Finite Group DatabasebetaFinite Group DatabasebetaOrder 20288.
810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D317 |
| commutator, socle, cosocle | C2×C317 | C25 |
| Frattini | C2 | C24×D317 |
| Fitting | C2×#41×C317 | C2 |
| hypercenter | C2×#41 | D317 |
| hypocenter | C317 | #585 |
| LCS | G ▷ C2×C317 ▷ C317 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23778828 | C2 | semidirect |
| 2. #23778827 | C2 | semidirect |
| 3. #23778826 | C2 | semidirect |
| 4. #23778825 | C2 | semidirect |
| 5. C2×#19099574 | C2 | semidirect |
| 6. #41×D317 | C2 | semidirect |
| 7. C2×#41×C317 | C2 | semidirect |
| 8. #19099575 | C22 | semidirect |
| 9. #19099574 | C22 | semidirect |
| 10. #19099574 | C22 | semidirect |
| 11. #19099574 | C22 | semidirect |
| 12. #19099574 | C22 | semidirect |
| 13. #19099573 | C22 | semidirect |
| 14. C2×Dic634 | C22 | semidirect |
| 15. Q8×D317 | C22 | semidirect |
| 16. C2×D1268 | C22 | semidirect |
| 17. C2×C22⋊D317 | C22 | semidirect |
| 18. D4×D317 | C22 | semidirect |
| 19. C2×C4×D317 | C22 | semidirect |
| 20. #41×C317 | C22 | semidirect |
| 21. #41×C317 | C22 | semidirect |
| 22. C2×Q8×C317 | C22 | semidirect |
| 23. C2×D4×C317 | C22 | semidirect |
| 24. C22×C4×C317 | C22 | semidirect |
| 55. C317 | #585 | semidirect |
| 63. C2×#41 | D317 | semidirect |
| 70. #41 | C2×D317 | semidirect |
| 71. #41 | C2×D317 | semidirect |
| 74. C2×Q8 | C2×D317 | semidirect |
| 78. C2×D4 | C2×D317 | semidirect |
| 80. C22×C4 | C2×D317 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778828 | 2 |
| #23778827 | 2 |
| #23778826 | 2 |
| #23778825 | 2 |
| C2×#19099574 | 2 |
| #41×D317 | 2 |
| C2×#41×C317 | 2 |
| #585 | 317 |