Finite Group DatabasebetaFinite Group DatabasebetaOrder 1000.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic125 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C125 | C4 | 1 | 1 | yes | characteristic |
| 3 | #6016 | — | 5 | 1 | no | maximal |
| 4 | C125 | C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 5 | Dic25 | — | 5 | 1 | no | — |
| 6 | C2×C25 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 7 | #170 | — | 25 | 1 | no | — |
| 8 | C25 | #170 | 1 | 1 | yes | characteristic |
| 9 | Dic5 | — | 25 | 1 | no | — |
| 10 | C2×C5 | #1035 | 1 | 1 | yes | characteristic |
| 11 | C8 | — | 125 | 1 | no | Sylow |
| 12 | C5 | #6016 | 1 | 1 | yes | characteristic |
| 13 | C4 | — | 125 | 1 | no | — |
| 14 | C2 | #92517 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #92517 |
| commutator, hypocenter | C125 | C8 |
| Frattini | C2×C25 | AGL(1,𝔽5) |
| Fitting | C2×C125 | C4 |
| socle, prime radical | C2×C5 | #1035 |
| cosocle | Dic125 | C2 |
| LCS | G ▷ C125 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C125 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Dic125 | 2 |
| #6016 | 5 |