Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
4778 conjugacy classes of subgroups; only the 807 normal subgroups are shown.
This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 807 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #21437751 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #21433498 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | #21433488 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #21432560 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C59×2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11745689 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | #16838071 | — | 2 | 12 | no | — |
| 8 | #16838063 | — | 2 | 12 | no | — |
| 9 | #16837966 | C22 | 1 | 18 | yes | complemented |
| 10 | #16837909 | C22 | 1 | 18 | yes | complemented |
| 11 | #16837880 | C22 | 1 | 3 | yes | complemented |
| 12 | #16837592 | C22 | 1 | 16 | yes | complemented |
| 13 | #16837589 | C22 | 1 | 48 | yes | complemented |
| 14 | #16837376 | C22 | 1 | 36 | yes | complemented |
| 15 | C59×#585 | — | 2 | 12 | no | — |
| 16 | C59×#585 | C22 | 1 | 12 | yes | — |
| 17 | C2×2−1+4×C59 | — | 2 | 12 | no | — |
| 18 | C2×2−1+4×C59 | C22 | 1 | 3 | yes | — |
| 19 | C2×#11745689 | — | 2 | 12 | no | — |
| 20 | C22×D236 | C23 | 1 | 3 | yes | — |
| 21 | C22×C4×D59 | C23 | 1 | 3 | yes | — |
| 22 | #14273723 | — | 2 | 18 | no | — |
| 23 | #14273716 | — | 2 | 18 | no | — |
| 24 | #14273708 | C23 | 1 | 12 | yes | — |
| 25 | #14273703 | — | 2 | 18 | no | — |
| 26 | #14273701 | — | 2 | 36 | no | — |
| 27 | #14273698 | — | 2 | 18 | no | — |
| 28 | #14273695 | C23 | 1 | 36 | yes | — |
| 29 | Q8×Dic59 | — | 2 | 12 | no | — |
| 30 | C4×C22⋊D59 | — | 2 | 36 | no | — |
| 31 | #14273686 | C23 | 1 | 18 | yes | — |
| 32 | #14273642 | C23 | 1 | 48 | yes | — |
| 33 | 2−1+4×C59 | — | 2 | 48 | no | — |
| 34 | 2−1+4×C59 | — | 2 | 36 | no | — |
| 35 | 2−1+4×C59 | — | 2 | 24 | no | — |
| 36 | H(5,𝔽2)×C59 | — | 2 | 16 | no | — |
| 37 | H(5,𝔽2)×C59 | — | 2 | 36 | no | — |
| 38 | H(5,𝔽2)×C59 | C23 | 1 | 16 | yes | — |
| 39 | C2×#41×C59 | — | 2 | 48 | no | — |
| 40 | C2×#41×C59 | — | 2 | 72 | no | — |
| 41 | C2×#41×C59 | D4 | 1 | 12 | yes | complemented |
| 42 | C2×#41×C59 | C23 | 1 | 18 | yes | — |
| 43 | C22×Q8×C59 | — | 2 | 16 | no | — |
| 44 | C22×Q8×C59 | D4 | 1 | 12 | yes | complemented |
| 45 | C22×Q8×C59 | C23 | 1 | 1 | yes | characteristic |
| 46 | C23×D59 | C24 | 1 | 3 | yes | — |
| 47 | #41×C59 | — | 2 | 144 | no | — |
| 48 | #41×C59 | — | 2 | 48 | no | — |
| 49 | #41×C59 | — | 2 | 48 | no | — |
| 50 | #41×C59 | — | 2 | 48 | no | — |
