Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
650 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 120 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 | C24×C52 | C22 |
| hypercenter | C24 | D52 |
| hypocenter | C52 | H(3,𝔽4) |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717273 | C2 | semidirect |
| 2. C5×#4099 | C2 | semidirect |
| 3. #11717101 | C2 | semidirect |
| 4. #11717077 | C2 | semidirect |
| 5. #11717071 | C2 | semidirect |
| 6. C24×C52 | C22 | semidirect |
| 7. C22×D52 | C22 | semidirect |
| 10. C2×#6039 | C22 | semidirect |
| 11. C22⋊#1039 | C22 | semidirect |
| 12. C5×C22⋊Dic5 | C22 | semidirect |
| 13. #87779 | C22 | semidirect |
| 14. #87778 | C22 | semidirect |
| 15. Dic52 | C22 | semidirect |
| 30. #4099 | D5 | semidirect |
| 45. C24×C5 | C2×D5 | semidirect |
| 56. C22⋊Dic5 | C2×D5 | semidirect |
| 88. C52 | H(3,𝔽4) | semidirect |
| 108. C24 | D52 | semidirect |
| 136. C5 | #65928 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717273 | 2 |
| C5×#4099 | 2 |
| #11717101 | 2 |
| #11717077 | 2 |
| #11717071 | 2 |
| #65928 | 5 |