← #13046021
Subgroups — #13046021
Order 1280.
11222 conjugacy classes of subgroups.
The subgroup lattice has 254 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | D5×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | AGL(1,𝔽5)×2−1+4 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | AGL(1,𝔽5)×H(5,𝔽2) | C2 | 1 | 20 | yes | maximal, complemented |
| 4 | #10619412 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C2×#66266 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | C5×#585 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #66287 | C4 | 1 | 6 | yes | complemented |
| 8 | D5×2−1+4 | C22 | 1 | 6 | yes | — |
| 9 | #66285 | C4 | 1 | 10 | yes | complemented |
| 10 | D5×H(5,𝔽2) | C22 | 1 | 10 | yes | — |
| 11 | #66283 | C4 | 1 | 15 | yes | complemented |
| 12 | C2×D5×#41 | C22 | 1 | 15 | yes | — |
| 13 | #66266 | C22 | 1 | 60 | yes | complemented |
| 14 | #66266 | C22 | 1 | 120 | yes | complemented |
| 15 | #66266 | C22 | 1 | 20 | yes | complemented |
| 16 | #41×AGL(1,𝔽5) | C22 | 1 | 120 | yes | complemented |
| 17 | #66262 | C22 | 1 | 30 | yes | complemented |
| 18 | C2×Q8×AGL(1,𝔽5) | C22 | 1 | 30 | yes | complemented |
| 19 | #66258 | C22 | 1 | 90 | yes | complemented |
| 20 | C2×D4×AGL(1,𝔽5) | C22 | 1 | 90 | yes | complemented |
| 21 | C2×#4130 | C22 | 1 | 45 | yes | complemented |
| 22 | C22×C4⋊AGL(1,𝔽5) | C22 | 1 | 15 | yes | complemented |
| 23 | #62654 | — | 5 | 1 | no | maximal, Sylow |
| 24 | C23×AGL(1,𝔽5) | C23 | 1 | 30 | yes | — |
| 25 | C5×2−1+4 | C2×C4 | 1 | 6 | yes | complemented |
| 26 | C5×H(5,𝔽2) | C2×C4 | 1 | 10 | yes | complemented |
| 27 | C2×C5×#41 | C2×C4 | 1 | 15 | yes | complemented |
| 28 | #4150 | C2×C4 | 1 | 60 | yes | complemented |
| 29 | #4149 | C2×C4 | 1 | 60 | yes | complemented |
| 30 | D5×#41 | C2×C4 | 1 | 60 | yes | complemented |
| 31 | D5×#41 | C2×C4 | 1 | 60 | yes | complemented |
| 32 | D5×#41 | C23 | 1 | 60 | yes | — |
| 33 | D5×#41 | C23 | 1 | 20 | yes | — |
| 34 | #4147 | C2×C4 | 1 | 30 | yes | complemented |
| 35 | C2×#713 | C2×C4 | 1 | 15 | yes | complemented |
| 36 | C2×Q8×D5 | C23 | 1 | 15 | yes | — |
| 37 | #4144 | C2×C4 | 1 | 90 | yes | complemented |
| 38 | C2×#711 | C2×C4 | 1 | 45 | yes | complemented |
| 39 | C2×D4×D5 | C23 | 1 | 45 | yes | — |
| 40 | C2×#709 | C2×C4 | 1 | 45 | yes | complemented |
| 41 | C22×C4×D5 | C23 | 1 | 15 | yes | — |
| 42 | C2×C22⋊AGL(1,𝔽5) | C23 | 1 | 90 | yes | — |
| 43 | Q8×AGL(1,𝔽5) | C23 | 1 | 120 | yes | — |
| 44 | Q8×AGL(1,𝔽5) | C23 | 1 | 40 | yes | — |
| 45 | D4×AGL(1,𝔽5) | C23 | 1 | 360 | yes | — |
| 46 | D4×AGL(1,𝔽5) | C23 | 1 | 120 | yes | — |
| 47 | #4130 | C23 | 1 | 60 | yes | — |
| 48 | #4130 | C23 | 1 | 180 | yes | — |
| 49 | C2×C4⋊AGL(1,𝔽5) | C23 | 1 | 45 | yes | — |
| 50 | C2×C4⋊AGL(1,𝔽5) | C23 | 1 | 90 | yes | — |
