Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
644 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D3×D7 |
| commutator, socle, cosocle | C23×C3×C7 | C23 |
| Fitting | C22×C3×C4×C7 | C22 |
| hypercenter | C22×C4 | D3×D7 |
| hypocenter | C3×C7 | #392 |
| LCS | G ▷ C23×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×D28 | C2 | semidirect |
| 2. C22×#4245 | C2 | semidirect |
| 3. C2×C7×(C2×D3)⋊C4 | C2 | semidirect |
| 4. C2×(C2×D7)⋊Dic3 | C2 | semidirect |
| 5. #10620670 | C2 | semidirect |
| 20. C22×D28 | D3 | semidirect |
| 30. C22×D3×C7 | D4 | semidirect |
| 37. C22×C3×D7 | D4 | semidirect |
| 39. C2×C3×C4×C7 | D4 | semidirect |
| 42. C2×Dic21 | D4 | semidirect |
| 62. C2×(C2×D3)⋊C4 | D7 | semidirect |
| 97. C22×D7 | #67 | semidirect |
| 101. C2×C4×C7 | #67 | semidirect |
| 163. C22×D3 | D28 | semidirect |
| 166. C2×C3×C4 | C22⋊D7 | semidirect |
| 173. C3×C7 | #392 | semidirect |
| 234. C2×C4 | #4245 | semidirect |
| 235. C7 | #5223 | semidirect |
| 263. C3 | #89732 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D28 | 2 |
| C22×#4245 | 2 |
| C2×C7×(C2×D3)⋊C4 | 2 |
| C2×(C2×D7)⋊Dic3 | 2 |
| #10620670 | 2 |
| #89732 | 3 |
| #5223 | 7 |