Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
684 conjugacy classes of subgroups; only the 320 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 320 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#6157 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C22 | C23×#6157 |
| Fitting | C4×Q8×C53 | C4 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C4×Q8 | #6157 |
| hypocenter | C53 | C42×Q8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. Q8×#6157 | C4 | direct (N × H) |
| 14. Q8×Dic53 | C4 | semidirect |
| 15. C4×Dic106 | C4 | semidirect |
| 16. #13133182 | C4 | semidirect |
| 19. C4×Q8×C53 | C4 | semidirect |
| 32. C4×#6157 | Q8 | direct (N × H) |
| 47. Q8×C53 | C42 | semidirect |
| 58. Dic106 | C42 | semidirect |
| 65. #6157 | C4×Q8 | direct (N × H) |
| 68. C4×C53 | C4×Q8 | semidirect |
| 71. Dic53 | C4×Q8 | semidirect |
| 83. C53 | C42×Q8 | semidirect |
| 89. C4×Q8 | #6157 | direct (N × H) |
| 120. Q8 | C4×#6157 | direct (N × H) |
| 149. C4 | Q8×#6157 | direct (N × H) |