Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
348 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 184 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×C7 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C7 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 16. #2843 | C7 | direct (N × H) |
| 27. C42×C7 | Q8 | semidirect |
| 72. C42 | C7×Q8 | semidirect |
| 87. C7 | #2843 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#400 | 2 |
| C7×#398 | 2 |
| C7×#395 | 2 |
| C7×#384 | 2 |
| C7×#383 | 2 |
| #2843 | 7 |