Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
1324 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 318 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C7 | C22×D3 |
| commutator, cosocle | C22×C3 | C24×C7 |
| Frattini | C22 | C23×D3×C7 |
| Fitting | C25×C3×C7 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C25×C7 | D3 |
| hypocenter | C3 | C2×C7×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C7 ◁ C25×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×C3×C7 | C2 | semidirect |
| 2. C22×C7×#67 | C2 | semidirect |
| 3. C7×#953 | C2 | direct (N × H) |
| 4. C2×C7×#217 | C2 | semidirect |
| 8. C24×C3×C7 | C22 | semidirect |
| 9. C23×D3×C7 | C22 | semidirect |
| 11. C2×C7×#67 | C22 | semidirect |
| 12. C22×C7×Dic3 | C22 | semidirect |
| 13. C7×#217 | C22 | semidirect |
| 14. C25×C7 | D3 | semidirect |
| 18. C2×#953 | C7 | direct (N × H) |
| 22. C23×C3×C7 | D4 | semidirect |
| 26. C22×D3×C7 | C23 | semidirect |
| 29. C2×C7×Dic3 | C23 | semidirect |
| 33. C24×C7 | C2×D3 | semidirect |
| 38. C25×C3 | C2×C7 | semidirect |
| 39. C22×#67 | C2×C7 | semidirect |
| 40. #953 | C2×C7 | direct (N × H) |
| 41. C2×#217 | C2×C7 | semidirect |
| 46. C22×C3×C7 | C2×D4 | semidirect |
| 57. C23×C7 | #67 | semidirect |
| 65. C24×C3 | C22×C7 | semidirect |
| 66. C23×D3 | C22×C7 | semidirect |
| 68. C2×#67 | C22×C7 | semidirect |
| 69. C22×Dic3 | C22×C7 | semidirect |
| 70. #217 | C22×C7 | semidirect |
| 72. C2×C3×C7 | #120 | semidirect |
| 75. C25 | D3×C7 | semidirect |
| 85. C22×C7 | C2×#67 | semidirect |
| 91. C23×C3 | C7×D4 | semidirect |
| 95. C22×D3 | C23×C7 | semidirect |
| 98. C2×Dic3 | C23×C7 | semidirect |
| 99. C3×C7 | C2×#120 | semidirect |
| 103. C24 | C2×D3×C7 | semidirect |
| 110. C2×C7 | #953 | direct (N × H) |
| 116. C22×C3 | C2×C7×D4 | semidirect |
| 126. C23 | C7×#67 | semidirect |
| 132. C7 | C2×#953 | direct (N × H) |
| 134. C2×C3 | C7×#120 | semidirect |
| 143. C22 | C2×C7×#67 | semidirect |
| 146. C3 | C2×C7×#120 | semidirect |
| 149. C2 | C7×#953 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C3×C7 | 2 |
| C22×C7×#67 | 2 |
| C7×#953 | 2 |
| C2×C7×#217 | 2 |
| C2×C7×#120 | 3 |
| C2×#953 | 7 |