Finite Group DatabasebetaFinite Group DatabasebetaOrder 1120.
188 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 188 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 | C2×C4×C5×C7 | C22 |
| hypercenter | C2×C4 | D5×D7 |
| hypocenter | C5×C7 | #121 |
| LCS | G ▷ C22×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×C4×C5×D7 | C2 | semidirect |
| 2. C2×#63307 | C2 | semidirect |
| 3. C2×#63306 | C2 | semidirect |
| 4. C2×#63305 | C2 | semidirect |
| 5. #10587902 | C2 | semidirect |
| 6. C7×#685 | C2 | semidirect |
| 7. Dic7⋊Dic5 | C2 | semidirect |
| 30. C2×C5×D7 | D4 | semidirect |
| 33. C5×Dic7 | D4 | semidirect |
| 38. C2×C4×D7 | D5 | semidirect |
| 47. #685 | D7 | semidirect |
| 77. C5×C7 | #121 | semidirect |
| 88. C2×D7 | D20 | semidirect |
| 92. Dic7 | C22⋊D5 | semidirect |
| 137. C7 | #4076 | semidirect |
| 138. C5 | #6447 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C5×D7 | 2 |
| C2×#63307 | 2 |
| C2×#63306 | 2 |
| C2×#63305 | 2 |
| #10587902 | 2 |
| C7×#685 | 2 |
| Dic7⋊Dic5 | 2 |
| #6447 | 5 |
| #4076 | 7 |