Finite Group DatabasebetaFinite Group DatabasebetaOrder 3125.
1158 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10597837 | C5 | 1 | 125 | yes | maximal, complemented |
| 2 | C5×5−1+2 | C5 | 1 | 30 | yes | maximal, complemented |
| 3 | C5×H(3,𝔽5) | C5 | 1 | 1 | yes | maximal, characteristic |
| 4 | C53 | C52 | 1 | 6 | yes | — |
| 5 | 5−1+2 | C52 | 1 | 25 | yes | — |
| 6 | 5−1+2 | C52 | 1 | 600 | yes | complemented |
| 7 | H(3,𝔽5) | C52 | 1 | 25 | yes | — |
| 8 | C5×C25 | C52 | 1 | 150 | yes | complemented |
| 9 | C52 | — | 25 | 6 | no | — |
| 10 | C52 | C53 | 1 | 30 | yes | — |
| 11 | C52 | C53 | 1 | 1 | yes | characteristic |
| 12 | C25 | C53 | 1 | 125 | yes | — |
| 13 | C5 | — | 5 | 1 | no | — |
| 14 | C5 | — | 5 | 30 | no | — |
| 15 | C5 | C54 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C5 | C54 |
| prime radical | C5×H(3,𝔽5) | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5 ◁ G | |
| LECS | G ▷ C5 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10597837 | C5 | semidirect |
| 2. C5×5−1+2 | C5 | semidirect |
| 6. 5−1+2 | C52 | semidirect |
| 8. C5×C25 | C52 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10597837 | 5 |
| C5×5−1+2 | 5 |
| C5×H(3,𝔽5) | 5 |