| 51 | C2×C4⋊AGL(1,𝔽5) | C23 | 1 | 15 | yes | — |
| 52 | C2×C4×AGL(1,𝔽5) | C23 | 1 | 90 | yes | — |
| 53 | C2×#585 | — | 5 | 1 | no | — |
| 54 | C4×2−1+4 | — | 5 | 12 | no | — |
| 55 | C4×H(5,𝔽2) | — | 5 | 20 | no | — |
| 56 | #3428 | — | 5 | 15 | no | — |
| 57 | C2×#520 | — | 5 | 15 | no | — |
| 58 | C23×D5 | C24 | 1 | 15 | yes | — |
| 59 | C22×AGL(1,𝔽5) | — | 8 | 30 | no | — |
| 60 | C22×AGL(1,𝔽5) | C24 | 1 | 30 | yes | — |
| 61 | C22×AGL(1,𝔽5) | C24 | 1 | 90 | yes | — |
| 62 | C5×#41 | C22×C4 | 1 | 20 | yes | complemented |
| 63 | C5×#41 | C22×C4 | 1 | 60 | yes | complemented |
| 64 | C2×C5×Q8 | C22×C4 | 1 | 15 | yes | complemented |
| 65 | C2×C5×D4 | C22×C4 | 1 | 45 | yes | complemented |
| 66 | C22×C4×C5 | C22×C4 | 1 | 15 | yes | complemented |
| 67 | C2×C22⋊D5 | C22×C4 | 1 | 90 | yes | complemented |
| 68 | C22×Dic5 | C22×C4 | 1 | 15 | yes | complemented |
| 69 | #713 | C22×C4 | 1 | 20 | yes | complemented |
| 70 | #713 | C22×C4 | 1 | 60 | yes | complemented |
| 71 | Q8×D5 | C22×C4 | 1 | 60 | yes | complemented |
| 72 | Q8×D5 | C24 | 1 | 20 | yes | — |
| 73 | #711 | C22×C4 | 1 | 60 | yes | complemented |
| 74 | #711 | C22×C4 | 1 | 180 | yes | complemented |
| 75 | D4×D5 | C22×C4 | 1 | 180 | yes | complemented |
| 76 | D4×D5 | C24 | 1 | 60 | yes | — |
| 77 | #709 | C22×C4 | 1 | 60 | yes | complemented |
| 78 | #709 | C22×C4 | 1 | 180 | yes | complemented |
| 79 | C2×D20 | C22×C4 | 1 | 45 | yes | complemented |
| 80 | C2×C4×D5 | C22×C4 | 1 | 45 | yes | complemented |
| 81 | C2×C4×D5 | C22×C4 | 1 | 45 | yes | complemented |
| 82 | C2×C4×D5 | C24 | 1 | 45 | yes | — |
| 83 | C2×C4×D5 | C24 | 1 | 15 | yes | — |
| 84 | C2×Dic10 | C22×C4 | 1 | 45 | yes | complemented |
| 85 | C22⋊AGL(1,𝔽5) | C24 | 1 | 120 | yes | — |
| 86 | C4⋊AGL(1,𝔽5) | C24 | 1 | 10 | yes | — |
| 87 | C4⋊AGL(1,𝔽5) | C24 | 1 | 6 | yes | — |
| 88 | C4⋊AGL(1,𝔽5) | C24 | 1 | 90 | yes | — |
| 89 | C4⋊AGL(1,𝔽5) | C24 | 1 | 30 | yes | — |
| 90 | C4×AGL(1,𝔽5) | C24 | 1 | 120 | yes | — |
| 91 | #585 | — | 5 | 6 | no | — |
| 92 | #585 | — | 5 | 10 | no | — |
| 93 | #585 | — | 5 | 15 | no | — |
| 94 | #585 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 95 | C2×2−1+4 | — | 5 | 6 | no | — |
| 96 | C2×H(5,𝔽2) | — | 5 | 10 | no | — |
| 97 | C22×#41 | — | 5 | 15 | no | — |
| 98 | #520 | — | 5 | 60 | no | — |
| 99 | #520 | — | 5 | 120 | no | — |
| 100 | #520 | — | 5 | 20 | no | — |
| 101 | #519 | — | 5 | 30 | no | — |
| 102 | #518 | — | 5 | 90 | no | — |
| 103 | C4×#41 | — | 5 | 120 | no | — |
| 104 | C2×C4×Q8 | — | 5 | 30 | no | — |
| 105 | C2×C4×D4 | — | 5 | 90 | no | — |
| 106 | C2×#117 | — | 5 | 45 | no | — |
| 107 | C22×C4⋊C4 | — | 5 | 15 | no | — |
| 108 | C23×C5 | C23×C4 | 1 | 15 | yes | — |
| 109 | C22×D5 | — | 4 | 15 | no | — |
| 110 | C22×D5 | — | 4 | 15 | no | — |
| 111 | C22×D5 | C23×C4 | 1 | 45 | yes | — |
