Finite Group DatabasebetaFinite Group DatabasebetaOrder 1500.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×D125 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #92517 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C3×C125 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×#1035 | — | 5 | 1 | no | maximal |
| 5 | D125 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | C3×D25 | — | 5 | 1 | no | — |
| 7 | C125 | C3×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | #1035 | — | 5 | 1 | no | — |
| 9 | C3×C25 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 10 | C3×AGL(1,𝔽5) | — | 25 | 1 | no | — |
| 11 | D25 | — | 5 | 1 | no | — |
| 12 | C3×D5 | — | 25 | 1 | no | — |
| 13 | C25 | C3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 14 | AGL(1,𝔽5) | — | 25 | 1 | no | — |
| 15 | C3×C5 | #1035 | 1 | 1 | yes | characteristic |
| 16 | C3×C4 | — | 125 | 1 | no | Hall |
| 17 | D5 | — | 25 | 1 | no | — |
| 18 | C2×C3 | — | 125 | 1 | no | — |
| 19 | C5 | C3×#1035 | 1 | 1 | yes | characteristic |
| 20 | C4 | — | 125 | 1 | no | Sylow |
| 21 | C3 | #92517 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 22 | C2 | — | 125 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #92517 |
| commutator, hypocenter | C125 | C3×C4 |
| Frattini | C25 | C3×AGL(1,𝔽5) |
| Fitting | C3×C125 | C4 |
| socle | C3×C5 | #1035 |
| prime radical | C3×D125 | C2 |
| cosocle | D125 | C2×C3 |
| LCS | G ▷ C125 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #92517 | C3 | direct (N × H) |
| 3. C3×C125 | C4 | semidirect |
| 7. C125 | C3×C4 | semidirect |
| 21. C3 | #92517 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D125 | 2 |
| #92517 | 3 |
| C3×#1035 | 5 |