Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
690 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 278 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×#127 |
| prime radical | C23×C7 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1723 | C2 | semidirect |
| 18. #19618 | C7 | direct (N × H) |
| 38. #1723 | C2×C7 | semidirect |
| 45. C2×C7×C8 | Q16 | semidirect |
| 110. C2×C8 | C7×Q16 | semidirect |
| 135. C7 | #19618 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3170 | 2 |
| C7×#3160 | 2 |
| C7×#1723 | 2 |
| C2×C7×#322 | 2 |
| #19618 | 7 |