Finite Group DatabasebetaFinite Group DatabasebetaOrder 21875.
2316 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#10597837 | C5 | 1 | 125 | yes | maximal, complemented |
| 2 | C5×C7×5−1+2 | C5 | 1 | 30 | yes | maximal, complemented |
| 3 | C5×C7×H(3,𝔽5) | C5 | 1 | 1 | yes | maximal, characteristic |
| 4 | 5−1+4 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 5 | C53×C7 | C52 | 1 | 6 | yes | — |
| 6 | C7×5−1+2 | C52 | 1 | 25 | yes | — |
| 7 | C7×5−1+2 | C52 | 1 | 600 | yes | complemented |
| 8 | C7×H(3,𝔽5) | C52 | 1 | 25 | yes | — |
| 9 | C5×C7×C25 | C52 | 1 | 150 | yes | complemented |
| 10 | #10597837 | C5×C7 | 1 | 125 | yes | complemented |
| 11 | C5×5−1+2 | C5×C7 | 1 | 30 | yes | complemented |
| 12 | C5×H(3,𝔽5) | C5×C7 | 1 | 1 | yes | characteristic |
| 13 | C52×C7 | — | 25 | 6 | no | — |
| 14 | C52×C7 | C53 | 1 | 30 | yes | — |
| 15 | C52×C7 | C53 | 1 | 1 | yes | characteristic |
| 16 | C7×C25 | C53 | 1 | 125 | yes | — |
| 17 | C53 | C52×C7 | 1 | 6 | yes | — |
| 18 | 5−1+2 | C52×C7 | 1 | 25 | yes | — |
| 19 | 5−1+2 | C52×C7 | 1 | 600 | yes | complemented |
| 20 | H(3,𝔽5) | C52×C7 | 1 | 25 | yes | — |
| 21 | C5×C25 | C52×C7 | 1 | 150 | yes | complemented |
| 22 | C5×C7 | — | 5 | 1 | no | — |
| 23 | C5×C7 | — | 5 | 30 | no | — |
| 24 | C5×C7 | C54 | 1 | 1 | yes | characteristic |
| 25 | C52 | — | 25 | 6 | no | — |
| 26 | C52 | C53×C7 | 1 | 30 | yes | — |
| 27 | C52 | C53×C7 | 1 | 1 | yes | characteristic |
| 28 | C25 | C53×C7 | 1 | 125 | yes | — |
| 29 | C7 | 5−1+4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 30 | C5 | — | 5 | 1 | no | — |
| 31 | C5 | — | 5 | 30 | no | — |
| 32 | C5 | C54×C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C54 |
| commutator, Frattini, cosocle | C5 | C54×C7 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| 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. C7×#10597837 | C5 | semidirect |
| 2. C5×C7×5−1+2 | C5 | semidirect |
| 4. 5−1+4 | C7 | direct (N × H) |
| 7. C7×5−1+2 | C52 | semidirect |
| 9. C5×C7×C25 | C52 | semidirect |
| 10. #10597837 | C5×C7 | semidirect |
| 11. C5×5−1+2 | C5×C7 | semidirect |
| 19. 5−1+2 | C52×C7 | semidirect |
| 21. C5×C25 | C52×C7 | semidirect |
| 29. C7 | 5−1+4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#10597837 | 5 |
| C5×C7×5−1+2 | 5 |
| C5×C7×H(3,𝔽5) | 5 |
| 5−1+4 | 7 |