Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
9486 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.
This group has 9486 conjugacy classes of subgroups — too many to draw. Only its 6366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| prime radical | C26×C7 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×#126 | C2 | direct (N × H) |
| 2. C23×C7×C4⋊C4 | C2 | semidirect |
| 4. C23×C42×C7 | C2 | semidirect |
| 8. C2×C7×#126 | C22 | direct (N × H) |
| 9. C22×C7×C4⋊C4 | C22 | semidirect |
| 11. C22×C42×C7 | C22 | semidirect |
| 12. C23×#126 | C7 | direct (N × H) |
| 20. C7×#126 | C23 | direct (N × H) |
| 21. C2×C7×C4⋊C4 | C23 | semidirect |
| 23. C2×C42×C7 | C23 | semidirect |
| 24. C22×#126 | C2×C7 | direct (N × H) |
| 25. C23×C4⋊C4 | C2×C7 | semidirect |
| 27. C23×C42 | C2×C7 | semidirect |
| 37. C7×C4⋊C4 | C24 | semidirect |
| 39. C42×C7 | C24 | semidirect |
| 43. C2×#126 | C22×C7 | direct (N × H) |
| 44. C22×C4⋊C4 | C22×C7 | semidirect |
| 46. C22×C42 | C22×C7 | semidirect |
| 49. C23×C7 | #126 | direct (N × H) |
| 64. #126 | C23×C7 | direct (N × H) |
| 65. C2×C4⋊C4 | C23×C7 | semidirect |
| 67. C2×C42 | C23×C7 | semidirect |
| 70. C22×C7 | C2×#126 | direct (N × H) |
| 84. C4⋊C4 | C24×C7 | semidirect |
| 86. C42 | C24×C7 | semidirect |
| 88. C2×C7 | C22×#126 | direct (N × H) |
| 92. C23 | C7×#126 | direct (N × H) |
| 100. C7 | C23×#126 | direct (N × H) |
| 103. C22 | C2×C7×#126 | direct (N × H) |
| 109. C2 | C22×C7×#126 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×#126 | 2 |
| C23×C7×C4⋊C4 | 2 |
| C23×C7×C22⋊C4 | 2 |
| C23×C42×C7 | 2 |
| C23×#126 | 7 |