Finite Group DatabasebetaFinite Group DatabasebetaOrder 2500.
47 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #6953⋊C5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | H(3,𝔽5)⋊C5 | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | C5×#1042 | — | 5 | 1 | no | maximal |
| 4 | #92539 | — | 5 | 1 | no | maximal |
| 5 | C52×D5 | — | 5 | 1 | no | — |
| 6 | #6953 | D5 | 1 | 1 | yes | characteristic |
| 7 | C53 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 8 | H(3,𝔽5) | — | 4 | 1 | no | — |
| 9 | H(3,𝔽5) | Dic5 | 1 | 1 | yes | characteristic, complemented |
| 10 | #1042 | — | 5 | 1 | no | — |
| 11 | C5×AGL(1,𝔽5) | — | 25 | 1 | no | — |
| 12 | C5×AGL(1,𝔽5) | — | 25 | 1 | no | — |
| 13 | C5×Dic5 | — | 25 | 1 | no | — |
| 14 | C2×C52 | — | 25 | 1 | no | — |
| 15 | C5×D5 | — | 5 | 1 | no | — |
| 16 | C5×D5 | — | 5 | 1 | no | — |
| 17 | C5×D5 | — | 10 | 1 | no | — |
| 18 | C5×D5 | — | 10 | 1 | no | — |
| 19 | C5×D5 | — | 25 | 1 | no | — |
| 20 | C52 | — | 5 | 1 | no | — |
| 21 | C52 | — | 5 | 1 | no | — |
| 22 | C52 | — | 5 | 1 | no | — |
| 23 | C52 | — | 10 | 1 | no | — |
| 24 | C52 | — | 10 | 1 | no | — |
| 25 | C52 | — | 20 | 1 | no | — |
| 26 | C52 | #1042 | 1 | 1 | yes | characteristic |
| 27 | AGL(1,𝔽5) | — | 25 | 1 | no | — |
| 28 | AGL(1,𝔽5) | — | 25 | 1 | no | — |
| 29 | C4×C5 | — | 125 | 1 | no | — |
| 30 | Dic5 | — | 25 | 1 | no | — |
| 31 | C2×C5 | — | 25 | 1 | no | — |
| 32 | C2×C5 | — | 25 | 1 | no | — |
| 33 | C2×C5 | — | 50 | 1 | no | — |
| 34 | C2×C5 | — | 50 | 1 | no | — |
| 35 | D5 | — | 5 | 1 | no | — |
| 36 | D5 | — | 25 | 1 | no | — |
| 37 | C5 | — | 5 | 1 | no | — |
| 38 | C5 | — | 5 | 1 | no | — |
| 39 | C5 | — | 10 | 1 | no | — |
| 40 | C5 | — | 10 | 1 | no | — |
| 41 | C5 | — | 25 | 1 | no | — |
| 42 | C5 | — | 100 | 1 | no | — |
| 43 | C5 | #92535 | 1 | 1 | yes | characteristic |
| 44 | C4 | — | 125 | 1 | no | Sylow |
| 45 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C5 | #92535 |
| commutator, Fitting, hypocenter | H(3,𝔽5)⋊C5 | C4 |
| Frattini | C52 | #1042 |
| prime radical, cosocle | #6953⋊C5 | C2 |
| LCS | G ▷ H(3,𝔽5)⋊C5 | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽5)⋊C5 | C4 | semidirect |
| 7. C53 | AGL(1,𝔽5) | semidirect |
| 9. H(3,𝔽5) | Dic5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6953⋊C5 | 2 |
| C5×#1042 | 5 |
| #92539 | 5 |