Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 216 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×C197 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C197 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C197 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#127×C197 | C2 | semidirect |
| 2. C2×#121×C197 | C2 | semidirect |
| 3. C2×#117×C197 | C2 | semidirect |
| 4. #394×C197 | C2 | semidirect |
| 5. #392×C197 | C2 | semidirect |
| 6. #390×C197 | C2 | semidirect |
| 7. #382×C197 | C2 | semidirect |
| 34. C22×C4×C197 | D4 | semidirect |
| 46. C42×C197 | D4 | semidirect |
| 97. C197 | #2619 | direct (N × H) |
| 98. #2619 | C197 | direct (N × H) |
| 99. C2×#127 | C2×C197 | semidirect |
| 100. C2×#121 | C2×C197 | semidirect |
| 101. C2×#117 | C2×C197 | semidirect |
| 102. #394 | C2×C197 | semidirect |
| 103. #392 | C2×C197 | semidirect |
| 104. #390 | C2×C197 | semidirect |
| 105. #382 | C2×C197 | semidirect |
| 132. C22×C4 | D4×C197 | semidirect |
| 144. C42 | D4×C197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C197 | 2 |
| C2×#121×C197 | 2 |
| C2×#117×C197 | 2 |
| #394×C197 | 2 |
| #392×C197 | 2 |
| #390×C197 | 2 |
| #382×C197 | 2 |
| #2619 | 197 |