Finite Group DatabasebetaFinite Group Databasebeta← C2×ΓL(1,𝔽8)×GL(3,𝔽2) #72228109
Order 7056.
188 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×GL(3,𝔽2) |
| commutator, hypocenter | C7×GL(3,𝔽2) | C2×C3 |
| Frattini | C1 | G |
| Fitting | C2×C7 | C3×GL(3,𝔽2) |
| socle | C2×C7×GL(3,𝔽2) | C3 |
| cosocle | C7 | C2×C3×GL(3,𝔽2) |
| LCS | G ▷ C7×GL(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×GL(3,𝔽2) | C3 | semidirect |
| 2. ΓL(1,𝔽8)×GL(3,𝔽2) | C2 | direct (N × H) |
| 3. C7×GL(3,𝔽2) | C2×C3 | semidirect |
| 13. C2×GL(3,𝔽2) | ΓL(1,𝔽8) | direct (N × H) |
| 26. GL(3,𝔽2) | C2×ΓL(1,𝔽8) | direct (N × H) |
| 74. C2×ΓL(1,𝔽8) | GL(3,𝔽2) | direct (N × H) |
| 98. ΓL(1,𝔽8) | C2×GL(3,𝔽2) | direct (N × H) |
| 107. C2×C7 | C3×GL(3,𝔽2) | semidirect |
| 124. C7 | C2×C3×GL(3,𝔽2) | semidirect |
| 142. C2 | ΓL(1,𝔽8)×GL(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×GL(3,𝔽2) | 3 |
| ΓL(1,𝔽8)×GL(3,𝔽2) | 2 |
| C2×ΓL(1,𝔽8)×S4 | 7 |
| C2×C3×GL(3,𝔽2) | 7 |
| C2×ΓL(1,𝔽8)2 | 8 |