| 51 | #41×C59 | — | 2 | 72 | no | — |
| 52 | C2×Q8×C59 | — | 2 | 72 | no | — |
| 53 | C2×Q8×C59 | — | 2 | 48 | no | — |
| 54 | C2×Q8×C59 | C2×D4 | 1 | 18 | yes | — |
| 55 | C2×Q8×C59 | C2×D4 | 1 | 12 | yes | — |
| 56 | C2×D4×C59 | — | 2 | 72 | no | — |
| 57 | C2×D4×C59 | — | 2 | 48 | no | — |
| 58 | C2×D4×C59 | C2×D4 | 1 | 18 | yes | — |
| 59 | C2×D4×C59 | C24 | 1 | 12 | yes | — |
| 60 | C22×C4×C59 | — | 2 | 48 | no | — |
| 61 | C22×C4×C59 | C2×D4 | 1 | 36 | yes | — |
| 62 | C22×C4×C59 | C24 | 1 | 3 | yes | — |
| 63 | C2×C22⋊D59 | — | 2 | 36 | no | — |
| 64 | C22×Dic59 | C24 | 1 | 1 | yes | characteristic |
| 65 | #11745689 | — | 4 | 16 | no | — |
| 66 | C2×D236 | — | 2 | 18 | no | — |
| 67 | C2×C4×D59 | — | 2 | 18 | no | — |
| 68 | C22⋊Dic59 | — | 2 | 12 | no | — |
| 69 | C22⋊Dic59 | C24 | 1 | 12 | yes | — |
| 70 | (C2×D59)⋊C4 | — | 2 | 36 | no | — |
| 71 | C4⋊Dic59 | — | 2 | 18 | no | — |
| 72 | Dic59⋊C4 | — | 2 | 12 | no | — |
| 73 | C4×Dic59 | — | 2 | 18 | no | — |
| 74 | C23×C59 | — | 2 | 16 | no | — |
| 75 | C23×C59 | C22×D4 | 1 | 12 | yes | — |
| 76 | C23×C59 | C25 | 1 | 1 | yes | characteristic |
| 77 | C22×D59 | — | 2 | 9 | no | — |
| 78 | C22×D59 | — | 8 | 3 | no | — |
| 79 | Q8×C59 | — | 2 | 16 | no | — |
| 80 | Q8×C59 | — | 2 | 36 | no | — |
| 81 | Q8×C59 | #120 | 1 | 16 | yes | — |
| 82 | D4×C59 | — | 2 | 48 | no | — |
| 83 | D4×C59 | — | 2 | 24 | no | — |
| 84 | D4×C59 | — | 2 | 36 | no | — |
| 85 | C2×C4×C59 | — | 2 | 48 | no | — |
| 86 | C2×C4×C59 | — | 2 | 72 | no | — |
| 87 | C2×C4×C59 | C22×D4 | 1 | 12 | yes | — |
| 88 | C2×C4×C59 | C22×D4 | 1 | 18 | yes | — |
| 89 | C22⋊D59 | — | 8 | 36 | no | — |
| 90 | C2×Dic59 | — | 2 | 12 | no | — |
| 91 | C2×Dic59 | — | 2 | 3 | no | — |
| 92 | D236 | — | 4 | 12 | no | — |
| 93 | C4×D59 | — | 4 | 12 | no | — |
| 94 | #60402 | — | 59 | 1 | no | maximal, Sylow |
| 95 | C22×C59 | — | 2 | 12 | no | — |
| 96 | C22×C59 | — | 4 | 1 | no | — |
| 97 | C22×C59 | — | 4 | 12 | no | — |
| 98 | C22×C59 | — | 8 | 16 | no | — |
| 99 | C22×C59 | C23×D4 | 1 | 3 | yes | — |
| 100 | C2×D59 | — | 4 | 18 | no | — |
| 101 | C2×D59 | — | 4 | 3 | no | — |
| 102 | C4×C59 | — | 2 | 12 | no | — |
| 103 | C4×C59 | C2×#120 | 1 | 12 | yes | — |
| 104 | Dic59 | — | 4 | 12 | no | — |
| 105 | Dic59 | — | 4 | 1 | no | — |
| 106 | 2−1+6 | D59 | 1 | 1 | yes | characteristic, complemented |
| 107 | #3487 | — | 59 | 3 | no | — |
| 108 | #3027 | — | 59 | 12 | no | — |
| 109 | #3025 | — | 59 | 12 | no | — |
| 110 | #2884 | — | 59 | 3 | no | — |
| 111 | C2×C59 | — | 2 | 3 | no | — |
