Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
472 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D5×D7 |
| commutator, cosocle | C2×C5×C7 | C24 |
| Frattini | C2 | C22×D5×D7 |
| Fitting | C23×C5×C7 | C22 |
| socle | C22×C5×C7 | C23 |
| hypercenter | C23 | D5×D7 |
| hypocenter | C5×C7 | C22×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C5×D7 | C2 | semidirect |
| 2. C22×D5×D7 | C2 | semidirect |
| 3. C2×C22⋊D35 | C2 | semidirect |
| 4. C2×C7×C22⋊D5 | C2 | semidirect |
| 5. D7×C22⋊D5 | C2 | direct (N × H) |
| 6. C2×#63306 | C2 | semidirect |
| 7. C2×#63305 | C2 | semidirect |
| 8. C2×D7×Dic5 | C2 | semidirect |
| 9. C23×C5×C7 | C22 | semidirect |
| 10. C22×D35 | C22 | semidirect |
| 11. C22×C7×D5 | C22 | semidirect |
| 14. C22×C5×D7 | C22 | semidirect |
| 15. C22×C5×D7 | C22 | semidirect |
| 17. C2×D5×D7 | C22 | semidirect |
| 18. C22⋊D35 | C22 | semidirect |
| 19. C2×Dic35 | C22 | semidirect |
| 20. C7×C22⋊D5 | C22 | semidirect |
| 21. C2×C7×Dic5 | C22 | semidirect |
| 22. #63306 | C22 | semidirect |
| 23. #63305 | C22 | semidirect |
| 24. D7×Dic5 | C22 | semidirect |
| 29. C22×C5×C7 | C23 | semidirect |
| 31. C2×D35 | C23 | semidirect |
| 33. C2×C7×D5 | C23 | semidirect |
| 36. C2×C5×D7 | D4 | semidirect |
| 38. C2×C5×D7 | C23 | semidirect |
| 40. Dic35 | C23 | semidirect |
| 41. C7×Dic5 | C23 | semidirect |
| 43. C23×D7 | D5 | semidirect |
| 56. C2×C22⋊D5 | D7 | direct (N × H) |
| 59. C2×C5×C7 | C2×D4 | semidirect |
| 63. C5×D7 | C2×D4 | semidirect |
| 66. C23×C7 | C2×D5 | semidirect |
| 72. C22×D7 | C2×D5 | semidirect |
| 73. C22×D7 | C2×D5 | semidirect |
| 85. C23×C5 | C2×D7 | semidirect |
| 89. C22×D5 | C2×D7 | semidirect |
| 93. C22⋊D5 | C2×D7 | direct (N × H) |
| 96. C2×Dic5 | C2×D7 | semidirect |
| 97. C5×C7 | C22×D4 | semidirect |
| 103. C22×C7 | C22×D5 | semidirect |
| 109. C2×D7 | C22⋊D5 | direct (N × H) |
| 111. C2×D7 | C22×D5 | semidirect |
| 121. C22×C5 | C22×D7 | semidirect |
| 126. C2×D5 | C22×D7 | semidirect |
| 128. Dic5 | C22×D7 | semidirect |
| 139. C2×C7 | C2×C22⋊D5 | semidirect |
| 143. D7 | C2×C22⋊D5 | direct (N × H) |
| 147. C2×C5 | D4×D7 | semidirect |
| 159. C23 | D5×D7 | semidirect |
| 167. C7 | C22×C22⋊D5 | semidirect |
| 168. C5 | C2×D4×D7 | semidirect |
| 181. C22 | C2×D5×D7 | semidirect |
| 189. C2 | D7×C22⋊D5 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C5×D7 | 2 |
| C22×D5×D7 | 2 |
| C2×C22⋊D35 | 2 |
| C2×C7×C22⋊D5 | 2 |
| D7×C22⋊D5 | 2 |
| C2×#63306 | 2 |
| C2×#63305 | 2 |
| C2×D7×Dic5 | 2 |
| C2×D4×D7 | 5 |
| C22×C22⋊D5 | 7 |