Finite Group DatabasebetaFinite Group DatabasebetaOrder 12300.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C41⋊((C5⋊D5)⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #16861522 | — | 3 | 1 | no | maximal, Hall |
| 3 | C41×C52⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C5⋊D205 | D3 | 1 | 1 | yes | characteristic |
| 5 | C52×C41 | Dic3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | #11717962 | — | 15 | 1 | no | — |
| 7 | #11717961 | — | 15 | 1 | no | — |
| 8 | C3⋊#4223 | — | 25 | 1 | no | maximal, Hall |
| 9 | D205 | — | 15 | 1 | no | — |
| 10 | D205 | — | 15 | 1 | no | — |
| 11 | #64480 | — | 41 | 1 | no | maximal, Hall |
| 12 | C3×D41 | — | 25 | 1 | no | — |
| 13 | C5×C41 | — | 3 | 1 | no | — |
| 14 | C5×C41 | — | 3 | 1 | no | — |
| 15 | #4223 | — | 75 | 1 | no | Hall |
| 16 | (C5⋊D5)⋊C3 | — | 41 | 1 | no | — |
| 17 | C3×C41 | — | 25 | 1 | no | Hall |
| 18 | #1044 | — | 123 | 1 | no | Hall |
| 19 | D41 | — | 25 | 1 | no | — |
| 20 | C52⋊C3 | #4223 | 1 | 1 | yes | characteristic, complemented, Hall |
| 21 | C5⋊D5 | — | 41 | 1 | no | — |
| 22 | C41 | #64480 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 23 | C52 | C3⋊#4223 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 24 | AGL(1,𝔽5) | — | 615 | 1 | no | — |
| 25 | AGL(1,𝔽5) | — | 615 | 1 | no | — |
| 26 | Dic3 | — | 1025 | 1 | no | Hall |
| 27 | D5 | — | 615 | 1 | no | — |
| 28 | D5 | — | 615 | 1 | no | — |
| 29 | C2×C3 | — | 1025 | 1 | no | — |
| 30 | C5 | — | 3 | 1 | no | — |
| 31 | C5 | — | 3 | 1 | no | — |
| 32 | C4 | — | 3075 | 1 | no | Sylow |
| 33 | C3 | — | 25 | 1 | no | Sylow |
| 34 | C2 | — | 1025 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C41×C52⋊C3 | C4 |
| Fitting, socle | C52×C41 | Dic3 |
| prime radical, cosocle | C41⋊((C5⋊D5)⋊C3) | C2 |
| LCS | G ▷ C41×C52⋊C3 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C41×C52⋊C3 | C4 | semidirect |
| 5. C52×C41 | Dic3 | semidirect |
| 20. C52⋊C3 | #4223 | semidirect |
| 22. C41 | #64480 | semidirect |
| 23. C52 | C3⋊#4223 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C41⋊((C5⋊D5)⋊C3) | 2 |
| #16861522 | 3 |
| C3⋊#4223 | 25 |
| #64480 | 41 |