Finite Group DatabasebetaFinite Group DatabasebetaOrder 12500.
126528 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×C54 | C5 | 1 | 781 | yes | maximal, direct product factor |
| 2 | C2×C55 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C55 | C4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 4 | C2×C54 | C2×C5 | 1 | 781 | yes | — |
| 5 | C54 | C4×C5 | 1 | 781 | yes | direct product factor |
| 6 | C4×C53 | C52 | 1 | 20306 | yes | direct product factor |
| 7 | C2×C53 | C2×C52 | 1 | 20306 | yes | — |
| 8 | C53 | C4×C52 | 1 | 20306 | yes | direct product factor |
| 9 | C4×C52 | C53 | 1 | 20306 | yes | direct product factor |
| 10 | C2×C52 | C2×C53 | 1 | 20306 | yes | — |
| 11 | C52 | C4×C53 | 1 | 20306 | yes | direct product factor |
| 12 | C4×C5 | C54 | 1 | 781 | yes | direct product factor |
| 13 | C2×C5 | C2×C54 | 1 | 781 | yes | — |
| 14 | C5 | C4×C54 | 1 | 781 | yes | direct product factor |
| 15 | C4 | C55 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 16 | C2 | C2×C55 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C2×C55 |
| socle, prime radical | C2×C55 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×C54 | C5 | direct (N × H) |
| 3. C55 | C4 | direct (N × H) |
| 5. C54 | C4×C5 | direct (N × H) |
| 6. C4×C53 | C52 | direct (N × H) |
| 8. C53 | C4×C52 | direct (N × H) |
| 9. C4×C52 | C53 | direct (N × H) |
| 11. C52 | C4×C53 | direct (N × H) |
| 12. C4×C5 | C54 | direct (N × H) |
| 14. C5 | C4×C54 | direct (N × H) |
| 15. C4 | C55 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C54 | 5 |
| C2×C55 | 2 |