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 | C2×C52×Q8 | C22 |
| hypercenter | C2×Q8 | D52 |
| hypocenter | C52 | H(3,𝔽4) |
| 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. #11717267 | C2 | semidirect |
| 2. C5×#4093 | C2 | semidirect |
| 3. #11717041 | C2 | semidirect |
| 4. #11717040 | C2 | semidirect |
| 5. #11717033 | C2 | semidirect |
| 6. #11716970 | C2 | semidirect |
| 7. C22×D52 | C22 | semidirect |
| 8. C2×C52×Q8 | C22 | semidirect |
| 11. C2×#6039 | C22 | semidirect |
| 14. #87808 | C22 | semidirect |
| 15. C4×#1039 | C22 | semidirect |
| 16. C5×#685 | C22 | semidirect |
| 17. C5×#685 | C22 | semidirect |
| 18. C4×C5×Dic5 | C22 | semidirect |
| 19. #87779 | C22 | semidirect |
| 20. #87778 | C22 | semidirect |
| 39. #4093 | D5 | semidirect |
| 59. C2×C5×Q8 | C2×D5 | semidirect |
| 77. #685 | C2×D5 | semidirect |
| 78. #685 | C2×D5 | semidirect |
| 81. C4×Dic5 | C2×D5 | semidirect |
| 119. C52 | H(3,𝔽4) | semidirect |
| 145. C2×Q8 | D52 | semidirect |
| 180. C5 | #66017 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717267 | 2 |
| C5×#4093 | 2 |
| #11717041 | 2 |
| #11717040 | 2 |
| #11717033 | 2 |
| #11716970 | 2 |
| #66017 | 5 |