Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
404 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 404 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 | C22×C197 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C197 |
| prime radical | #550×C197 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C197 ◁ C4×Q8×C197 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #557×C197 | C2 | semidirect |
| 3. #492×C197 | C2 | semidirect |
| 4. #461×C197 | C2 | semidirect |
| 5. #460×C197 | C2 | semidirect |
| 6. C8×Q8×C197 | C2 | semidirect |
| 7. C4×SD16×C197 | C2 | semidirect |
| 47. Q8×C197 | SD16 | semidirect |
| 60. C8×C197 | #41 | semidirect |
| 76. C197 | #3392 | direct (N × H) |
| 77. #3392 | C197 | direct (N × H) |
| 78. #557 | C2×C197 | semidirect |
| 80. #492 | C2×C197 | semidirect |
| 81. #461 | C2×C197 | semidirect |
| 82. #460 | C2×C197 | semidirect |
| 83. C8×Q8 | C2×C197 | semidirect |
| 84. C4×SD16 | C2×C197 | semidirect |
| 124. Q8 | SD16×C197 | semidirect |
| 137. C8 | #41×C197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #557×C197 | 2 |
| #550×C197 | 2 |
| #492×C197 | 2 |
| #461×C197 | 2 |
| #460×C197 | 2 |
| C8×Q8×C197 | 2 |
| C4×SD16×C197 | 2 |
| #3392 | 197 |