| 112 | C2×C59 | — | 4 | 12 | no | — |
| 113 | C2×C59 | C22×#120 | 1 | 1 | yes | characteristic |
| 114 | D59 | — | 8 | 3 | no | — |
| 115 | #585 | — | 2 | 12 | no | — |
| 116 | #585 | C2×D59 | 1 | 12 | yes | — |
| 117 | C2×2−1+4 | — | 2 | 12 | no | — |
| 118 | C2×2−1+4 | C2×D59 | 1 | 3 | yes | — |
| 119 | C22×#41 | — | 59 | 1 | no | — |
| 120 | #550 | — | 118 | 12 | no | — |
| 121 | #548 | — | 118 | 12 | no | — |
| 122 | #535 | — | 59 | 18 | no | — |
| 123 | #525 | — | 59 | 18 | no | — |
| 124 | #519 | — | 59 | 3 | no | — |
| 125 | #458 | — | 59 | 16 | no | — |
| 126 | B(4,𝔽2) | — | 59 | 48 | no | — |
| 127 | #410 | — | 59 | 36 | no | — |
| 128 | C59 | #60402 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | 2−1+4 | — | 2 | 48 | no | — |
| 130 | 2−1+4 | — | 2 | 36 | no | — |
| 131 | 2−1+4 | — | 2 | 24 | no | — |
| 132 | H(5,𝔽2) | — | 2 | 16 | no | — |
| 133 | H(5,𝔽2) | — | 2 | 36 | no | — |
| 134 | H(5,𝔽2) | C22×D59 | 1 | 16 | yes | — |
| 135 | C2×#41 | — | 2 | 48 | no | — |
| 136 | C2×#41 | — | 2 | 72 | no | — |
| 137 | C2×#41 | — | 118 | 12 | no | — |
| 138 | C2×#41 | C22⋊D59 | 1 | 12 | yes | complemented |
| 139 | C2×#41 | C22×D59 | 1 | 18 | yes | — |
| 140 | C22×Q8 | — | 2 | 16 | no | — |
| 141 | C22×Q8 | C22⋊D59 | 1 | 12 | yes | complemented |
| 142 | C22×Q8 | C22×D59 | 1 | 1 | yes | characteristic |
| 143 | C22×D4 | — | 59 | 3 | no | — |
| 144 | C23×C4 | — | 59 | 3 | no | — |
| 145 | #127 | — | 118 | 18 | no | — |
| 146 | #124 | — | 118 | 18 | no | — |
| 147 | #123 | — | 59 | 12 | no | — |
| 148 | C22⋊Q8 | — | 118 | 18 | no | — |
| 149 | #121 | — | 118 | 18 | no | — |
| 150 | #121 | — | 118 | 36 | no | — |
| 151 | #120 | — | 59 | 36 | no | — |
| 152 | C4×Q8 | — | 118 | 12 | no | — |
| 153 | C4×D4 | — | 118 | 36 | no | — |
| 154 | #117 | — | 59 | 18 | no | — |
| 155 | #99 | — | 59 | 48 | no | — |
| 156 | C24 | — | 59 | 3 | no | — |
| 157 | #41 | — | 2 | 144 | no | — |
| 158 | #41 | — | 2 | 48 | no | — |
| 159 | #41 | — | 2 | 48 | no | — |
| 160 | #41 | — | 2 | 48 | no | — |
| 161 | #41 | — | 2 | 72 | no | — |
| 162 | #41 | — | 236 | 16 | no | — |
| 163 | C2×Q8 | — | 2 | 72 | no | — |
| 164 | C2×Q8 | — | 2 | 48 | no | — |
| 165 | C2×Q8 | C2×C22⋊D59 | 1 | 18 | yes | — |
| 166 | C2×Q8 | C2×C22⋊D59 | 1 | 12 | yes | — |
| 167 | C2×D4 | — | 2 | 72 | no | — |
| 168 | C2×D4 | — | 2 | 48 | no | — |
| 169 | C2×D4 | — | 118 | 36 | no | — |
| 170 | C2×D4 | — | 118 | 18 | no | — |
| 171 | C2×D4 | C2×C22⋊D59 | 1 | 18 | yes | — |
| 172 | C2×D4 | C23×D59 | 1 | 12 | yes | — |
| 173 | C22×C4 | — | 2 | 48 | no | — |
| 174 | C22×C4 | — | 59 | 1 | no | — |
