Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C59 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C59 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C59 ◁ C22×Q8×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C59×2−1+6 | C2 | semidirect |
| 2. C59×#3487 | C2 | semidirect |
| 3. C59×#3027 | C2 | semidirect |
| 4. C59×#3025 | C2 | semidirect |
| 5. C59×#2884 | C2 | semidirect |
| 10. C22×#41×C59 | C22 | semidirect |
| 13. C59×#535 | C22 | semidirect |
| 14. C59×#525 | C22 | semidirect |
| 15. C59×#519 | C22 | semidirect |
| 16. C59×#458 | C22 | semidirect |
| 17. C59×B(4,𝔽2) | C22 | semidirect |
| 18. C59×#410 | C22 | semidirect |
| 28. C2×#41×C59 | D4 | semidirect |
| 31. C22×Q8×C59 | D4 | semidirect |
| 94. #60402 | C59 | direct (N × H) |
| 106. 2−1+6 | C2×C59 | semidirect |
| 107. #3487 | C2×C59 | semidirect |
| 108. #3027 | C2×C59 | semidirect |
| 109. #3025 | C2×C59 | semidirect |
| 110. #2884 | C2×C59 | semidirect |
| 119. C22×#41 | C22×C59 | semidirect |
| 122. #535 | C22×C59 | semidirect |
| 123. #525 | C22×C59 | semidirect |
| 124. #519 | C22×C59 | semidirect |
| 125. #458 | C22×C59 | semidirect |
| 126. B(4,𝔽2) | C22×C59 | semidirect |
| 127. #410 | C22×C59 | semidirect |
| 128. C59 | #60402 | direct (N × H) |
| 138. C2×#41 | D4×C59 | semidirect |
| 141. C22×Q8 | D4×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C59×2−1+6 | 2 |
| C59×#3487 | 2 |
| C59×#3027 | 2 |
| C59×#3025 | 2 |
| C59×#2884 | 2 |
| #60402 | 59 |