Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
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×D284 |
| commutator, cosocle | C4×C71 | C26 |
| Frattini | C4 | C25×D71 |
| Fitting | C71×#3573 | C2 |
| socle | C2×C71 | C24×D4 |
| hypercenter | #3573 | D71 |
| hypocenter | C71 | #63091 |
| LCS | G ▷ C4×C71 ▷ C2×C71 ▷ C71 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ #3573 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22603151 | C2 | semidirect |
| 2. #22603121 | C2 | semidirect |
| 3. #22603114 | C2 | semidirect |
| 4. C2×#17986804 | C2 | semidirect |
| 5. C71×#3573 | C2 | semidirect |
| 7. #17986835 | C22 | semidirect |
| 8. C2×#14556174 | C22 | semidirect |
| 10. #17986816 | C22 | semidirect |
| 11. #17986811 | C22 | semidirect |
| 12. #17986804 | C22 | semidirect |
| 13. #17986804 | C22 | semidirect |
| 14. #17986799 | C22 | semidirect |
| 15. C2×#14556156 | C22 | semidirect |
| 16. C2×#14556150 | C22 | semidirect |
| 17. C22×D568 | C22 | semidirect |
| 19. #568×C71 | C22 | semidirect |
| 20. C2×#131×C71 | C22 | semidirect |
| 22. #14556175 | C23 | semidirect |
| 24. #14556174 | C23 | semidirect |
| 27. C2×#11765865 | C23 | semidirect |
| 29. C22×D284 | C23 | semidirect |
| 31. #14556161 | C23 | semidirect |
| 32. #14556156 | C23 | semidirect |
| 33. #14556150 | C23 | semidirect |
| 34. #14556150 | C23 | semidirect |
| 35. C2×Dic284 | C23 | semidirect |
| 36. C2×#11765854 | C23 | semidirect |
| 37. C2×D568 | C23 | semidirect |
| 38. C2×D568 | C23 | semidirect |
| 42. #131×C71 | C23 | semidirect |
| 43. #131×C71 | C23 | semidirect |
| 44. C2×M16×C71 | C23 | semidirect |
| 45. C22×C8×C71 | C23 | semidirect |
| 102. #3573 | D71 | semidirect |
| 103. C71 | #63091 | semidirect |
| 117. #568 | C2×D71 | semidirect |
| 118. C2×#131 | C2×D71 | semidirect |
| 140. #131 | C22×D71 | semidirect |
| 141. #131 | C22×D71 | semidirect |
| 142. C2×M16 | C22×D71 | semidirect |
| 143. C22×C8 | C22×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22603151 | 2 |
| #22603121 | 2 |
| #22603114 | 2 |
| C2×#17986804 | 2 |
| C71×#3573 | 2 |
| #63091 | 71 |