Finite Group DatabasebetaFinite Group DatabasebetaOrder 21875.
70 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C52⋊C25 | C5 | 1 | 3 | yes | maximal |
| 2 | C7×C52⋊C25 | C5 | 1 | 3 | yes | maximal |
| 3 | #15691067 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 4 | C53×C7 | C52 | 1 | 1 | yes | characteristic |
| 5 | C5×C7×C25 | — | 5 | 3 | no | — |
| 6 | C5×C7×C25 | — | 5 | 3 | no | — |
| 7 | C52⋊C25 | C5×C7 | 1 | 3 | yes | — |
| 8 | C52⋊C25 | C5×C7 | 1 | 3 | yes | — |
| 9 | C52×C7 | — | 5 | 3 | no | — |
| 10 | C52×C7 | — | 5 | 3 | no | — |
| 11 | C52×C7 | H(3,𝔽5) | 1 | 1 | yes | characteristic |
| 12 | C7×C25 | — | 25 | 3 | no | — |
| 13 | C7×C25 | — | 25 | 3 | no | — |
| 14 | C53 | C52×C7 | 1 | 1 | yes | characteristic |
| 15 | C5×C25 | — | 5 | 3 | no | — |
| 16 | C5×C25 | — | 5 | 3 | no | — |
| 17 | C5×C7 | — | 25 | 1 | no | — |
| 18 | C5×C7 | #10597832 | 1 | 3 | yes | — |
| 19 | C5×C7 | #10597833 | 1 | 3 | yes | — |
| 20 | C52 | — | 5 | 3 | no | — |
| 21 | C52 | — | 5 | 3 | no | — |
| 22 | C52 | C7×H(3,𝔽5) | 1 | 1 | yes | characteristic |
| 23 | C25 | — | 25 | 3 | no | — |
| 24 | C25 | — | 25 | 3 | no | — |
| 25 | C7 | #15691067 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 26 | C5 | — | 25 | 1 | no | — |
| 27 | C5 | C7×#10597832 | 1 | 3 | yes | — |
| 28 | C5 | C7×#10597833 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C7 |
| prime radical | C53×C7 | C52 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ C53×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #15691067 | C7 | direct (N × H) |
| 25. C7 | #15691067 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C52⋊C25 | 5 |
| #15691067 | 7 |