Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
554 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 554 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C7 | S4 |
| commutator, hypocenter | A4 | C22×C4×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting, socle | C25×C7 | D3 |
| prime radical | C23×C7×A4 | C2 |
| cosocle | C2×A4 | C23×C7 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C23×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×SL(2,ℤ/4) | C2 | direct (N × H) |
| 4. C22×C7×A4 | C4 | semidirect |
| 7. C7×SL(2,ℤ/4) | C22 | direct (N × H) |
| 13. C22×SL(2,ℤ/4) | C7 | direct (N × H) |
| 15. C2×C7×A4 | C2×C4 | semidirect |
| 21. C24×C7 | Dic3 | semidirect |
| 31. C2×SL(2,ℤ/4) | C2×C7 | direct (N × H) |
| 34. C7×A4 | C22×C4 | semidirect |
| 46. C23×C7 | C2×Dic3 | semidirect |
| 54. C22×A4 | C4×C7 | semidirect |
| 57. SL(2,ℤ/4) | C22×C7 | direct (N × H) |
| 74. C22×C7 | SL(2,ℤ/4) | direct (N × H) |
| 75. C22×C7 | C22×Dic3 | semidirect |
| 80. C2×A4 | C2×C4×C7 | semidirect |
| 87. C24 | C7×Dic3 | semidirect |
| 99. C2×C7 | C2×SL(2,ℤ/4) | direct (N × H) |
| 103. A4 | C22×C4×C7 | semidirect |
| 114. C23 | C2×C7×Dic3 | semidirect |
| 122. C7 | C22×SL(2,ℤ/4) | direct (N × H) |
| 134. C22 | C7×SL(2,ℤ/4) | direct (N × H) |
| 135. C22 | C22×C7×Dic3 | semidirect |
| 143. C2 | C2×C7×SL(2,ℤ/4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C7×A4 | 2 |
| C2×C7×SL(2,ℤ/4) | 2 |
| C22×C7×C22⋊C4 | 3 |
| C22×C7×Dic3 | 4 |
| C22×SL(2,ℤ/4) | 7 |