Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
480 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×#117 | C2 |
| hypercenter | #117 | D21 |
| hypocenter | C3×C7 | #540 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7×#117 | C2 | semidirect |
| 2. #10621413 | C2 | semidirect |
| 3. #10621377 | C2 | semidirect |
| 4. #10621367 | C2 | semidirect |
| 5. #10621357 | C2 | semidirect |
| 6. C4×D84 | C2 | semidirect |
| 7. #10621353 | C2 | semidirect |
| 22. C7×#117 | D3 | semidirect |
| 55. C3×#117 | D7 | semidirect |
| 109. #117 | D21 | semidirect |
| 133. C3×C7 | #540 | semidirect |
| 174. C7 | #5535 | semidirect |
| 190. C3 | #90226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×#117 | 2 |
| #10621413 | 2 |
| #10621377 | 2 |
| #10621367 | 2 |
| #10621357 | 2 |
| C4×D84 | 2 |
| #10621353 | 2 |
| #90226 | 3 |
| #5535 | 7 |