Finite Group DatabasebetaFinite Group DatabasebetaOrder 500.
29 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #6953 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | H(3,𝔽5) | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | C5×AGL(1,𝔽5) | — | 5 | 2 | no | maximal |
| 4 | C5×D5 | — | 5 | 2 | no | — |
| 5 | C5×D5 | — | 10 | 2 | no | — |
| 6 | C52 | — | 2 | 2 | no | — |
| 7 | C52 | AGL(1,𝔽5) | 1 | 2 | yes | complemented |
| 8 | AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 9 | C4×C5 | — | 25 | 1 | no | — |
| 10 | C2×C5 | — | 25 | 1 | no | — |
| 11 | D5 | — | 5 | 2 | no | — |
| 12 | D5 | — | 10 | 2 | no | — |
| 13 | C5 | — | 5 | 2 | no | — |
| 14 | C5 | — | 10 | 2 | no | — |
| 15 | C5 | #1044 | 1 | 1 | yes | characteristic |
| 16 | C4 | — | 25 | 1 | no | Sylow |
| 17 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C5 | #1044 |
| commutator, Fitting, hypocenter | H(3,𝔽5) | C4 |
| prime radical, cosocle | #6953 | C2 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽5) | C4 | semidirect |
| 7. C52 | AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6953 | 2 |
| C5×AGL(1,𝔽5) | 5 |