Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
512 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 90 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 | C22×Q8 | C23×C7 |
| prime radical | C7×#2733 | C2 |
| LCS | G ▷ C22×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 |
|---|---|---|
| 2. C2×C7×#355 | C2 | semidirect |
| 3. C7×#1888 | C2 | semidirect |
| 4. C7×#1857 | C2 | semidirect |
| 5. C7×#1394 | C2 | semidirect |
| 20. #13426 | C7 | direct (N × H) |
| 50. C7×#101 | D4 | semidirect |
| 54. C2×#355 | C2×C7 | semidirect |
| 55. #1888 | C2×C7 | semidirect |
| 56. #1857 | C2×C7 | semidirect |
| 57. #1394 | C2×C7 | semidirect |
| 180. #101 | C7×D4 | semidirect |
| 289. C7 | #13426 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2733 | 2 |
| C2×C7×#355 | 2 |
| C7×#1888 | 2 |
| C7×#1857 | 2 |
| C7×#1394 | 2 |
| #13426 | 7 |