Finite Group DatabasebetaFinite Group DatabasebetaOrder 5696.
298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D89 |
| commutator | C4×C89 | C24 |
| Frattini | C4 | C23×D89 |
| Fitting | C2×D8×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C2×D8 | D89 |
| hypocenter | C89 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13083573 | C2 | semidirect |
| 2. C2×D4×D89 | C2 | semidirect |
| 3. #15668036 | C2 | direct (N × H) |
| 4. C2×#13083563 | C2 | semidirect |
| 5. C2×#13083562 | C2 | semidirect |
| 6. C2×#13083559 | C2 | semidirect |
| 7. C2×#13083557 | C2 | semidirect |
| 8. C2×D8×C89 | C2 | semidirect |
| 11. #13083573 | C22 | semidirect |
| 12. C2×Dic178 | C22 | semidirect |
| 17. D4×D89 | C22 | semidirect |
| 20. #13083563 | C22 | semidirect |
| 21. #13083562 | C22 | semidirect |
| 22. #13083559 | C22 | semidirect |
| 23. #13083557 | C22 | semidirect |
| 24. C2×#10623494 | C22 | semidirect |
| 26. C2×D4×C89 | C22 | semidirect |
| 27. D8×C89 | C22 | semidirect |
| 28. C2×C8×C89 | C22 | semidirect |
| 38. D4×C89 | C23 | semidirect |
| 50. Dic178 | C23 | semidirect |
| 51. C8×C89 | C23 | semidirect |
| 52. #10623494 | C23 | semidirect |
| 69. C2×C89 | AGL(1,ℤ/8) | semidirect |
| 73. C89 | C2×AGL(1,ℤ/8) | semidirect |
| 81. C2×D8 | D89 | semidirect |
| 93. C2×D4 | C2×D89 | semidirect |
| 99. D8 | C2×D89 | semidirect |
| 102. C2×C8 | C2×D89 | semidirect |
| 118. D4 | C22×D89 | semidirect |
| 126. C8 | C22×D89 | semidirect |
| 145. C2 | #15668036 | direct (N × H) |