| 112 | C22×D5 | C25 | 1 | 15 | yes | — |
| 113 | C2×AGL(1,𝔽5) | — | 4 | 90 | no | — |
| 114 | C2×AGL(1,𝔽5) | — | 4 | 30 | no | — |
| 115 | C2×AGL(1,𝔽5) | C25 | 1 | 12 | yes | — |
| 116 | C2×AGL(1,𝔽5) | C25 | 1 | 20 | yes | — |
| 117 | C5×Q8 | C23×C4 | 1 | 20 | yes | — |
| 118 | C5×D4 | C23×C4 | 1 | 60 | yes | — |
| 119 | C2×C4×C5 | C23×C4 | 1 | 45 | yes | — |
| 120 | C2×C4×C5 | C23×C4 | 1 | 15 | yes | — |
| 121 | C22⋊D5 | C23×C4 | 1 | 120 | yes | — |
| 122 | C2×Dic5 | C23×C4 | 1 | 45 | yes | — |
| 123 | C2×Dic5 | C23×C4 | 1 | 15 | yes | — |
| 124 | D20 | C23×C4 | 1 | 60 | yes | — |
| 125 | C4×D5 | C23×C4 | 1 | 15 | yes | — |
| 126 | C4×D5 | C23×C4 | 1 | 15 | yes | — |
| 127 | C4×D5 | C23×C4 | 1 | 90 | yes | — |
| 128 | C4×D5 | C25 | 1 | 15 | yes | — |
| 129 | C4×D5 | C25 | 1 | 1 | yes | characteristic |
| 130 | Dic10 | C23×C4 | 1 | 60 | yes | — |
| 131 | 2−1+4 | — | 5 | 60 | no | — |
| 132 | 2−1+4 | — | 5 | 30 | no | — |
| 133 | 2−1+4 | C2×AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 134 | H(5,𝔽2) | — | 5 | 60 | no | — |
| 135 | H(5,𝔽2) | — | 5 | 90 | no | — |
| 136 | H(5,𝔽2) | C2×AGL(1,𝔽5) | 1 | 10 | yes | complemented |
| 137 | C2×#41 | — | 5 | 60 | no | — |
| 138 | C2×#41 | — | 5 | 20 | no | — |
| 139 | C2×#41 | — | 5 | 60 | no | — |
| 140 | C2×#41 | — | 5 | 60 | no | — |
| 141 | C2×#41 | — | 5 | 15 | no | — |
| 142 | C2×#41 | — | 5 | 45 | no | — |
| 143 | C2×#41 | — | 5 | 45 | no | — |
| 144 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 15 | yes | complemented |
| 145 | C22×Q8 | — | 5 | 15 | no | — |
| 146 | C22×D4 | — | 5 | 45 | no | — |
| 147 | C23×C4 | — | 5 | 15 | no | — |
| 148 | C23×C4 | — | 5 | 30 | no | — |
| 149 | C4×Q8 | — | 5 | 120 | no | — |
| 150 | C4×Q8 | — | 5 | 40 | no | — |
| 151 | C4×D4 | — | 5 | 360 | no | — |
| 152 | C4×D4 | — | 5 | 120 | no | — |
| 153 | #117 | — | 5 | 60 | no | — |
| 154 | #117 | — | 5 | 180 | no | — |
| 155 | C2×C4⋊C4 | — | 5 | 45 | no | — |
| 156 | C2×C4⋊C4 | — | 5 | 90 | no | — |
| 157 | C2×C4⋊C4 | — | 5 | 15 | no | — |
| 158 | C2×C22⋊C4 | — | 5 | 90 | no | — |
| 159 | C2×C42 | — | 5 | 90 | no | — |
| 160 | C22×C5 | — | 4 | 15 | no | — |
| 161 | C22×C5 | C24×C4 | 1 | 15 | yes | — |
| 162 | C2×D5 | — | 2 | 15 | no | — |
| 163 | C2×D5 | — | 2 | 15 | no | — |
| 164 | C2×D5 | — | 4 | 45 | no | — |
| 165 | C2×D5 | C24×C4 | 1 | 15 | yes | — |
| 166 | C2×D5 | C26 | 1 | 1 | yes | characteristic |
| 167 | AGL(1,𝔽5) | — | 2 | 20 | no | — |
| 168 | AGL(1,𝔽5) | — | 2 | 12 | no | — |
| 169 | C4×C5 | C24×C4 | 1 | 15 | yes | — |
| 170 | C4×C5 | C24×C4 | 1 | 1 | yes | characteristic |
| 171 | Dic5 | C24×C4 | 1 | 15 | yes | — |
| 172 | Dic5 | C24×C4 | 1 | 1 | yes | characteristic |
| 173 | C24 | — | 5 | 15 | no | — |
| 174 | #41 | — | 5 | 20 | no | — |
| 175 | #41 | — | 5 | 60 | no | — |
