Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
238 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 86. #2097 | C53 | direct (N × H) |
| 99. C53 | #2097 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#107×C53 | 2 |
| C2×#106×C53 | 2 |
| C2×#103×C53 | 2 |
| C53×#391 | 2 |
| C53×#382 | 2 |
| C53×#336 | 2 |
| #2097 | 53 |