Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C2×C7×#126 | C2 |
| hypercenter | C2×#126 | D7 |
| hypocenter | C7 | #2818 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#126 | C2 | semidirect |
| 2. C2×#6437 | C2 | semidirect |
| 3. #89763 | C2 | semidirect |
| 4. #89732 | C2 | semidirect |
| 5. #89709 | C2 | semidirect |
| 26. C42×C7 | D4 | semidirect |
| 29. C2×#126 | D7 | semidirect |
| 76. C42 | C22⋊D7 | semidirect |
| 94. C7 | #2818 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#126 | 2 |
| C2×#6437 | 2 |
| #89763 | 2 |
| #89732 | 2 |
| #89709 | 2 |
| #2818 | 7 |