Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
3550 conjugacy classes of subgroups; only the 579 normal subgroups are shown.
This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 579 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D71 |
| commutator, socle, cosocle | C23×C71 | C25 |
| Fitting | C71×#2840 | C2 |
| prime radical | C25×D71 | C22 |
| hypercenter | #2840 | D71 |
| hypocenter | C71 | C2×#2840 |
| LCS | G ▷ C23×C71 ▷ C71 | |
| UCS | C1 ◁ C23 ◁ #2840 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22596593 | C2 | semidirect |
| 2. #22596592 | C2 | semidirect |
| 3. #393×D71 | C2 | semidirect |
| 5. C71×#2840 | C2 | semidirect |
| 7. C22×Q8×D71 | C22 | semidirect |
| 9. #17985912 | C22 | semidirect |
| 10. #17985911 | C22 | semidirect |
| 11. #17985910 | C22 | semidirect |
| 14. #95×D71 | C22 | semidirect |
| 15. #393×C71 | C22 | semidirect |
| 18. C24×D71 | Q8 | semidirect |
| 21. C22×Dic142 | C23 | semidirect |
| 27. C71⋊#95 | C23 | semidirect |
| 29. C22×Q8×C71 | C23 | semidirect |
| 31. #95×C71 | C23 | semidirect |
| 37. C23×D71 | C2×Q8 | semidirect |
| 47. C24×C71 | C2×Q8 | semidirect |
| 89. D71 | #2840 | direct (N × H) |
| 92. #2840 | D71 | direct (N × H) |
| 95. C71 | C2×#2840 | semidirect |
| 102. #393 | C2×D71 | semidirect |
| 114. C22×Q8 | C22×D71 | semidirect |
| 121. #95 | C22×D71 | semidirect |
| 131. C24 | Q8×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22596593 | 2 |
| #22596592 | 2 |
| #393×D71 | 2 |
| #379×D71 | 2 |
| C71×#2840 | 2 |
| C2×#2840 | 71 |