Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
328 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 116 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 | C2×C4×C5×C7 | C22 |
| socle | C22×C5×C7 | C23 |
| hypercenter | C2×C4 | D5×D7 |
| hypocenter | C5×C7 | C2×#41 |
| LCS | G ▷ C2×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. C2×Dic70 | C2 | semidirect |
| 2. C2×C7×D20 | C2 | semidirect |
| 3. C2×C4×C5×D7 | C2 | semidirect |
| 4. C2×#63305 | C2 | semidirect |
| 5. C2×D5×Dic7 | C2 | semidirect |
| 6. #10587945 | C2 | direct (N × H) |
| 11. Dic70 | C22 | semidirect |
| 12. C7×D20 | C22 | semidirect |
| 14. C4×C5×D7 | C22 | semidirect |
| 15. #63305 | C22 | semidirect |
| 16. D5×Dic7 | C22 | semidirect |
| 31. C2×C4×D7 | D5 | semidirect |
| 36. C2×D20 | D7 | semidirect |
| 39. C2×C5×C7 | #41 | semidirect |
| 51. C4×D7 | C2×D5 | semidirect |
| 60. D20 | C2×D7 | semidirect |
| 63. C5×C7 | C2×#41 | semidirect |
| 88. C2×C7 | #709 | semidirect |
| 92. C2×C5 | #1159 | semidirect |
| 105. C7 | C2×#709 | semidirect |
| 106. C5 | C2×#1159 | semidirect |
| 117. C2 | #10587945 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Dic70 | 2 |
| C2×C7×D20 | 2 |
| C2×C4×C5×D7 | 2 |
| C2×#63305 | 2 |
| C2×D5×Dic7 | 2 |
| #10587945 | 2 |
| C2×#1159 | 5 |
| C2×#709 | 7 |