Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1000 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 227 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×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | C5⋊D5 |
| hypocenter | C52 | #538 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×C22⋊Q8 | C2 | semidirect |
| 2. #11717267 | C2 | semidirect |
| 3. #11717250 | C2 | semidirect |
| 4. C2×#87808 | C2 | semidirect |
| 5. #11717218 | C2 | semidirect |
| 6. #11717215 | C2 | semidirect |
| 7. #11717214 | C2 | semidirect |
| 8. #11717202 | C2 | semidirect |
| 9. #11717201 | C2 | semidirect |
| 11. C2×C52×Q8 | C22 | semidirect |
| 17. C2×C4⋊(C5⋊D5) | C22 | semidirect |
| 18. C2×C4×C5⋊D5 | C22 | semidirect |
| 19. C52×C4⋊C4 | C22 | semidirect |
| 20. C52×C4⋊C4 | C22 | semidirect |
| 26. #87808 | C22 | semidirect |
| 27. #87808 | C22 | semidirect |
| 28. C4⋊#1039 | C22 | semidirect |
| 29. #87806 | C22 | semidirect |
| 30. C4×#1039 | C22 | semidirect |
| 52. C5×C22⋊Q8 | D5 | semidirect |
| 79. C2×C5×Q8 | C2×D5 | semidirect |
| 87. C5×C4⋊C4 | C2×D5 | semidirect |
| 88. C5×C4⋊C4 | C2×D5 | semidirect |
| 129. C22⋊Q8 | C5⋊D5 | semidirect |
| 135. C52 | #538 | semidirect |
| 154. C2×Q8 | C2×C5⋊D5 | semidirect |
| 164. C4⋊C4 | C2×C5⋊D5 | semidirect |
| 165. C4⋊C4 | C2×C5⋊D5 | semidirect |
| 200. C5 | #65978 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×C22⋊Q8 | 2 |
| #11717267 | 2 |
| #11717250 | 2 |
| C2×#87808 | 2 |
| #11717218 | 2 |
| #11717215 | 2 |
| #11717214 | 2 |
| #11717202 | 2 |
| #11717201 | 2 |
| #65978 | 5 |