Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
10894 conjugacy classes of subgroups; only the 3815 normal subgroups are shown.
This group has 10894 conjugacy classes of subgroups — too many to draw. Only its 3815 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D316 |
| commutator, cosocle | C4×C79 | C26 |
| Frattini | C4 | C25×D79 |
| Fitting | C79×#3573 | C2 |
| socle | C2×C79 | C24×D4 |
| hypercenter | #3573 | D79 |
| hypocenter | C79 | #63091 |
| LCS | G ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ #3573 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23778211 | C2 | semidirect |
| 2. #23778181 | C2 | semidirect |
| 3. #23778174 | C2 | semidirect |
| 4. C2×#19099368 | C2 | semidirect |
| 5. C79×#3573 | C2 | semidirect |
| 7. #19099399 | C22 | semidirect |
| 8. C2×#14579207 | C22 | semidirect |
| 10. #19099380 | C22 | semidirect |
| 11. #19099375 | C22 | semidirect |
| 12. #19099368 | C22 | semidirect |
| 13. #19099368 | C22 | semidirect |
| 14. #19099363 | C22 | semidirect |
| 15. C2×#14579189 | C22 | semidirect |
| 16. C2×#14579183 | C22 | semidirect |
| 17. C22×D632 | C22 | semidirect |
| 19. #568×C79 | C22 | semidirect |
| 20. C2×#131×C79 | C22 | semidirect |
| 22. #14579208 | C23 | semidirect |
| 24. #14579207 | C23 | semidirect |
| 27. C2×#11929624 | C23 | semidirect |
| 29. C22×D316 | C23 | semidirect |
| 31. #14579194 | C23 | semidirect |
| 32. #14579189 | C23 | semidirect |
| 33. #14579183 | C23 | semidirect |
| 34. #14579183 | C23 | semidirect |
| 35. C2×Dic316 | C23 | semidirect |
| 36. C2×#11929613 | C23 | semidirect |
| 37. C2×D632 | C23 | semidirect |
| 38. C2×D632 | C23 | semidirect |
| 42. #131×C79 | C23 | semidirect |
| 43. #131×C79 | C23 | semidirect |
| 44. C2×M16×C79 | C23 | semidirect |
| 45. C22×C8×C79 | C23 | semidirect |
| 102. #3573 | D79 | semidirect |
| 103. C79 | #63091 | semidirect |
| 117. #568 | C2×D79 | semidirect |
| 118. C2×#131 | C2×D79 | semidirect |
| 140. #131 | C22×D79 | semidirect |
| 141. #131 | C22×D79 | semidirect |
| 142. C2×M16 | C22×D79 | semidirect |
| 143. C22×C8 | C22×D79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778211 | 2 |
| #23778181 | 2 |
| #23778174 | 2 |
| C2×#19099368 | 2 |
| C79×#3573 | 2 |
| #63091 | 79 |