Finite Group DatabasebetaFinite Group DatabasebetaOrder 15625.
10832 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C55 | C5 | 1 | 1 | yes | maximal, characteristic |
| 2 | C52×5−1+2 | C5 | 1 | 5 | yes | maximal, complemented |
| 3 | #15691103 | C5 | 1 | 125 | yes | maximal, direct product factor |
| 4 | C5×C52⋊C25 | C5 | 1 | 25 | yes | maximal, complemented |
| 5 | C54 | — | 5 | 25 | no | — |
| 6 | C54 | — | 5 | 125 | no | — |
| 7 | C54 | C52 | 1 | 25 | yes | — |
| 8 | C54 | C52 | 1 | 1 | yes | characteristic |
| 9 | C54 | C52 | 1 | 5 | yes | — |
| 10 | C5×5−1+2 | — | 5 | 125 | no | — |
| 11 | C5×5−1+2 | C52 | 1 | 125 | yes | complemented |
| 12 | C52×C25 | C52 | 1 | 25 | yes | complemented |
| 13 | C52⋊C25 | C52 | 1 | 625 | yes | complemented |
| 14 | C53 | — | 5 | 3125 | no | — |
| 15 | C53 | — | 5 | 125 | no | — |
| 16 | C53 | — | 5 | 25 | no | — |
| 17 | C53 | — | 5 | 25 | no | — |
| 18 | C53 | — | 5 | 100 | no | — |
| 19 | C53 | — | 5 | 625 | no | — |
| 20 | C53 | H(3,𝔽5) | 1 | 25 | yes | — |
| 21 | C53 | 5−1+2 | 1 | 125 | yes | complemented |
| 22 | C53 | C53 | 1 | 5 | yes | — |
| 23 | C53 | C53 | 1 | 25 | yes | — |
| 24 | C53 | C53 | 1 | 1 | yes | characteristic |
| 25 | 5−1+2 | — | 5 | 625 | no | — |
| 26 | C5×C25 | — | 5 | 125 | no | — |
| 27 | C5×C25 | C53 | 1 | 125 | yes | complemented |
| 28 | C52 | — | 5 | 3125 | no | — |
| 29 | C52 | — | 5 | 125 | no | — |
| 30 | C52 | — | 5 | 625 | no | — |
| 31 | C52 | — | 5 | 25 | no | — |
| 32 | C52 | — | 5 | 25 | no | — |
| 33 | C52 | — | 5 | 100 | no | — |
| 34 | C52 | C5×H(3,𝔽5) | 1 | 25 | yes | — |
| 35 | C52 | C5×H(3,𝔽5) | 1 | 5 | yes | — |
| 36 | C52 | C5×5−1+2 | 1 | 25 | yes | — |
| 37 | C52 | C5×5−1+2 | 1 | 125 | yes | complemented |
| 38 | C52 | C54 | 1 | 1 | yes | characteristic |
| 39 | C25 | — | 5 | 125 | no | — |
| 40 | C5 | — | 5 | 25 | no | — |
| 41 | C5 | — | 5 | 125 | no | — |
| 42 | C5 | #15691103 | 1 | 25 | yes | direct product factor |
| 43 | C5 | C52×H(3,𝔽5) | 1 | 1 | yes | characteristic |
| 44 | C5 | C52×5−1+2 | 1 | 5 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C53 | C53 |
| commutator, Frattini, cosocle | C52 | C54 |
| prime radical | C55 | C5 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C53 ◁ G | |
| LECS | G ▷ C52 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C52×5−1+2 | C5 | semidirect |
| 3. #15691103 | C5 | direct (N × H) |
| 4. C5×C52⋊C25 | C5 | semidirect |
| 11. C5×5−1+2 | C52 | semidirect |
| 12. C52×C25 | C52 | semidirect |
| 13. C52⋊C25 | C52 | semidirect |
| 21. C53 | 5−1+2 | semidirect |
| 27. C5×C25 | C53 | semidirect |
| 37. C52 | C5×5−1+2 | semidirect |
| 42. C5 | #15691103 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C55 | 5 |
| C52×5−1+2 | 5 |
| #15691103 | 5 |
| C5×C52⋊C25 | 5 |