Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1284 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1284 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×H(3,𝔽4) | C2 | semidirect |
| 2. C2×C7×#454 | C2 | semidirect |
| 3. C7×#2202 | C2 | direct (N × H) |
| 4. C2×C7×#355 | C2 | semidirect |
| 5. C2×C7×2−1+4 | C22 | semidirect |
| 8. C7×H(3,𝔽4) | C22 | semidirect |
| 13. C7×#454 | C22 | semidirect |
| 15. C2×C7×#101 | C22 | semidirect |
| 16. C7×#355 | C22 | semidirect |
| 17. C2×#2202 | C7 | direct (N × H) |
| 21. C7×2−1+4 | C23 | semidirect |
| 46. C7×#101 | C23 | semidirect |
| 47. C2×H(3,𝔽4) | C2×C7 | semidirect |
| 48. C2×#454 | C2×C7 | semidirect |
| 49. #2202 | C2×C7 | direct (N × H) |
| 50. C2×#355 | C2×C7 | semidirect |
| 92. C2×2−1+4 | C22×C7 | semidirect |
| 95. H(3,𝔽4) | C22×C7 | semidirect |
| 100. #454 | C22×C7 | semidirect |
| 102. C2×#101 | C22×C7 | semidirect |
| 103. #355 | C22×C7 | semidirect |
| 139. 2−1+4 | C23×C7 | semidirect |
| 164. #101 | C23×C7 | semidirect |
| 225. C2×C7 | #2202 | direct (N × H) |
| 259. C7 | C2×#2202 | direct (N × H) |
| 279. C2 | C7×#2202 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×H(3,𝔽4) | 2 |
| C2×C7×#454 | 2 |
| C7×#2202 | 2 |
| C2×C7×#355 | 2 |
| C2×#2202 | 7 |