Finite Group DatabasebetaFinite Group DatabasebetaOrder 8125.
260 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C53×C13 | C5 | 1 | 6 | yes | maximal, complemented |
| 2 | C13×H(3,𝔽5) | C5 | 1 | 25 | yes | maximal, direct product factor |
| 3 | C5×H(3,𝔽5) | C13 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 4 | C52×C13 | — | 5 | 30 | no | — |
| 5 | C52×C13 | C52 | 1 | 1 | yes | characteristic |
| 6 | C52×C13 | C52 | 1 | 30 | yes | complemented |
| 7 | C53 | C5×C13 | 1 | 6 | yes | complemented |
| 8 | H(3,𝔽5) | C5×C13 | 1 | 25 | yes | direct product factor |
| 9 | C5×C13 | — | 5 | 30 | no | — |
| 10 | C5×C13 | H(3,𝔽5) | 1 | 5 | yes | direct product factor |
| 11 | C5×C13 | C53 | 1 | 1 | yes | characteristic |
| 12 | C52 | — | 5 | 30 | no | — |
| 13 | C52 | C52×C13 | 1 | 1 | yes | characteristic |
| 14 | C52 | C52×C13 | 1 | 30 | yes | complemented |
| 15 | C13 | C5×H(3,𝔽5) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 16 | C5 | — | 5 | 30 | no | — |
| 17 | C5 | C13×H(3,𝔽5) | 1 | 5 | yes | direct product factor |
| 18 | C5 | C53×C13 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C13 | C52 |
| commutator, Frattini | C5 | C53×C13 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C13 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×C13 | C5 | semidirect |
| 2. C13×H(3,𝔽5) | C5 | direct (N × H) |
| 3. C5×H(3,𝔽5) | C13 | direct (N × H) |
| 6. C52×C13 | C52 | semidirect |
| 7. C53 | C5×C13 | semidirect |
| 8. H(3,𝔽5) | C5×C13 | direct (N × H) |
| 10. C5×C13 | H(3,𝔽5) | direct (N × H) |
| 14. C52 | C52×C13 | semidirect |
| 15. C13 | C5×H(3,𝔽5) | direct (N × H) |
| 17. C5 | C13×H(3,𝔽5) | direct (N × H) |