Finite Group DatabasebetaFinite Group DatabasebetaOrder 8128.
322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D127 |
| commutator | C22×C127 | C24 |
| Frattini | C22 | C23×D127 |
| Fitting | C2×C22⋊C4×C127 | C2 |
| socle | C23×C127 | C23 |
| hypercenter | C2×C22⋊C4 | D127 |
| hypocenter | C127 | C2×C22⋊Q8 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Dic254 | C2 | semidirect |
| 3. #16861186 | C2 | direct (N × H) |
| 4. C2×C4⋊Dic127 | C2 | semidirect |
| 5. C2×Dic127⋊C4 | C2 | semidirect |
| 6. C2×Dic127⋊C4 | C2 | semidirect |
| 10. C2×Dic254 | C22 | semidirect |
| 17. C4⋊Dic127 | C22 | semidirect |
| 18. Dic127⋊C4 | C22 | semidirect |
| 19. Dic127⋊C4 | C22 | semidirect |
| 37. C23×C127 | Q8 | semidirect |
| 49. C22×C127 | C2×Q8 | semidirect |
| 59. C2×C127 | C22⋊Q8 | semidirect |
| 63. C127 | C2×C22⋊Q8 | semidirect |
| 91. C23 | Dic254 | semidirect |
| 113. C22 | C2×Dic254 | semidirect |
| 123. C2 | #16861186 | direct (N × H) |