Finite Group DatabasebetaFinite Group DatabasebetaOrder 24804.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C53⋊(D13⋊C9) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#14583753 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C53×C13⋊C9 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×D689 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | #14583753 | C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C3×C13×C53 | C3×C4 | 1 | 1 | yes | characteristic |
| 7 | C9×#6157 | — | 13 | 1 | no | maximal, Hall |
| 8 | D689 | C2×C9 | 1 | 1 | yes | characteristic |
| 9 | C9×D53 | — | 13 | 1 | no | — |
| 10 | C13×C53 | C4×C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | C3×#6157 | — | 13 | 1 | no | — |
| 12 | C9×C53 | — | 13 | 1 | no | Hall |
| 13 | #261⋊C9 | — | 53 | 1 | no | maximal, Hall |
| 14 | C3×D53 | — | 39 | 1 | no | — |
| 15 | D13⋊C9 | — | 53 | 1 | no | — |
| 16 | #6157 | — | 13 | 1 | no | Hall |
| 17 | C3×C53 | AGL(1,𝔽13) | 1 | 1 | yes | characteristic |
| 18 | C3×#261 | — | 53 | 1 | no | — |
| 19 | C13⋊C9 | #6157 | 1 | 1 | yes | characteristic, complemented, Hall |
| 20 | D53 | — | 39 | 1 | no | — |
| 21 | C3×D13 | — | 53 | 1 | no | — |
| 22 | C53 | #261⋊C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 23 | #261 | — | 53 | 1 | no | Hall |
| 24 | C3×C13 | C3×#6157 | 1 | 1 | yes | characteristic |
| 25 | C4×C9 | — | 689 | 1 | no | Hall |
| 26 | D13 | — | 53 | 1 | no | — |
| 27 | C2×C9 | — | 689 | 1 | no | — |
| 28 | C13 | C9×#6157 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 29 | C3×C4 | — | 2067 | 1 | no | — |
| 30 | C9 | — | 13 | 1 | no | Sylow |
| 31 | C2×C3 | — | 2067 | 1 | no | — |
| 32 | C4 | — | 2067 | 1 | no | Sylow |
| 33 | C3 | #14583753⋊C3 | 1 | 1 | yes | characteristic |
| 34 | C2 | — | 2067 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #14583753⋊C3 |
| commutator, hypocenter | C13×C53 | C4×C9 |
| Fitting, socle | C3×C13×C53 | C3×C4 |
| prime radical, cosocle | C3×D689 | C2×C3 |
| LCS | G ▷ C13×C53 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×C13⋊C9 | C4 | semidirect |
| 5. #14583753 | C9 | semidirect |
| 10. C13×C53 | C4×C9 | semidirect |
| 19. C13⋊C9 | #6157 | semidirect |
| 22. C53 | #261⋊C9 | semidirect |
| 28. C13 | C9×#6157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53⋊(D13⋊C9) | 2 |
| C3×#14583753 | 3 |
| C9×#6157 | 13 |
| #261⋊C9 | 53 |