Finite Group DatabasebetaFinite Group DatabasebetaOrder 250.
30 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(3,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 2 | C2×C52 | C5 | 1 | 6 | yes | maximal, complemented |
| 3 | C52 | C2×C5 | 1 | 6 | yes | complemented |
| 4 | C2×C5 | — | 5 | 6 | no | — |
| 5 | C2×C5 | C52 | 1 | 1 | yes | characteristic |
| 6 | C5 | — | 5 | 6 | no | — |
| 7 | C5 | C2×C52 | 1 | 1 | yes | characteristic |
| 8 | C2 | H(3,𝔽5) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C52 |
| commutator, Frattini, cosocle | C5 | C2×C52 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽5) | C2 | direct (N × H) |
| 2. C2×C52 | C5 | semidirect |
| 3. C52 | C2×C5 | semidirect |
| 8. C2 | H(3,𝔽5) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽5) | 2 |
| C2×C52 | 5 |