Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
134 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D71 |
| commutator, cosocle | C32×C71 | C23 |
| Frattini | C32 | C22×D71 |
| Fitting | C71×D64 | C2 |
| socle | C2×C71 | C2×D32 |
| hypercenter | D64 | D71 |
| hypocenter | C71 | #13742 |
| LCS | G ▷ C32×C71 ▷ C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22590646 | C2 | semidirect |
| 2. D32×D71 | C2 | semidirect |
| 3. #22586556 | C2 | semidirect |
| 4. #22586555 | C2 | semidirect |
| 5. #22586552 | C2 | semidirect |
| 6. #22586550 | C2 | semidirect |
| 7. C71×D64 | C2 | semidirect |
| 10. Dic1136 | C22 | semidirect |
| 15. C71⋊C64 | C22 | semidirect |
| 17. D32×C71 | C22 | semidirect |
| 18. C64×C71 | C22 | semidirect |
| 71. D64 | D71 | semidirect |
| 73. C71 | #13742 | semidirect |
| 81. D32 | C2×D71 | semidirect |
| 84. C64 | C2×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22590646 | 2 |
| D32×D71 | 2 |
| #22586556 | 2 |
| #22586555 | 2 |
| #22586552 | 2 |
| #22586550 | 2 |
| C71×D64 | 2 |
| #13742 | 71 |