Finite Group DatabasebetaFinite Group DatabasebetaOrder 24896.
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×D389 |
| commutator, cosocle | C2×C389 | C25 |
| Frattini | C2 | C24×D389 |
| Fitting | C2×#41×C389 | C2 |
| socle | C22×C389 | C24 |
| hypercenter | C2×#41 | D389 |
| hypocenter | C389 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C389 ▷ C389 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27159430 | C2 | direct (N × H) |
| 2. C2×#20261967 | C2 | semidirect |
| 3. C2×#20261966 | C2 | semidirect |
| 4. C22×D1556 | C2 | semidirect |
| 5. C2×D4×D389 | C2 | semidirect |
| 6. C2×#41×C389 | C2 | semidirect |
| 7. C23×D389 | C22 | semidirect |
| 8. #20261967 | C22 | semidirect |
| 9. #20261966 | C22 | semidirect |
| 10. C2×Dic778 | C22 | semidirect |
| 11. C2×D1556 | C22 | semidirect |
| 12. C2×D1556 | C22 | semidirect |
| 13. C2×D1556 | C22 | semidirect |
| 14. C2×C22⋊D389 | C22 | semidirect |
| 15. D4×D389 | C22 | semidirect |
| 16. C2×C4×D389 | C22 | semidirect |
| 17. #41×C389 | C22 | semidirect |
| 18. C2×Q8×C389 | C22 | semidirect |
| 19. C2×D4×C389 | C22 | semidirect |
| 20. C22×C4×C389 | C22 | semidirect |
| 23. C22×D389 | C23 | semidirect |
| 24. Dic778 | C23 | semidirect |
| 25. D1556 | C23 | semidirect |
| 26. D1556 | C23 | semidirect |
| 27. C22⋊D389 | C23 | semidirect |
| 29. C4×D389 | C23 | semidirect |
| 31. Q8×C389 | C23 | semidirect |
| 32. D4×C389 | C23 | semidirect |
| 35. C2×C4×C389 | C23 | semidirect |
| 46. C2×C389 | H(5,𝔽2) | semidirect |
| 49. C389 | C2×H(5,𝔽2) | semidirect |
| 54. C2×#41 | D389 | semidirect |
| 60. #41 | C2×D389 | semidirect |
| 62. C2×Q8 | C2×D389 | semidirect |
| 68. C2×D4 | C2×D389 | semidirect |
| 70. C22×C4 | C2×D389 | semidirect |
| 76. Q8 | C22×D389 | semidirect |
| 80. D4 | C22×D389 | semidirect |
| 85. C2×C4 | C22×D389 | semidirect |
| 97. C2 | #27159430 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27159430 | 2 |
| C2×#20261967 | 2 |
| C2×#20261966 | 2 |
| C22×D1556 | 2 |
| C2×D4×D389 | 2 |
| C2×#41×C389 | 2 |
| C2×H(5,𝔽2) | 389 |