Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1200 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C7 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#555 | C2 | semidirect |
| 2. C2×C7×#456 | C2 | semidirect |
| 3. C2×C7×#454 | C2 | semidirect |
| 4. C7×#2204 | C2 | direct (N × H) |
| 5. C2×C7×#356 | C2 | semidirect |
| 6. C2×C7×#355 | C2 | semidirect |
| 12. C7×#555 | C22 | semidirect |
| 20. C7×#456 | C22 | semidirect |
| 22. C7×#454 | C22 | semidirect |
| 27. C7×#356 | C22 | semidirect |
| 28. C7×#355 | C22 | semidirect |
| 29. C2×#2204 | C7 | direct (N × H) |
| 83. C2×#555 | C2×C7 | semidirect |
| 84. C2×#456 | C2×C7 | semidirect |
| 85. C2×#454 | C2×C7 | semidirect |
| 86. #2204 | C2×C7 | direct (N × H) |
| 87. C2×#356 | C2×C7 | semidirect |
| 88. C2×#355 | C2×C7 | semidirect |
| 164. #555 | C22×C7 | semidirect |
| 172. #456 | C22×C7 | semidirect |
| 174. #454 | C22×C7 | semidirect |
| 179. #356 | C22×C7 | semidirect |
| 180. #355 | C22×C7 | semidirect |
| 376. C2×C7 | #2204 | direct (N × H) |
| 427. C7 | C2×#2204 | direct (N × H) |
| 451. C2 | C7×#2204 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#555 | 2 |
| C2×C7×#456 | 2 |
| C2×C7×#454 | 2 |
| C7×#2204 | 2 |
| C2×C7×#356 | 2 |
| C2×C7×#355 | 2 |
| C2×#2204 | 7 |