Finite Group DatabasebetaFinite Group DatabasebetaOrder 424.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×C53 | — | 2 | 1 | no | — |
| 5 | C2×C53 | C22 | 1 | 1 | yes | characteristic |
| 6 | D53 | — | 2 | 1 | no | — |
| 7 | C53 | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | D4 | — | 53 | 1 | no | maximal, Sylow |
| 9 | C22 | — | 53 | 1 | no | — |
| 10 | C22 | D53 | 1 | 1 | yes | characteristic, complemented |
| 11 | C4 | — | 53 | 1 | no | — |
| 12 | C2 | — | 2 | 1 | no | — |
| 13 | C2 | — | 106 | 1 | no | — |
| 14 | C2 | C2×D53 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D53 |
| commutator, socle, cosocle | C2×C53 | C22 |
| Fitting | C22×C53 | C2 |
| hypercenter | C22 | D53 |
| hypocenter | C53 | D4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C53 | C2 | semidirect |
| 2. C2×D53 | C2 | semidirect |
| 3. Dic53 | C2 | semidirect |
| 7. C53 | D4 | semidirect |
| 10. C22 | D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C53 | 2 |
| C2×D53 | 2 |
| Dic53 | 2 |
| D4 | 53 |