Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
512 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×C5⋊D5 |
| commutator, socle, cosocle | C2×C53 | C23 |
| Fitting | C22×C53 | C22 |
| hypercenter | C22 | D5×C5⋊D5 |
| hypocenter | C53 | #41 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊(C5⋊(C5⋊D5)) | C2 | semidirect |
| 2. C5×C22⋊(C5⋊D5) | C2 | semidirect |
| 3. C2×C52×Dic5 | C2 | semidirect |
| 4. #11758755 | C2 | semidirect |
| 5. #11758754 | C2 | semidirect |
| 6. #11758751 | C2 | semidirect |
| 7. Dic5×C5⋊D5 | C2 | semidirect |
| 25. C22⋊(C5⋊D5) | D5 | semidirect |
| 31. C2×C5×Dic5 | D5 | semidirect |
| 36. C53 | #41 | semidirect |
| 75. C2×Dic5 | C5⋊D5 | semidirect |
| 83. C52 | #709 | semidirect |
| 84. C52 | #711 | semidirect |
| 122. C5 | #87879 | semidirect |
| 123. C5 | #87900 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22⋊(C5⋊(C5⋊D5)) | 2 |
| C5×C22⋊(C5⋊D5) | 2 |
| C2×C52×Dic5 | 2 |
| #11758755 | 2 |
| #11758754 | 2 |
| #11758751 | 2 |
| Dic5×C5⋊D5 | 2 |
| #87900 | 5 |
| #87879 | 5 |