Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
448 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 120 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 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C42×C52 | C22 |
| prime radical | C2×C5×D20 | C22 |
| hypercenter | C42 | D52 |
| hypocenter | C52 | #563 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717190 | C2 | semidirect |
| 3. C5×#4016 | C2 | semidirect |
| 4. #11717014 | C2 | semidirect |
| 5. #11717007 | C2 | semidirect |
| 7. #11716977 | C2 | semidirect |
| 8. #11716976 | C2 | semidirect |
| 9. #11716957 | C2 | semidirect |
| 64. #4016 | D5 | semidirect |
| 169. C52 | #563 | semidirect |
| 246. C5 | #66006 | semidirect |
| 247. C5 | #66033 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717190 | 2 |
| C5×#4021 | 2 |
| C5×#4016 | 2 |
| #11717014 | 2 |
| #11717007 | 2 |
| #11717002 | 2 |
| #11716977 | 2 |
| #11716976 | 2 |
| #11716957 | 2 |
| #66033 | 5 |
| #66006 | 5 |