| 175 | C22×C4 | — | 118 | 18 | no | — |
| 176 | C22×C4 | C2×C22⋊D59 | 1 | 36 | yes | — |
| 177 | C22×C4 | C23×D59 | 1 | 3 | yes | — |
| 178 | C4⋊C4 | — | 118 | 18 | no | — |
| 179 | C4⋊C4 | — | 118 | 12 | no | — |
| 180 | C22⋊C4 | — | 59 | 12 | no | — |
| 181 | C22⋊C4 | — | 118 | 12 | no | — |
| 182 | C22⋊C4 | — | 118 | 36 | no | — |
| 183 | C42 | — | 118 | 18 | no | — |
| 184 | C23 | — | 2 | 16 | no | — |
| 185 | C23 | — | 118 | 9 | no | — |
| 186 | C23 | — | 472 | 3 | no | — |
| 187 | C23 | C22×C22⋊D59 | 1 | 12 | yes | — |
| 188 | C23 | C24×D59 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 2 | 16 | no | — |
| 190 | Q8 | — | 2 | 36 | no | — |
| 191 | Q8 | #14273696 | 1 | 16 | yes | — |
| 192 | D4 | — | 2 | 48 | no | — |
| 193 | D4 | — | 2 | 24 | no | — |
| 194 | D4 | — | 2 | 36 | no | — |
| 195 | D4 | — | 472 | 36 | no | — |
| 196 | D4 | — | 236 | 12 | no | — |
| 197 | C2×C4 | — | 2 | 48 | no | — |
| 198 | C2×C4 | — | 2 | 72 | no | — |
| 199 | C2×C4 | — | 118 | 12 | no | — |
| 200 | C2×C4 | — | 118 | 3 | no | — |
| 201 | C2×C4 | — | 236 | 12 | no | — |
| 202 | C2×C4 | C22×C22⋊D59 | 1 | 12 | yes | — |
| 203 | C2×C4 | C22×C22⋊D59 | 1 | 18 | yes | — |
| 204 | C22 | — | 2 | 12 | no | — |
| 205 | C22 | — | 4 | 1 | no | — |
| 206 | C22 | — | 4 | 12 | no | — |
| 207 | C22 | — | 8 | 16 | no | — |
| 208 | C22 | — | 236 | 18 | no | — |
| 209 | C22 | — | 236 | 3 | no | — |
| 210 | C22 | C23×C22⋊D59 | 1 | 3 | yes | — |
| 211 | C4 | — | 2 | 12 | no | — |
| 212 | C4 | — | 236 | 12 | no | — |
| 213 | C4 | — | 236 | 1 | no | — |
| 214 | C4 | C2×#14273696 | 1 | 12 | yes | — |
| 215 | C2 | — | 2 | 3 | no | — |
| 216 | C2 | — | 4 | 12 | no | — |
| 217 | C2 | — | 472 | 3 | no | — |
| 218 | C2 | C22×#14273696 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21437751 | C2 | semidirect |
| 2. #21433498 | C2 | semidirect |
| 3. #21433488 | C2 | semidirect |
| 4. #21432560 | C2 | semidirect |
| 5. C59×2−1+6 | C2 | semidirect |
| 6. C22×#11745689 | C22 | semidirect |
| 9. #16837966 | C22 | semidirect |
| 10. #16837909 | C22 | semidirect |
| 11. #16837880 | C22 | semidirect |
| 12. #16837592 | C22 | semidirect |
| 13. #16837589 | C22 | semidirect |
| 14. #16837376 | C22 | semidirect |
| 41. C2×#41×C59 | D4 | semidirect |
| 44. C22×Q8×C59 | D4 | semidirect |
| 106. 2−1+6 | D59 | semidirect |
| 128. C59 | #60402 | semidirect |
| 138. C2×#41 | C22⋊D59 | semidirect |
| 141. C22×Q8 | C22⋊D59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21437751 | 2 |
| #21433498 | 2 |
| #21433488 | 2 |
| #21432560 | 2 |
| C59×2−1+6 | 2 |
| #60402 | 59 |