| 176 | #41 | — | 5 | 60 | no | — |
| 177 | #41 | — | 5 | 180 | no | — |
| 178 | #41 | — | 5 | 60 | no | — |
| 179 | #41 | — | 5 | 180 | no | — |
| 180 | #41 | C22×AGL(1,𝔽5) | 1 | 20 | yes | complemented |
| 181 | #41 | C22×AGL(1,𝔽5) | 1 | 60 | yes | complemented |
| 182 | C2×Q8 | — | 5 | 20 | no | — |
| 183 | C2×Q8 | — | 5 | 60 | no | — |
| 184 | C2×Q8 | — | 5 | 45 | no | — |
| 185 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 15 | yes | complemented |
| 186 | C2×D4 | — | 5 | 90 | no | — |
| 187 | C2×D4 | — | 5 | 60 | no | — |
| 188 | C2×D4 | — | 5 | 180 | no | — |
| 189 | C2×D4 | — | 5 | 45 | no | — |
| 190 | C2×D4 | C22×AGL(1,𝔽5) | 1 | 45 | yes | complemented |
| 191 | C22×C4 | — | 5 | 15 | no | — |
| 192 | C22×C4 | — | 5 | 45 | no | — |
| 193 | C22×C4 | — | 5 | 15 | no | — |
| 194 | C22×C4 | — | 5 | 45 | no | — |
| 195 | C22×C4 | — | 5 | 45 | no | — |
| 196 | C22×C4 | — | 5 | 30 | no | — |
| 197 | C22×C4 | — | 5 | 90 | no | — |
| 198 | C22×C4 | — | 40 | 30 | no | — |
| 199 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 15 | yes | complemented |
| 200 | C4⋊C4 | — | 5 | 10 | no | — |
| 201 | C4⋊C4 | — | 5 | 6 | no | — |
| 202 | C4⋊C4 | — | 5 | 90 | no | — |
| 203 | C4⋊C4 | — | 5 | 30 | no | — |
| 204 | C22⋊C4 | — | 5 | 120 | no | — |
| 205 | C42 | — | 5 | 120 | no | — |
| 206 | C2×C5 | — | 2 | 15 | no | — |
| 207 | C2×C5 | C25×C4 | 1 | 1 | yes | characteristic |
| 208 | D5 | — | 2 | 15 | no | — |
| 209 | D5 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 210 | D5 | 2−1+6 | 1 | 1 | yes | characteristic |
| 211 | C23 | — | 5 | 15 | no | — |
| 212 | C23 | — | 5 | 45 | no | — |
| 213 | C23 | — | 20 | 15 | no | — |
| 214 | C23 | — | 20 | 15 | no | — |
| 215 | C23 | C23×AGL(1,𝔽5) | 1 | 15 | yes | — |
| 216 | Q8 | — | 5 | 60 | no | — |
| 217 | Q8 | C23×AGL(1,𝔽5) | 1 | 20 | yes | — |
| 218 | D4 | — | 5 | 120 | no | — |
| 219 | D4 | — | 5 | 60 | no | — |
| 220 | D4 | C23×AGL(1,𝔽5) | 1 | 60 | yes | — |
| 221 | C2×C4 | — | 5 | 15 | no | — |
| 222 | C2×C4 | — | 5 | 45 | no | — |
| 223 | C2×C4 | — | 5 | 15 | no | — |
| 224 | C2×C4 | — | 5 | 1 | no | — |
| 225 | C2×C4 | — | 5 | 15 | no | — |
| 226 | C2×C4 | — | 5 | 15 | no | — |
| 227 | C2×C4 | — | 5 | 90 | no | — |
| 228 | C2×C4 | — | 5 | 12 | no | — |
| 229 | C2×C4 | — | 5 | 20 | no | — |
| 230 | C2×C4 | — | 20 | 90 | no | — |
| 231 | C2×C4 | — | 20 | 30 | no | — |
| 232 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 15 | yes | — |
| 233 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 45 | yes | — |
| 234 | C5 | #62654 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 235 | C22 | — | 4 | 15 | no | — |
| 236 | C22 | — | 5 | 1 | no | — |
| 237 | C22 | — | 5 | 15 | no | — |
| 238 | C22 | — | 10 | 15 | no | — |
| 239 | C22 | — | 10 | 15 | no | — |
| 240 | C22 | — | 20 | 45 | no | — |
