Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
3550 conjugacy classes of subgroups; only the 974 normal subgroups are shown.
This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 974 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| prime radical | C26×C7 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2840 | C2 | direct (N × H) |
| 2. C2×C7×#393 | C2 | semidirect |
| 5. C23×C7×Q8 | C22 | semidirect |
| 7. C7×#393 | C22 | semidirect |
| 9. C2×C7×#95 | C22 | semidirect |
| 10. C2×#2840 | C7 | direct (N × H) |
| 12. C25×C7 | Q8 | semidirect |
| 16. C22×C7×Q8 | C23 | semidirect |
| 20. C7×#95 | C23 | semidirect |
| 21. #2840 | C2×C7 | direct (N × H) |
| 22. C2×#393 | C2×C7 | semidirect |
| 29. C24×C7 | C2×Q8 | semidirect |
| 38. C23×Q8 | C22×C7 | semidirect |
| 40. #393 | C22×C7 | semidirect |
| 42. C2×#95 | C22×C7 | semidirect |
| 57. C25 | C7×Q8 | semidirect |
| 61. C22×Q8 | C23×C7 | semidirect |
| 65. #95 | C23×C7 | semidirect |
| 80. C24 | C2×C7×Q8 | semidirect |
| 89. C2×C7 | #2840 | direct (N × H) |
| 105. C7 | C2×#2840 | direct (N × H) |
| 116. C2 | C7×#2840 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2840 | 2 |
| C2×C7×#393 | 2 |
| C2×C7×#379 | 2 |
| C2×#2840 | 7 |