Finite Group DatabasebetaFinite Group DatabasebetaOrder 300.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | (C5⋊D5)⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1043 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C52⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C5⋊D5 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | C52 | C3×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | AGL(1,𝔽5) | — | 15 | 2 | no | — |
| 7 | C3×C4 | — | 25 | 1 | no | maximal, Hall |
| 8 | D5 | — | 15 | 2 | no | — |
| 9 | C2×C3 | — | 25 | 1 | no | — |
| 10 | C5 | — | 3 | 2 | no | — |
| 11 | C4 | — | 25 | 1 | no | Sylow |
| 12 | C3 | — | 25 | 1 | no | Sylow |
| 13 | C2 | — | 25 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52 | C3×C4 |
| prime radical | (C5⋊D5)⋊C3 | C2 |
| cosocle | C5⋊D5 | C2×C3 |
| LCS | G ▷ C52 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1043 | C3 | semidirect |
| 3. C52⋊C3 | C4 | semidirect |
| 5. C52 | C3×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| (C5⋊D5)⋊C3 | 2 |
| #1043 | 3 |
| C3×C4 | 25 |