Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
904 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 904 conjugacy classes of subgroups — too many to draw. Only its 243 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×#125 | C2 |
| prime radical | C22×C5⋊D5 | C23 |
| hypercenter | #125 | C5⋊D5 |
| hypocenter | C52 | C2×#125 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×#125 | C2 | semidirect |
| 4. #11717210 | C2 | semidirect |
| 5. #11717209 | C2 | semidirect |
| 7. #11717190 | C2 | semidirect |
| 32. C5×#125 | D5 | semidirect |
| 64. C5⋊D5 | #125 | direct (N × H) |
| 78. #125 | C5⋊D5 | direct (N × H) |
| 82. C52 | C2×#125 | semidirect |
| 118. C5 | D5×#125 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×#125 | 2 |
| C4⋊C4×C5⋊D5 | 2 |
| #11717210 | 2 |
| #11717209 | 2 |
| C42×C5⋊D5 | 2 |
| #11717190 | 2 |
| D5×#125 | 5 |