Finite Group DatabasebetaFinite Group DatabasebetaOrder 17536.
480 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 192 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×C137 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C137 |
| prime radical | C22×D4×C137 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C137 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #400×C137 | C2 | semidirect |
| 2. #393×C137 | C2 | semidirect |
| 3. #391×C137 | C2 | semidirect |
| 6. #385×C137 | C2 | semidirect |
| 27. C2×D4×C137 | Q8 | semidirect |
| 86. C137 | #2804 | direct (N × H) |
| 87. #2804 | C137 | direct (N × H) |
| 88. #400 | C2×C137 | semidirect |
| 89. #393 | C2×C137 | semidirect |
| 90. #391 | C2×C137 | semidirect |
| 93. #385 | C2×C137 | semidirect |
| 114. C2×D4 | Q8×C137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #400×C137 | 2 |
| #393×C137 | 2 |
| #391×C137 | 2 |
| #390×C137 | 2 |
| #386×C137 | 2 |
| #385×C137 | 2 |
| #2804 | 137 |