Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
548 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 548 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 |
| hypercenter | C2×Q8 | D52 |
| hypocenter | C52 | #555 |
| 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. #11717267 | C2 | semidirect |
| 2. C5×#4093 | C2 | semidirect |
| 3. C5×#4090 | C2 | semidirect |
| 4. #11717028 | C2 | semidirect |
| 5. #11717025 | C2 | semidirect |
| 6. #11717005 | C2 | semidirect |
| 7. #11716998 | C2 | semidirect |
| 8. #11716996 | C2 | semidirect |
| 9. #11716954 | C2 | semidirect |
| 56. #4093 | D5 | semidirect |
| 57. #4090 | D5 | semidirect |
| 185. C52 | #555 | semidirect |
| 272. C5 | #66004 | semidirect |
| 273. C5 | #66068 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717267 | 2 |
| C5×#4093 | 2 |
| C5×#4090 | 2 |
| #11717028 | 2 |
| #11717025 | 2 |
| #11717005 | 2 |
| #11716998 | 2 |
| #11716996 | 2 |
| #11716954 | 2 |
| #66068 | 5 |
| #66004 | 5 |