| 241 | C22 | C24×AGL(1,𝔽5) | 1 | 15 | yes | — |
| 242 | C4 | — | 5 | 1 | no | — |
| 243 | C4 | — | 5 | 15 | no | — |
| 244 | C4 | — | 10 | 20 | no | — |
| 245 | C4 | — | 10 | 12 | no | — |
| 246 | C4 | C24×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 247 | C4 | C24×AGL(1,𝔽5) | 1 | 15 | yes | — |
| 248 | C2 | — | 2 | 15 | no | — |
| 249 | C2 | — | 5 | 1 | no | — |
| 250 | C2 | — | 5 | 1 | no | — |
| 251 | C2 | — | 10 | 15 | no | — |
| 252 | C2 | C25×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 2. AGL(1,𝔽5)×2−1+4 | C2 | semidirect |
| 3. AGL(1,𝔽5)×H(5,𝔽2) | C2 | semidirect |
| 4. #10619412 | C2 | semidirect |
| 5. C2×#66266 | C2 | semidirect |
| 6. C5×#585 | C4 | semidirect |
| 7. #66287 | C4 | semidirect |
| 9. #66285 | C4 | semidirect |
| 11. #66283 | C4 | semidirect |
| 13. #66266 | C22 | semidirect |
| 14. #66266 | C22 | semidirect |
| 15. #66266 | C22 | semidirect |
| 16. #41×AGL(1,𝔽5) | C22 | semidirect |
| 17. #66262 | C22 | semidirect |
| 18. C2×Q8×AGL(1,𝔽5) | C22 | semidirect |
| 19. #66258 | C22 | semidirect |
| 20. C2×D4×AGL(1,𝔽5) | C22 | semidirect |
| 21. C2×#4130 | C22 | semidirect |
| 22. C22×C4⋊AGL(1,𝔽5) | C22 | semidirect |
| 25. C5×2−1+4 | C2×C4 | semidirect |
| 26. C5×H(5,𝔽2) | C2×C4 | semidirect |
| 27. C2×C5×#41 | C2×C4 | semidirect |
| 28. #4150 | C2×C4 | semidirect |
| 29. #4149 | C2×C4 | semidirect |
| 30. D5×#41 | C2×C4 | semidirect |
| 31. D5×#41 | C2×C4 | semidirect |
| 34. #4147 | C2×C4 | semidirect |
| 35. C2×#713 | C2×C4 | semidirect |
| 37. #4144 | C2×C4 | semidirect |
| 38. C2×#711 | C2×C4 | semidirect |
| 40. C2×#709 | C2×C4 | semidirect |
| 62. C5×#41 | C22×C4 | semidirect |
| 63. C5×#41 | C22×C4 | semidirect |
| 64. C2×C5×Q8 | C22×C4 | semidirect |
| 65. C2×C5×D4 | C22×C4 | semidirect |
| 66. C22×C4×C5 | C22×C4 | semidirect |
| 67. C2×C22⋊D5 | C22×C4 | semidirect |
| 68. C22×Dic5 | C22×C4 | semidirect |
| 69. #713 | C22×C4 | semidirect |
| 70. #713 | C22×C4 | semidirect |
| 71. Q8×D5 | C22×C4 | semidirect |
| 73. #711 | C22×C4 | semidirect |
| 74. #711 | C22×C4 | semidirect |
| 75. D4×D5 | C22×C4 | semidirect |
| 77. #709 | C22×C4 | semidirect |
| 78. #709 | C22×C4 | semidirect |
| 79. C2×D20 | C22×C4 | semidirect |
| 80. C2×C4×D5 | C22×C4 | semidirect |
| 81. C2×C4×D5 | C22×C4 | semidirect |
| 84. C2×Dic10 | C22×C4 | semidirect |
| 94. #585 | AGL(1,𝔽5) | semidirect |
| 133. 2−1+4 | C2×AGL(1,𝔽5) | semidirect |
| 136. H(5,𝔽2) | C2×AGL(1,𝔽5) | semidirect |
| 144. C2×#41 | C2×AGL(1,𝔽5) | semidirect |
| 180. #41 | C22×AGL(1,𝔽5) | semidirect |
| 181. #41 | C22×AGL(1,𝔽5) | semidirect |
| 185. C2×Q8 | C22×AGL(1,𝔽5) | semidirect |
| 190. C2×D4 | C22×AGL(1,𝔽5) | semidirect |
| 199. C22×C4 | C22×AGL(1,𝔽5) | semidirect |
| 234. C5 | #62654 | semidirect |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta