Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 148 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, prime radical | C23×C5×C7 | C22 |
| hypercenter | C23 | D5×D7 |
| hypocenter | C5×C7 | C22⋊Q8 |
| 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. C2×#63308 | C2 | semidirect |
| 5. #10587859 | C2 | semidirect |
| 6. #10587858 | C2 | semidirect |
| 7. Dic7⋊Dic5 | C2 | semidirect |
| 21. C22×C5×C7 | Q8 | semidirect |
| 65. C5×C7 | C22⋊Q8 | semidirect |
| 69. C22×C7 | Dic10 | semidirect |
| 79. C22×C5 | Dic14 | semidirect |
| 111. C7 | #4070 | semidirect |
| 112. C5 | #6441 | semidirect |
| 115. C22 | #63308 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C5×Dic7 | 2 |
| C2×#63308 | 2 |
| C22⋊Dic35 | 2 |
| C7×C22⋊Dic5 | 2 |
| #10587859 | 2 |
| #10587858 | 2 |
| Dic7⋊Dic5 | 2 |
| #6441 | 5 |
| #4070 | 7 |