Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
644 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 121 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 | C22×D21 |
| commutator, socle, cosocle | C23×C3×C7 | C23 |
| Fitting | C3×C7×#95 | C2 |
| hypercenter | #95 | D21 |
| hypocenter | C3×C7 | #392 |
| LCS | G ▷ C23×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D84 | C2 | semidirect |
| 2. C2×#66631 | C2 | semidirect |
| 3. C3×C7×#95 | C2 | semidirect |
| 13. C7×#95 | D3 | semidirect |
| 20. C22×D21 | D4 | semidirect |
| 22. C2×C3×C4×C7 | D4 | semidirect |
| 33. C3×#95 | D7 | semidirect |
| 52. C2×C4×C7 | D12 | semidirect |
| 67. #95 | D21 | semidirect |
| 85. C2×C3×C4 | D28 | semidirect |
| 88. C3×C7 | #392 | semidirect |
| 118. C2×C4 | D84 | semidirect |
| 119. C7 | #4671 | semidirect |
| 134. C3 | #89443 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D84 | 2 |
| C2×#66631 | 2 |
| C3×C7×#95 | 2 |
| #89443 | 3 |
| #4671 | 7 |