Finite Group DatabasebetaFinite Group DatabasebetaOrder 212.
6 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53 | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | C4 | — | 53 | 1 | no | maximal, Sylow |
| 4 | C2 | — | 53 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C53 | C4 |
| prime radical, cosocle | D53 | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D53 | 2 |
| C4 | 53 |