Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
2536 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 2536 conjugacy classes of subgroups — too many to draw. Only its 581 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D197 |
| commutator, cosocle | C22×C197 | C25 |
| Frattini | C22 | C24×D197 |
| Fitting | C2×#127×C197 | C2 |
| socle | C23×C197 | C24 |
| hypercenter | C2×#127 | D197 |
| hypocenter | C197 | C2×D42 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D4×D197 | C2 | semidirect |
| 2. #27161714 | C2 | direct (N × H) |
| 3. C2×#20263297 | C2 | semidirect |
| 4. C2×#20263294 | C2 | semidirect |
| 5. C2×C4×D788 | C2 | semidirect |
| 6. C2×#127×C197 | C2 | semidirect |
| 7. C24×D197 | C22 | semidirect |
| 9. C22×C22⋊D197 | C22 | semidirect |
| 11. C2×D4×D197 | C22 | semidirect |
| 12. C22×C4×D197 | C22 | semidirect |
| 13. #20263297 | C22 | semidirect |
| 14. #20263294 | C22 | semidirect |
| 15. C4×D788 | C22 | semidirect |
| 16. C2×C4⋊Dic197 | C22 | semidirect |
| 17. C2×C22⋊Dic197 | C22 | semidirect |
| 18. C2×#15691335 | C22 | semidirect |
| 19. C22×D4×C197 | C22 | semidirect |
| 20. C22×D4×C197 | C22 | semidirect |
| 21. #127×C197 | C22 | semidirect |
| 22. C2×C42×C197 | C22 | semidirect |
| 26. C23×D197 | C23 | semidirect |
| 27. C2×D788 | D4 | semidirect |
| 30. C2×C22⋊D197 | C23 | semidirect |
| 34. C2×C4×D197 | C23 | semidirect |
| 35. C4⋊Dic197 | C23 | semidirect |
| 36. C22⋊Dic197 | C23 | semidirect |
| 37. #15691335 | C23 | semidirect |
| 41. C2×D4×C197 | C23 | semidirect |
| 42. C2×D4×C197 | C23 | semidirect |
| 45. C42×C197 | C23 | semidirect |
| 51. C22×D197 | C2×D4 | semidirect |
| 53. D788 | C2×D4 | semidirect |
| 65. C2×C4×C197 | C2×D4 | semidirect |
| 78. C2×D197 | C22×D4 | semidirect |
| 80. C4×C197 | C22×D4 | semidirect |
| 83. C2×C197 | D42 | semidirect |
| 87. C197 | C2×D42 | semidirect |
| 91. C2×#127 | D197 | semidirect |
| 100. C22×D4 | C2×D197 | semidirect |
| 101. C22×D4 | C2×D197 | semidirect |
| 103. #127 | C2×D197 | semidirect |
| 110. C2×C42 | C2×D197 | semidirect |
| 123. C2×D4 | C22×D197 | semidirect |
| 124. C2×D4 | C22×D197 | semidirect |
| 133. C42 | C22×D197 | semidirect |
| 153. C2×C4 | D4×D197 | semidirect |
| 169. C4 | C2×D4×D197 | semidirect |
| 172. C2 | #27161714 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×D197 | 2 |
| #27161714 | 2 |
| C2×#20263297 | 2 |
| C2×#20263294 | 2 |
| C2×C4×D788 | 2 |
| C2×#127×C197 | 2 |
| C2×D42 | 197 |