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 | C23×C5×C7 | C22 |
| hypercenter | C23 | D5×D7 |
| hypocenter | C5×C7 | #120 |
| LCS | G ▷ C22×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D5×D7 | C2 | semidirect |
| 2. C2×C22⋊D35 | C2 | semidirect |
| 3. C2×C7×C22⋊D5 | C2 | semidirect |
| 4. C2×C5×C22⋊D7 | C2 | semidirect |
| 5. #10587856 | C2 | semidirect |
| 6. #10587855 | C2 | semidirect |
| 7. (C2×D7)⋊Dic5 | C2 | semidirect |
| 8. C23×C5×C7 | C22 | semidirect |
| 16. C2×Dic35 | C22 | semidirect |
| 18. C2×C7×Dic5 | C22 | semidirect |
| 20. C2×C5×Dic7 | C22 | semidirect |
| 29. C2×D35 | D4 | semidirect |
| 32. C2×C7×D5 | D4 | semidirect |
| 35. C2×C5×D7 | D4 | semidirect |
| 43. C2×C22⋊D7 | D5 | semidirect |
| 50. C2×C22⋊D5 | D7 | semidirect |
| 62. C23×C7 | C2×D5 | semidirect |
| 73. C2×Dic7 | C2×D5 | semidirect |
| 75. C23×C5 | C2×D7 | semidirect |
| 86. C2×Dic5 | C2×D7 | semidirect |
| 87. C5×C7 | #120 | semidirect |
| 102. C2×D7 | C22⋊D5 | semidirect |
| 119. C2×D5 | C22⋊D7 | semidirect |
| 150. C23 | D5×D7 | semidirect |
| 157. C7 | #4083 | semidirect |
| 158. C5 | #6500 | 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×C22⋊D35 | 2 |
| C2×C7×C22⋊D5 | 2 |
| C2×C5×C22⋊D7 | 2 |
| #10587856 | 2 |
| #10587855 | 2 |
| (C2×D7)⋊Dic5 | 2 |
| #6500 | 5 |
| #4083 | 7 |