Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
3876 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 3876 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#2847 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C24×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C7×Q8 | C2 | semidirect |
| 2. C2×C7×#448 | C2 | semidirect |
| 4. C7×#1787 | C2 | semidirect |
| 5. C7×#1779 | C2 | semidirect |
| 11. C22×C7×SD16 | C22 | semidirect |
| 15. C2×C7×#103 | C22 | semidirect |
| 19. #32879 | C7 | direct (N × H) |
| 27. C22×C7×Q8 | D4 | semidirect |
| 46. C24×Q8 | C2×C7 | semidirect |
| 47. C2×#448 | C2×C7 | semidirect |
| 49. #1787 | C2×C7 | semidirect |
| 50. #1779 | C2×C7 | semidirect |
| 54. C24×C7 | SD16 | semidirect |
| 89. C22×SD16 | C22×C7 | semidirect |
| 93. C2×#103 | C22×C7 | semidirect |
| 111. C7×Q8 | #120 | semidirect |
| 132. C22×Q8 | C7×D4 | semidirect |
| 159. C22×C7 | #448 | semidirect |
| 170. C24 | C7×SD16 | semidirect |
| 219. Q8 | C7×#120 | semidirect |
| 233. C7 | #32879 | direct (N × H) |
| 242. C22 | C7×#448 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C7×Q8 | 2 |
| C2×C7×#448 | 2 |
| C7×#2847 | 2 |
| C7×#1787 | 2 |
| C7×#1779 | 2 |
| #32879 | 7 |