Finite Group DatabasebetaFinite Group DatabasebetaOrder 23373.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C49×C53 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×C7⋊C9 | — | 7 | 1 | no | maximal |
| 3 | C49×C53 | C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×C7×C53 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 5 | C9×C53 | — | 49 | 1 | no | Hall |
| 6 | C49⋊C9 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | C7×C53 | C7⋊C9 | 1 | 1 | yes | characteristic |
| 8 | C3×C53 | C49⋊C3 | 1 | 1 | yes | characteristic |
| 9 | C3×C49 | C3×C53 | 1 | 1 | yes | characteristic |
| 10 | C7⋊C9 | — | 7 | 1 | no | — |
| 11 | C53 | C49⋊C9 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | C49 | C9×C53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C3×C7 | ΓL(1,𝔽8)×C53 | 1 | 1 | yes | characteristic |
| 14 | C9 | — | 49 | 1 | no | Sylow |
| 15 | C7 | C53×C7⋊C9 | 1 | 1 | yes | characteristic |
| 16 | C3 | C53×C49⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C53 | C49⋊C3 |
| commutator, hypocenter | C49 | C9×C53 |
| Frattini | C3×C7 | ΓL(1,𝔽8)×C53 |
| Fitting | C3×C49×C53 | C3 |
| socle, prime radical | C3×C7×C53 | ΓL(1,𝔽8) |
| cosocle | C3×C49 | C3×C53 |
| LCS | G ▷ C49 | |
| UCS | C1 ◁ C3×C53 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C49×C53 | C9 | semidirect |
| 6. C49⋊C9 | C53 | direct (N × H) |
| 11. C53 | C49⋊C9 | direct (N × H) |
| 12. C49 | C9×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C49×C53 | 3 |
| C53×C7⋊C9 | 7 |
| C49⋊C9 | 53 |