Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
556 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 556 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 | C22×C4×C52 | C22 |
| hypercenter | C22×C4 | D52 |
| hypocenter | C52 | #540 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717246 | C2 | semidirect |
| 2. C5×#4076 | C2 | semidirect |
| 3. #11717088 | C2 | semidirect |
| 4. #11717032 | C2 | semidirect |
| 5. #11717017 | C2 | semidirect |
| 6. #11717003 | C2 | semidirect |
| 7. #11716991 | C2 | semidirect |
| 39. #4076 | D5 | semidirect |
| 123. C52 | #540 | semidirect |
| 184. C5 | #65953 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717246 | 2 |
| C5×#4076 | 2 |
| #11717088 | 2 |
| #11717032 | 2 |
| #11717017 | 2 |
| #11717003 | 2 |
| #11716991 | 2 |
| #65953 | 5 |