Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3 | C22×D7 |
| commutator, cosocle | C23×C7 | C23×C3 |
| Frattini | C23 | C22×C3×D7 |
| Fitting | C3×C7×#95 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C3×#95 | D7 |
| hypocenter | C7 | C3×#392 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×D28 | C2 | semidirect |
| 2. C2×C3×(C2×D7)⋊C4 | C2 | semidirect |
| 3. C3×C7×#95 | C2 | semidirect |
| 4. #89443 | C3 | direct (N × H) |
| 11. C22×D28 | C2×C3 | semidirect |
| 12. C2×(C2×D7)⋊C4 | C2×C3 | semidirect |
| 13. C7×#95 | C2×C3 | semidirect |
| 20. C22×C3×D7 | D4 | semidirect |
| 22. C2×C3×C4×C7 | D4 | semidirect |
| 33. C3×#95 | D7 | semidirect |
| 50. C22×D7 | C3×D4 | semidirect |
| 52. C2×C4×C7 | C3×D4 | semidirect |
| 67. #95 | C3×D7 | semidirect |
| 87. C2×C3×C4 | D28 | semidirect |
| 88. C3×C7 | #392 | semidirect |
| 118. C2×C4 | C3×D28 | semidirect |
| 119. C7 | C3×#392 | semidirect |
| 134. C3 | #89443 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D28 | 2 |
| C2×C3×(C2×D7)⋊C4 | 2 |
| C3×C7×#95 | 2 |
| #89443 | 3 |
| C3×#392 | 7 |