Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
388 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#595 |
| commutator, socle | C22×C17 | C23×C4 |
| Fitting, prime radical | C2×#41×C17 | C4 |
| cosocle | C2×Dic17 | C24 |
| hypercenter | C2×#41 | #595 |
| hypocenter | C17 | #2989 |
| LCS | G ▷ C22×C17 ▷ C17 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763860 | C2 | semidirect |
| 2. #11763844 | C2 | semidirect |
| 3. #11763840 | C2 | semidirect |
| 4. #11763830 | C2 | semidirect |
| 5. #11763825 | C2 | semidirect |
| 6. #11763812 | C2 | semidirect |
| 7. C2×C4⋊#3624 | C2 | semidirect |
| 66. C2×C4×C17 | M16 | semidirect |
| 138. C17 | #2989 | semidirect |
| 186. C2×C4 | #63247 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763860 | 2 |
| #11763844 | 2 |
| #11763840 | 2 |
| #11763830 | 2 |
| #11763825 | 2 |
| #11763812 | 2 |
| C2×C4⋊#3624 | 2 |
| #41×Dic17 | 2 |
| #2989 | 17 |