Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 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×C97 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C97 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C97 ◁ C42×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 43. C8×C97 | Q16 | semidirect |
| 56. #1674 | C97 | direct (N × H) |
| 57. C97 | #1674 | direct (N × H) |
| 100. C8 | Q16×C97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #500×C97 | 2 |
| #462×C97 | 2 |
| C8×Q8×C97 | 2 |
| #335×C97 | 2 |
| #332×C97 | 2 |
| #1674 | 97 |