Finite Group DatabasebetaFinite Group DatabasebetaOrder 2240.
1324 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5×D7 |
| commutator, socle, cosocle | C22×C5×C7 | C24 |
| Fitting | C24×C5×C7 | C22 |
| hypercenter | C24 | D5×D7 |
| hypocenter | C5×C7 | C2×#120 |
| LCS | G ▷ C22×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C5×D7 | C2 | semidirect |
| 2. C2×D7×C22⋊D5 | C2 | semidirect |
| 3. #11765283 | C2 | semidirect |
| 4. C7×#4099 | C2 | semidirect |
| 5. #11765008 | C2 | semidirect |
| 6. #11765006 | C2 | semidirect |
| 7. D7×C22⋊Dic5 | C2 | semidirect |
| 8. C24×C5×C7 | C22 | semidirect |
| 11. C23×C5×D7 | C22 | semidirect |
| 12. C22×D5×D7 | C22 | semidirect |
| 13. C2×C22⋊D35 | C22 | semidirect |
| 14. C2×C7×C22⋊D5 | C22 | semidirect |
| 16. C2×#63306 | C22 | semidirect |
| 17. C2×#63305 | C22 | semidirect |
| 18. C2×D7×Dic5 | C22 | semidirect |
| 19. C22⋊Dic35 | C22 | semidirect |
| 20. C7×C22⋊Dic5 | C22 | semidirect |
| 21. (C2×D7)⋊Dic5 | C22 | semidirect |
| 26. C22×D35 | C23 | semidirect |
| 27. C22×C7×D5 | C23 | semidirect |
| 32. C22×C5×D7 | D4 | semidirect |
| 37. C2×Dic35 | C23 | semidirect |
| 39. C2×C7×Dic5 | C23 | semidirect |
| 43. C24×D7 | D5 | semidirect |
| 55. #4099 | D7 | direct (N × H) |
| 59. C22×C5×C7 | C2×D4 | semidirect |
| 67. C2×C5×D7 | C2×D4 | semidirect |
| 71. C24×C7 | C2×D5 | semidirect |
| 75. C23×D7 | C2×D5 | semidirect |
| 88. C24×C5 | C2×D7 | semidirect |
| 94. C2×C22⋊D5 | C2×D7 | semidirect |
| 98. C22⋊Dic5 | C2×D7 | semidirect |
| 103. C5×D7 | #120 | semidirect |
| 115. C22×D7 | C22⋊D5 | semidirect |
| 136. C22×D5 | C22×D7 | semidirect |
| 143. C2×Dic5 | C22×D7 | semidirect |
| 144. C5×C7 | C2×#120 | semidirect |
| 155. C22×C7 | C2×C22⋊D5 | semidirect |
| 163. C2×D7 | C2×C22⋊D5 | semidirect |
| 173. C22×C5 | D4×D7 | semidirect |
| 184. C24 | D5×D7 | semidirect |
| 199. D7 | #4099 | direct (N × H) |
| 225. C7 | C2×#4099 | semidirect |
| 226. C5 | D7×#120 | semidirect |
| 237. C22 | D7×C22⋊D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C5×D7 | 2 |
| C2×D7×C22⋊D5 | 2 |
| #11765283 | 2 |
| C7×#4099 | 2 |
| #11765008 | 2 |
| #11765006 | 2 |
| D7×C22⋊Dic5 | 2 |
| D7×#120 | 5 |
| C2×#4099 | 7 |