Finite Group DatabasebetaFinite Group DatabasebetaOrder 20288.
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×D317 |
| commutator, cosocle | C2×C317 | C25 |
| Frattini | C2 | C24×D317 |
| Fitting | C2×#41×C317 | C2 |
| socle | C22×C317 | C24 |
| hypercenter | C2×#41 | D317 |
| hypocenter | C317 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C317 ▷ C317 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23778826 | C2 | direct (N × H) |
| 2. C2×#19099575 | C2 | semidirect |
| 3. C2×#19099574 | C2 | semidirect |
| 4. C22×D1268 | C2 | semidirect |
| 5. C2×D4×D317 | C2 | semidirect |
| 6. C2×#41×C317 | C2 | semidirect |
| 7. C23×D317 | C22 | semidirect |
| 8. #19099575 | C22 | semidirect |
| 9. #19099574 | C22 | semidirect |
| 10. C2×Dic634 | C22 | semidirect |
| 11. C2×D1268 | C22 | semidirect |
| 12. C2×D1268 | C22 | semidirect |
| 13. C2×D1268 | C22 | semidirect |
| 14. C2×C22⋊D317 | C22 | semidirect |
| 15. D4×D317 | C22 | semidirect |
| 16. C2×C4×D317 | C22 | semidirect |
| 17. #41×C317 | C22 | semidirect |
| 18. C2×Q8×C317 | C22 | semidirect |
| 19. C2×D4×C317 | C22 | semidirect |
| 20. C22×C4×C317 | C22 | semidirect |
| 23. C22×D317 | C23 | semidirect |
| 24. Dic634 | C23 | semidirect |
| 25. D1268 | C23 | semidirect |
| 26. D1268 | C23 | semidirect |
| 27. C22⋊D317 | C23 | semidirect |
| 29. C4×D317 | C23 | semidirect |
| 31. Q8×C317 | C23 | semidirect |
| 32. D4×C317 | C23 | semidirect |
| 35. C2×C4×C317 | C23 | semidirect |
| 46. C2×C317 | H(5,𝔽2) | semidirect |
| 49. C317 | C2×H(5,𝔽2) | semidirect |
| 54. C2×#41 | D317 | semidirect |
| 60. #41 | C2×D317 | semidirect |
| 62. C2×Q8 | C2×D317 | semidirect |
| 68. C2×D4 | C2×D317 | semidirect |
| 70. C22×C4 | C2×D317 | semidirect |
| 76. Q8 | C22×D317 | semidirect |
| 80. D4 | C22×D317 | semidirect |
| 85. C2×C4 | C22×D317 | semidirect |
| 97. C2 | #23778826 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778826 | 2 |
| C2×#19099575 | 2 |
| C2×#19099574 | 2 |
| C22×D1268 | 2 |
| C2×D4×D317 | 2 |
| C2×#41×C317 | 2 |
| C2×H(5,𝔽2) | 317 |