Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
452 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C2×C52×Q8 | C22 |
| prime radical | C22×C5×D5 | C23 |
| hypercenter | C2×Q8 | D52 |
| hypocenter | C52 | #541 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717264 | C2 | semidirect |
| 3. C5×Q8×Dic5 | C2 | semidirect |
| 4. #11716991 | C2 | semidirect |
| 5. #11716986 | C2 | semidirect |
| 10. #11716964 | C2 | semidirect |
| 11. #11716962 | C2 | semidirect |
| 54. Q8×Dic5 | D5 | semidirect |
| 190. C52 | #541 | semidirect |
| 274. C5 | #65917 | semidirect |
| 275. C5 | #65959 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717264 | 2 |
| C5×#4092 | 2 |
| C5×Q8×Dic5 | 2 |
| #11716991 | 2 |
| #11716986 | 2 |
| #11716978 | 2 |
| #11716976 | 2 |
| #11716975 | 2 |
| #11716973 | 2 |
| #11716964 | 2 |
| #11716962 | 2 |
| #65959 | 5 |
| #65917 | 5 |