Finite Group DatabasebetaFinite Group DatabasebetaOrder 21248.
240 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23×C83 |
| prime radical | #493×C83 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C42×C83 ◁ C82×C83 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C83×#2209 | C2 | semidirect |
| 4. C8×Q16×C83 | C2 | semidirect |
| 5. C83×#1370 | C2 | semidirect |
| 6. C83×#1362 | C2 | semidirect |
| 108. #12429 | C83 | direct (N × H) |
| 113. #2209 | C2×C83 | semidirect |
| 116. C8×Q16 | C2×C83 | semidirect |
| 117. #1370 | C2×C83 | semidirect |
| 118. #1362 | C2×C83 | semidirect |
| 120. C83 | #12429 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C83×#2209 | 2 |
| C83×#2207 | 2 |
| C83×#1709 | 2 |
| C8×Q16×C83 | 2 |
| C83×#1370 | 2 |
| C83×#1362 | 2 |
| C83×#1361 | 2 |
| #12429 | 83 |