Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
430 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22×A4 |
| commutator, cosocle | C23 | C23×C3×C7 |
| Frattini | C2 | C23×C7×A4 |
| Fitting | C22×C7×#41 | C3 |
| socle | C23×C7 | C23×C3 |
| hypercenter | C7×#41 | A4 |
| hypocenter | C22 | C3×C7×#41 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×Q8×A4 | C2 | semidirect |
| 2. C7×D4×A4 | C2 | semidirect |
| 3. C2×C4×C7×A4 | C2 | semidirect |
| 4. C22×C7×#41 | C3 | semidirect |
| 13. C22×C7×Q8 | C2×C3 | semidirect |
| 14. C22×C7×D4 | C2×C3 | semidirect |
| 15. C23×C4×C7 | C2×C3 | semidirect |
| 16. A4×#41 | C7 | direct (N × H) |
| 29. C7×#41 | A4 | direct (N × H) |
| 41. Q8×A4 | C2×C7 | semidirect |
| 42. D4×A4 | C2×C7 | semidirect |
| 43. C2×C4×A4 | C2×C7 | semidirect |
| 45. C7×A4 | #41 | direct (N × H) |
| 48. C22×#41 | C3×C7 | semidirect |
| 56. C7×Q8 | C2×A4 | semidirect |
| 60. C7×D4 | C2×A4 | semidirect |
| 69. C2×C4×C7 | C2×A4 | semidirect |
| 80. C22×Q8 | C2×C3×C7 | semidirect |
| 81. C22×D4 | C2×C3×C7 | semidirect |
| 82. C23×C4 | C2×C3×C7 | semidirect |
| 90. C22×C7 | C3×#41 | semidirect |
| 109. #41 | C7×A4 | direct (N × H) |
| 127. A4 | C7×#41 | direct (N × H) |
| 137. Q8 | C2×C7×A4 | semidirect |
| 141. D4 | C2×C7×A4 | semidirect |
| 150. C2×C4 | C2×C7×A4 | semidirect |
| 151. C7 | A4×#41 | direct (N × H) |
| 161. C22 | C3×C7×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Q8×A4 | 2 |
| C7×D4×A4 | 2 |
| C2×C4×C7×A4 | 2 |
| C22×C7×#41 | 3 |
| C3×C7×#41 | 4 |
| A4×#41 | 7 |