Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
176 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C17 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C17 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C42×C17 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 51. C8×C17 | Q16 | semidirect |
| 52. #1723 | C17 | direct (N × H) |
| 84. C17 | #1723 | direct (N × H) |
| 115. C8 | Q16×C17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C17×#500 | 2 |
| C17×#498 | 2 |
| C17×#494 | 2 |
| C17×#487 | 2 |
| C17×#322 | 2 |
| #1723 | 17 |