Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1704 conjugacy classes of subgroups; only the 363 normal subgroups are shown.
This group has 1704 conjugacy classes of subgroups — too many to draw. Only its 363 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×C5⋊D5 |
| commutator, cosocle | C22×C52 | C24 |
| Frattini | C22 | C23×C5⋊D5 |
| Fitting | C22×C52×D4 | C2 |
| socle | C23×C52 | C23 |
| hypercenter | C22×D4 | C5⋊D5 |
| hypocenter | C52 | C2×#121 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C52×D4 | C2 | semidirect |
| 2. C22×C22⋊(C5⋊D5) | C2 | semidirect |
| 3. C22×C22⋊(C5⋊D5) | C2 | semidirect |
| 4. C23×#1039 | C2 | semidirect |
| 5. C2×C22⋊#1039 | C2 | semidirect |
| 6. #11717259 | C2 | direct (N × H) |
| 7. C2×#87808 | C2 | semidirect |
| 8. C2×#87806 | C2 | semidirect |
| 13. C2×C52×D4 | C22 | semidirect |
| 17. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 18. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 23. C22×#1039 | C22 | semidirect |
| 24. C22⋊#1039 | C22 | semidirect |
| 25. #87808 | C22 | semidirect |
| 26. #87806 | C22 | semidirect |
| 33. C23×C52 | D4 | semidirect |
| 49. C2×#1039 | D4 | semidirect |
| 53. C22×C5×D4 | D5 | semidirect |
| 70. C22×C52 | C2×D4 | semidirect |
| 80. #1039 | C2×D4 | semidirect |
| 85. C2×C5×D4 | C2×D5 | semidirect |
| 102. C2×C52 | #121 | semidirect |
| 112. C23×C5 | C22⋊D5 | semidirect |
| 134. C22×D4 | C5⋊D5 | semidirect |
| 140. C52 | C2×#121 | semidirect |
| 151. C22×C5 | C2×C22⋊D5 | semidirect |
| 169. C2×D4 | C2×C5⋊D5 | semidirect |
| 181. C2×C5 | #4085 | semidirect |
| 195. C23 | C22⋊(C5⋊D5) | semidirect |
| 211. C5 | C2×#4085 | semidirect |
| 226. C22 | C2×C22⋊(C5⋊D5) | semidirect |
| 235. C2 | #11717259 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C52×D4 | 2 |
| C22×C22⋊(C5⋊D5) | 2 |
| C23×#1039 | 2 |
| C2×C22⋊#1039 | 2 |
| #11717259 | 2 |
| C2×#87808 | 2 |
| C2×#87806 | 2 |
| C2×#4085 | 5 |