Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
454 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 454 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. Q8×#1039 | C2 | semidirect |
| 3. C5×#4090 | C2 | semidirect |
| 4. #11716989 | C2 | semidirect |
| 5. #11716988 | C2 | semidirect |
| 10. #11716963 | C2 | semidirect |
| 11. #11716961 | C2 | semidirect |
| 59. #4090 | D5 | semidirect |
| 189. C52 | #541 | semidirect |
| 273. C5 | #65959 | semidirect |
| 274. C5 | #66065 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#1039 | 2 |
| C5×#4092 | 2 |
| C5×#4090 | 2 |
| #11716989 | 2 |
| #11716988 | 2 |
| #11716980 | 2 |
| #11716977 | 2 |
| #11716974 | 2 |
| #11716973 | 2 |
| #11716963 | 2 |
| #11716961 | 2 |
| #66065 | 5 |
| #65959 | 5 |