Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
728 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #532 |
| 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. C2×#87880 | C2 | semidirect |
| 2. C2×C4×#1039 | C2 | semidirect |
| 3. C5×#4076 | C2 | semidirect |
| 4. #11717098 | C2 | semidirect |
| 5. #11717037 | C2 | semidirect |
| 6. #11717033 | C2 | semidirect |
| 7. #11717032 | C2 | semidirect |
| 8. #11717020 | C2 | semidirect |
| 9. #11717017 | C2 | semidirect |
| 35. C2×#1039 | D4 | semidirect |
| 52. #4076 | D5 | semidirect |
| 67. C4×C52 | #41 | semidirect |
| 71. #1039 | #41 | semidirect |
| 152. C52 | #532 | semidirect |
| 171. C4×C5 | #711 | semidirect |
| 228. C5 | #65930 | semidirect |
| 242. C4 | #87880 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87880 | 2 |
| C2×C4×#1039 | 2 |
| C5×#4076 | 2 |
| #11717098 | 2 |
| #11717037 | 2 |
| #11717033 | 2 |
| #11717032 | 2 |
| #11717020 | 2 |
| #11717017 | 2 |
| #65930 | 5 |