Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
156 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 52 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 |
| prime radical | C2×C22⋊D35 | C2 |
| hypercenter | C23 | D5×D7 |
| hypocenter | C5×C7 | #123 |
| 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 |
|---|---|---|
| 2. C22×C5×Dic7 | C2 | semidirect |
| 3. C7×C22⋊Dic5 | C2 | semidirect |
| 4. #10587859 | C2 | semidirect |
| 5. #10587856 | C2 | semidirect |
| 24. C22×Dic7 | D5 | semidirect |
| 30. C22⋊Dic5 | D7 | semidirect |
| 51. C5×C7 | #123 | semidirect |
| 89. C7 | #4075 | semidirect |
| 90. C5 | #6449 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊D35 | 2 |
| C22×C5×Dic7 | 2 |
| C7×C22⋊Dic5 | 2 |
| #10587859 | 2 |
| #10587856 | 2 |
| #6449 | 5 |
| #4075 | 7 |