Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
348 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2127 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | #127 | C23×C7 |
| prime radical | C7×#3220 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#535 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2128 | C2 | semidirect |
| 3. C7×#2127 | C2 | semidirect |
| 4. C7×#1412 | C2 | semidirect |
| 5. C7×#1408 | C2 | semidirect |
| 6. C7×#540 | C4 | semidirect |
| 8. C7×#496 | C4 | semidirect |
| 11. C7×#488 | C4 | semidirect |
| 16. C7×#433 | C4 | semidirect |
| 23. #13609 | C7 | direct (N × H) |
| 30. C7×#127 | C2×C4 | semidirect |
| 33. C7×#125 | C2×C4 | semidirect |
| 55. C7×#97 | C2×C4 | semidirect |
| 56. C4×C7×C8 | C2×C4 | semidirect |
| 58. #2128 | C2×C7 | semidirect |
| 59. #2127 | C2×C7 | semidirect |
| 60. #1412 | C2×C7 | semidirect |
| 61. #1408 | C2×C7 | semidirect |
| 101. #540 | C4×C7 | semidirect |
| 103. #496 | C4×C7 | semidirect |
| 106. #488 | C4×C7 | semidirect |
| 111. #433 | C4×C7 | semidirect |
| 159. #127 | C2×C4×C7 | semidirect |
| 162. #125 | C2×C4×C7 | semidirect |
| 184. #97 | C2×C4×C7 | semidirect |
| 185. C4×C8 | C2×C4×C7 | semidirect |
| 289. C7 | #13609 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3220 | 2 |
| C7×#2128 | 2 |
| C7×#2127 | 2 |
| C7×#1412 | 2 |
| C7×#1408 | 2 |
| #13609 | 7 |