Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
542 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 130 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 | #11717032 | C2 |
| hypercenter | C42 | D52 |
| hypocenter | C52 | #559 |
| 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. C4×#6045 | C2 | semidirect |
| 2. C5×#4021 | C2 | semidirect |
| 3. C4×C5×D20 | C2 | semidirect |
| 5. #11717031 | C2 | semidirect |
| 6. #11717023 | C2 | semidirect |
| 7. #11717017 | C2 | semidirect |
| 8. #11717003 | C2 | semidirect |
| 9. #11716997 | C2 | semidirect |
| 10. #11716981 | C2 | semidirect |
| 11. #11716961 | C2 | semidirect |
| 74. #4021 | D5 | semidirect |
| 75. C4×D20 | D5 | semidirect |
| 211. C52 | #559 | semidirect |
| 308. C5 | #65893 | semidirect |
| 309. C5 | #66015 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#6045 | 2 |
| C5×#4021 | 2 |
| C4×C5×D20 | 2 |
| #11717032 | 2 |
| #11717031 | 2 |
| #11717023 | 2 |
| #11717017 | 2 |
| #11717003 | 2 |
| #11716997 | 2 |
| #11716981 | 2 |
| #11716961 | 2 |
| #66015 | 5 |
| #65893 | 5 |