Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
912 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | #6038 |
| commutator, cosocle | C4×C52 | C24 |
| Frattini | C4 | C22×D52 |
| Fitting | C22×C4×C52 | C22 |
| socle | C23×C52 | D4 |
| hypercenter | C22×C4 | D52 |
| hypocenter | C52 | C22×D8 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4⋊(C5⋊D5) | C2 | semidirect |
| 2. C22×C5×D20 | C2 | semidirect |
| 3. C22×C5×#168 | C2 | semidirect |
| 4. C2×#87771 | C2 | direct (N × H) |
| 9. C2×C4⋊(C5⋊D5) | C22 | semidirect |
| 11. C2×C5×D20 | C22 | semidirect |
| 12. C2×C5×#168 | C22 | semidirect |
| 13. #87771 | C22 | direct (N × H) |
| 24. C4⋊(C5⋊D5) | C23 | semidirect |
| 26. C5×D20 | C23 | semidirect |
| 27. C5×#168 | C23 | semidirect |
| 32. C22×D20 | D5 | semidirect |
| 35. C22×#168 | D5 | semidirect |
| 37. C22×C52 | D8 | semidirect |
| 62. C2×D20 | C2×D5 | semidirect |
| 65. C2×#168 | C2×D5 | semidirect |
| 68. C2×C52 | C2×D8 | semidirect |
| 100. D20 | C22×D5 | semidirect |
| 102. #168 | C22×D5 | semidirect |
| 107. C52 | C22×D8 | semidirect |
| 112. C22×C5 | D40 | semidirect |
| 113. C22×C5 | #686 | semidirect |
| 142. C2×C5 | C2×D40 | semidirect |
| 143. C2×C5 | C2×#686 | semidirect |
| 163. C5 | C22×D40 | semidirect |
| 164. C5 | C22×#686 | semidirect |
| 169. C22 | #87771 | direct (N × H) |
| 175. C2 | C2×#87771 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊(C5⋊D5) | 2 |
| C22×C5×D20 | 2 |
| C22×C5×#168 | 2 |
| C2×#87771 | 2 |
| C22×#686 | 5 |
| C22×D40 | 5 |