Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D167 |
| commutator, cosocle | C2×C167 | C25 |
| Frattini | C2 | C24×D167 |
| Fitting | C2×#41×C167 | C2 |
| socle | C22×C167 | C24 |
| hypercenter | C2×#41 | D167 |
| hypocenter | C167 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128878 | C2 | direct (N × H) |
| 2. C2×#14581951 | C2 | semidirect |
| 3. C2×#14581950 | C2 | semidirect |
| 4. C22×D668 | C2 | semidirect |
| 5. C2×D4×D167 | C2 | semidirect |
| 6. C2×#41×C167 | C2 | semidirect |
| 7. C23×D167 | C22 | semidirect |
| 8. #14581951 | C22 | semidirect |
| 9. #14581950 | C22 | semidirect |
| 10. C2×Dic334 | C22 | semidirect |
| 11. C2×D668 | C22 | semidirect |
| 12. C2×D668 | C22 | semidirect |
| 13. C2×D668 | C22 | semidirect |
| 14. C2×C22⋊D167 | C22 | semidirect |
| 15. D4×D167 | C22 | semidirect |
| 16. C2×C4×D167 | C22 | semidirect |
| 17. #41×C167 | C22 | semidirect |
| 18. C2×Q8×C167 | C22 | semidirect |
| 19. C2×D4×C167 | C22 | semidirect |
| 20. C22×C4×C167 | C22 | semidirect |
| 23. C22×D167 | C23 | semidirect |
| 24. Dic334 | C23 | semidirect |
| 25. D668 | C23 | semidirect |
| 26. D668 | C23 | semidirect |
| 27. C22⋊D167 | C23 | semidirect |
| 29. C4×D167 | C23 | semidirect |
| 31. Q8×C167 | C23 | semidirect |
| 32. D4×C167 | C23 | semidirect |
| 35. C2×C4×C167 | C23 | semidirect |
| 46. C2×C167 | H(5,𝔽2) | semidirect |
| 49. C167 | C2×H(5,𝔽2) | semidirect |
| 54. C2×#41 | D167 | semidirect |
| 60. #41 | C2×D167 | semidirect |
| 62. C2×Q8 | C2×D167 | semidirect |
| 68. C2×D4 | C2×D167 | semidirect |
| 70. C22×C4 | C2×D167 | semidirect |
| 76. Q8 | C22×D167 | semidirect |
| 80. D4 | C22×D167 | semidirect |
| 85. C2×C4 | C22×D167 | semidirect |
| 97. C2 | #19128878 | direct (N × H) |