Finite Group DatabasebetaFinite Group Databasebeta| order | 3136 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C7 |
| direct product | #120×C7⋊D7 |
| Fitting | C72×#120 ⋊ C2 |
| Sylow | C72 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C7⋊D7 |
| commutator, socle, cosocle | C22×C72 | C24 |
| Fitting | C72×#120 | C2 |
| hypercenter | #120 | C7⋊D7 |
| hypocenter | C72 | C2×#120 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ #120 | |
3310 conjugacy classes of subgroups; only the 339 normal subgroups are shown.
This group has 3310 conjugacy classes of subgroups — too many to draw. Only its 339 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #1020×C7⋊D7, #120×#64424, #72215173, … |
| maximal normal subgroup of | #120×#64424, #72215173, #120×ΓL(1,𝔽8)⋊D7, … |
| maximal quotient of | B(4,𝔽2)×C7⋊D7, #453×C7⋊D7 |
Past the SmallGroups library coverage — no SmallGroups id.