Finite Group DatabasebetaFinite Group DatabasebetaOrder 21875.
2866 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C7×H(3,𝔽5) | C5 | 1 | 156 | yes | maximal, complemented |
| 2 | H(5,𝔽5) | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 3 | C53×C7 | C52 | 1 | 156 | yes | complemented |
| 4 | C7×H(3,𝔽5) | C52 | 1 | 650 | yes | complemented |
| 5 | C5×H(3,𝔽5) | C5×C7 | 1 | 156 | yes | complemented |
| 6 | C52×C7 | — | 25 | 156 | no | — |
| 7 | C52×C7 | C53 | 1 | 156 | yes | — |
| 8 | C53 | C52×C7 | 1 | 156 | yes | complemented |
| 9 | H(3,𝔽5) | C52×C7 | 1 | 650 | yes | complemented |
| 10 | C5×C7 | — | 5 | 156 | no | — |
| 11 | C5×C7 | C54 | 1 | 1 | yes | characteristic |
| 12 | C52 | — | 25 | 156 | no | — |
| 13 | C52 | C53×C7 | 1 | 156 | yes | — |
| 14 | C7 | H(5,𝔽5) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 15 | C5 | — | 5 | 156 | no | — |
| 16 | C5 | C54×C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C54 |
| commutator, Frattini, cosocle | C5 | C54×C7 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C7×H(3,𝔽5) | C5 | semidirect |
| 2. H(5,𝔽5) | C7 | direct (N × H) |
| 3. C53×C7 | C52 | semidirect |
| 4. C7×H(3,𝔽5) | C52 | semidirect |
| 5. C5×H(3,𝔽5) | C5×C7 | semidirect |
| 8. C53 | C52×C7 | semidirect |
| 9. H(3,𝔽5) | C52×C7 | semidirect |
| 14. C7 | H(5,𝔽5) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C7×H(3,𝔽5) | 5 |
| H(5,𝔽5) | 7 |