Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
260 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 54 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 | C2×D5×D7 |
| commutator, socle, cosocle | C22×C5×C7 | C23 |
| Fitting | C2×C4×C5×C7 | C22 |
| hypercenter | C2×C4 | D5×D7 |
| hypocenter | C5×C7 | #120 |
| LCS | G ▷ C22×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D5×D7 | C2 | semidirect |
| 2. C2×D140 | C2 | semidirect |
| 3. C2×#63307 | C2 | semidirect |
| 4. C2×#63306 | C2 | semidirect |
| 5. C7×#685 | C2 | semidirect |
| 6. C5×(C2×D7)⋊C4 | C2 | semidirect |
| 7. #10587856 | C2 | semidirect |
| 9. C22×D35 | C22 | semidirect |
| 15. C2×C4×C5×C7 | C22 | semidirect |
| 17. C2×C7×Dic5 | C22 | semidirect |
| 18. C2×C5×Dic7 | C22 | semidirect |
| 29. C2×D35 | D4 | semidirect |
| 32. C2×C7×D5 | D4 | semidirect |
| 35. C2×C5×D7 | D4 | semidirect |
| 46. (C2×D7)⋊C4 | D5 | semidirect |
| 53. #685 | D7 | semidirect |
| 69. C2×C4×C7 | C2×D5 | semidirect |
| 71. C2×Dic7 | C2×D5 | semidirect |
| 82. C2×C4×C5 | C2×D7 | semidirect |
| 84. C2×Dic5 | C2×D7 | semidirect |
| 87. C5×C7 | #120 | semidirect |
| 102. C2×D7 | D20 | semidirect |
| 119. C2×D5 | D28 | semidirect |
| 156. C2×C4 | D5×D7 | semidirect |
| 157. C7 | #4028 | semidirect |
| 158. C5 | #6445 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D5×D7 | 2 |
| C2×D140 | 2 |
| C2×#63307 | 2 |
| C2×#63306 | 2 |
| C7×#685 | 2 |
| C5×(C2×D7)⋊C4 | 2 |
| #10587856 | 2 |
| #6445 | 5 |
| #4028 | 7 |