Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
710 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #90408 |
| commutator, cosocle | C42×C7 | C24 |
| Frattini | C42 | C23×D7 |
| Fitting | C7×#1706 | C2 |
| socle | C22×C7 | D42 |
| hypercenter | #1706 | D7 |
| hypocenter | C7 | #19888 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1706 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11739218 | C2 | semidirect |
| 2. #11739029 | C2 | semidirect |
| 3. C7×#1706 | C2 | semidirect |
| 4. #11729130 | C2 | semidirect |
| 5. #11729039 | C2 | semidirect |
| 6. #11729022 | C2 | semidirect |
| 7. #11728963 | C2 | semidirect |
| 8. C8×Dic28 | C2 | semidirect |
| 9. #11725806 | C2 | semidirect |
| 86. #1706 | D7 | semidirect |
| 133. Dic28 | D8 | semidirect |
| 341. C7 | #19888 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739218 | 2 |
| #11739029 | 2 |
| C7×#1706 | 2 |
| #11729130 | 2 |
| #11729039 | 2 |
| #11729022 | 2 |
| #11728963 | 2 |
| C8×Dic28 | 2 |
| #11725806 | 2 |
| #19888 | 7 |