 | D4 | direct (N × H) |
| 41. D4×C167 | D4 | semidirect |
| 49. C22×C167 | C2×D4 | semidirect |
| 57. C2×D167 | C2×D4 | semidirect |
| 60. C4×C167 | C2×D4 | semidirect |
| 61. C4×C167 | C2×D4 | semidirect |
| 69. C167 | D42 | semidirect |
| 78. C4×D4 | D167 | semidirect |
| 90. C22×C4 | C2×D167 | semidirect |
| 92. C4⋊C4 | C2×D167 | semidirect |
| 95. C22⋊C4 | C2×D167 | semidirect |
| 96. C42 | C2×D167 | semidirect |
| 109. D4 | D668 | direct (N × H) |
| 126. C22 | C2×D668 | semidirect |
| 131. C4 | D4×D167 | semidirect |
| 132. C4 | C2×D668 | semidirect |