Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 220 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 | #442 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C2×C7×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C7×#322 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1593 | C2 | semidirect |
| 4. C7×#1368 | C2 | semidirect |
| 5. C7×#1334 | C2 | semidirect |
| 7. C7×#1313 | C2 | semidirect |
| 23. #12077 | C7 | direct (N × H) |
| 46. #1593 | C2×C7 | semidirect |
| 48. #1368 | C2×C7 | semidirect |
| 49. #1334 | C2×C7 | semidirect |
| 51. #1313 | C2×C7 | semidirect |
| 189. C7 | #12077 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2175 | 2 |
| C7×#1593 | 2 |
| C7×#1580 | 2 |
| C7×#1368 | 2 |
| C7×#1334 | 2 |
| C7×#1333 | 2 |
| C7×#1313 | 2 |
| #12077 | 7 |