Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
596 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 596 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 | C42×C52 | C22 |
| hypercenter | C42 | D52 |
| hypocenter | C52 | H(3,𝔽4) |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717195 | C2 | semidirect |
| 2. C5×#4021 | C2 | semidirect |
| 3. #11717041 | C2 | semidirect |
| 4. #11717040 | C2 | semidirect |
| 5. #11717031 | C2 | semidirect |
| 6. #11717025 | C2 | semidirect |
| 7. C22×D52 | C22 | semidirect |
| 9. C2×#6045 | C22 | semidirect |
| 11. C2×C5×Dic10 | C22 | semidirect |
| 14. C42×C52 | C22 | semidirect |
| 15. #87808 | C22 | semidirect |
| 16. C5×#685 | C22 | semidirect |
| 17. C5×#685 | C22 | semidirect |
| 18. #87779 | C22 | semidirect |
| 19. #87778 | C22 | semidirect |
| 20. Dic52 | C22 | semidirect |
| 45. #4021 | D5 | semidirect |
| 68. C2×Dic10 | C2×D5 | semidirect |
| 72. C42×C5 | C2×D5 | semidirect |
| 79. #685 | C2×D5 | semidirect |
| 80. #685 | C2×D5 | semidirect |
| 120. C52 | H(3,𝔽4) | semidirect |
| 157. C42 | D52 | semidirect |
| 182. C5 | #66017 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717195 | 2 |
| C5×#4021 | 2 |
| #11717041 | 2 |
| #11717040 | 2 |
| #11717031 | 2 |
| #11717025 | 2 |
| #66017 | 5 |