Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
260 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 56 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. C23×D35 | C2 | semidirect |
| 2. C2×C7×C22⋊D5 | C2 | semidirect |
| 3. C2×#63307 | C2 | semidirect |
| 4. C5×C22⋊Dic7 | C2 | semidirect |
| 5. #10587856 | C2 | semidirect |
| 10. C22×C7×D5 | C22 | semidirect |
| 12. C2×C7×Dic5 | C22 | semidirect |
| 13. C2×C5×Dic7 | C22 | semidirect |
| 18. C22×C5×C7 | D4 | semidirect |
| 23. C2×D35 | D4 | semidirect |
| 31. C22⋊Dic7 | D5 | semidirect |
| 34. C2×C22⋊D5 | D7 | semidirect |
| 51. C2×Dic7 | C2×D5 | semidirect |
| 56. C22×D5 | C2×D7 | semidirect |
| 59. C2×Dic5 | C2×D7 | semidirect |
| 61. C5×C7 | #120 | semidirect |
| 66. C22×C7 | D20 | semidirect |
| 75. C22×C5 | C22⋊D7 | semidirect |
| 109. C7 | #4028 | semidirect |
| 110. C5 | #6500 | semidirect |
| 118. C22 | #63307 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D35 | 2 |
| C2×C7×C22⋊D5 | 2 |
| C2×#63307 | 2 |
| C5×C22⋊Dic7 | 2 |
| #10587856 | 2 |
| #6500 | 5 |
| #4028 | 7 |