Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
992 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 215 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 | C23×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C52×#123 | C2 |
| prime radical | #11717257 | C2 |
| hypercenter | #123 | C5⋊D5 |
| hypocenter | C52 | #551 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×#123 | C2 | semidirect |
| 3. #11717256 | C2 | semidirect |
| 4. C4×C22⋊(C5⋊D5) | C2 | semidirect |
| 5. #11717216 | C2 | semidirect |
| 6. #11717212 | C2 | semidirect |
| 7. #11717203 | C2 | semidirect |
| 8. C22⋊C4×C5⋊D5 | C2 | semidirect |
| 9. #11717199 | C2 | semidirect |
| 12. C22×C4×C52 | C22 | semidirect |
| 16. C2×#6045 | C22 | semidirect |
| 17. C52×C4⋊C4 | C22 | semidirect |
| 19. C52×C22⋊C4 | C22 | semidirect |
| 22. C22⋊#1039 | C22 | semidirect |
| 26. C4⋊#1039 | C22 | semidirect |
| 27. #87806 | C22 | semidirect |
| 28. #87806 | C22 | semidirect |
| 29. C4×#1039 | C22 | semidirect |
| 30. C4×#1039 | C22 | semidirect |
| 49. C5×#123 | D5 | semidirect |
| 75. C22×C4×C5 | C2×D5 | semidirect |
| 80. C5×C4⋊C4 | C2×D5 | semidirect |
| 82. C5×C22⋊C4 | C2×D5 | semidirect |
| 120. #123 | C5⋊D5 | semidirect |
| 126. C52 | #551 | semidirect |
| 148. C22×C4 | C2×C5⋊D5 | semidirect |
| 152. C4⋊C4 | C2×C5⋊D5 | semidirect |
| 160. C22⋊C4 | C2×C5⋊D5 | semidirect |
| 185. C5 | #65993 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×#123 | 2 |
| #11717257 | 2 |
| #11717256 | 2 |
| C4×C22⋊(C5⋊D5) | 2 |
| #11717216 | 2 |
| #11717212 | 2 |
| #11717203 | 2 |
| C22⋊C4×C5⋊D5 | 2 |
| #11717199 | 2 |
| #65993 | 5 |