Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1474 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 153 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 | C2×C5×#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11756908 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 3 | C2×#91955 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#91954 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#66170 | — | 3 | 1 | no | maximal, Hall |
| 6 | C2×Q8×D15 | — | 4 | 1 | no | maximal |
| 7 | C5×#994 | C22 | 1 | 4 | yes | complemented |
| 8 | C2×C5×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 9 | #91955 | C22 | 1 | 12 | yes | complemented |
| 10 | #91954 | C22 | 1 | 12 | yes | complemented |
| 11 | C2×C5⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×C5⋊#226 | C22 | 1 | 3 | yes | complemented |
| 13 | C2×#5924 | — | 5 | 1 | no | maximal, Hall |
| 14 | C2×C5×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×#4150 | — | 3 | 1 | no | — |
| 16 | #66170 | — | 3 | 8 | no | — |
| 17 | #66170 | — | 3 | 8 | no | — |
| 18 | C2×#4097 | — | 3 | 3 | no | — |
| 19 | C2×#4096 | — | 3 | 3 | no | — |
| 20 | C2×#4094 | — | 3 | 1 | no | — |
| 21 | C22×#687 | — | 3 | 3 | no | — |
| 22 | C2×#4078 | — | 3 | 3 | no | — |
| 23 | C2×C3×C5×Q8 | — | 4 | 1 | no | — |
| 24 | Q8×D15 | — | 4 | 8 | no | — |
| 25 | C2×C4×D15 | — | 4 | 3 | no | — |
| 26 | C2×Dic30 | — | 4 | 3 | no | — |
| 27 | C5×#41⋊C3 | C23 | 1 | 6 | yes | — |
| 28 | C2×C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 29 | C5⋊GL(2,𝔽3) | C23 | 1 | 2 | yes | complemented |
| 30 | C5⋊#226 | C23 | 1 | 6 | yes | complemented |
| 31 | C2×#994 | D5 | 1 | 1 | yes | characteristic, complemented |
| 32 | #5924 | — | 5 | 8 | no | — |
| 33 | C2×#985 | — | 5 | 3 | no | — |
| 34 | C2×#984 | — | 5 | 3 | no | — |
| 35 | C5×H(5,𝔽2) | — | 3 | 4 | no | — |
| 36 | C5×H(5,𝔽2) | C2×D3 | 1 | 4 | yes | complemented |
| 37 | C2×C5×#41 | — | 3 | 1 | no | — |
| 38 | C2×C5×#41 | C2×D3 | 1 | 3 | yes | — |
| 39 | C22×C5×D4 | — | 3 | 3 | no | — |
| 40 | #4150 | — | 3 | 8 | no | — |
| 41 | #4150 | — | 6 | 4 | no | — |
| 42 | C2×Q8×D5 | — | 6 | 1 | no | — |
| 43 | C2×#711 | — | 6 | 3 | no | — |
| 44 | C2×#709 | — | 3 | 3 | no | — |
| 45 | C22×Dic10 | — | 3 | 3 | no | — |
| 46 | #4097 | — | 3 | 12 | no | — |
| 47 | #4097 | — | 3 | 12 | no | — |
| 48 | #4096 | — | 3 | 12 | no | — |
| 49 | #4096 | — | 3 | 12 | no | — |
| 50 | #4094 | — | 3 | 8 | no | — |
| 51 | C2×#689 | — | 3 | 3 | no | — |
| 52 | C2×#688 | — | 3 | 1 | no | — |
| 53 | C2×#687 | — | 3 | 6 | no | — |
| 54 | C2×#687 | — | 3 | 3 | no | — |
| 55 | C2×#687 | — | 3 | 24 | no | — |
| 56 | #4078 | — | 3 | 24 | no | — |
| 57 | C2×#686 | — | 3 | 3 | no | — |
| 58 | C2×#681 | — | 3 | 3 | no | — |
| 59 | C22×#168 | — | 3 | 3 | no | — |
| 60 | C2×#577 | — | 15 | 1 | no | Sylow |
| 61 | C22×D15 | — | 4 | 1 | no | — |
| 62 | C3×C5×Q8 | — | 4 | 4 | no | — |
| 63 | C2×C3×C4×C5 | — | 4 | 3 | no | — |
| 64 | C2×Dic15 | — | 4 | 3 | no | — |
| 65 | C4×D15 | — | 4 | 12 | no | — |
| 66 | Dic30 | — | 4 | 12 | no | — |
| 67 | C5×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 68 | C2×D3×Q8 | — | 20 | 1 | no | — |
| 69 | #994 | C2×D5 | 1 | 4 | yes | complemented |
| 70 | C2×#41⋊C3 | C2×D5 | 1 | 3 | yes | — |
| 71 | #985 | — | 5 | 12 | no | — |
| 72 | #984 | — | 5 | 12 | no | — |
| 73 | C2×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 74 | C2×#226 | — | 5 | 3 | no | — |
| 75 | C24×C5 | — | 6 | 1 | no | — |
| 76 | C5×#41 | — | 3 | 4 | no | — |
| 77 | C5×#41 | — | 3 | 4 | no | — |
| 78 | C5×#41 | — | 3 | 6 | no | — |
| 79 | C5×#41 | C22×D3 | 1 | 6 | yes | — |
| 80 | C2×C5×Q8 | S4 | 1 | 1 | yes | characteristic |
| 81 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 82 | C2×C5×D4 | — | 3 | 12 | no | — |
| 83 | C2×C5×D4 | — | 3 | 3 | no | — |
| 84 | C2×C5×D4 | — | 3 | 3 | no | — |
| 85 | C2×C5×D4 | — | 6 | 6 | no | — |
| 86 | C22×C4×C5 | — | 3 | 3 | no | — |
| 87 | C2×C22⋊D5 | — | 6 | 3 | no | — |
| 88 | C22×Dic5 | — | 6 | 3 | no | — |
| 89 | Q8×D5 | — | 6 | 8 | no | — |
| 90 | #711 | — | 6 | 24 | no | — |
| 91 | #709 | — | 3 | 12 | no | — |
| 92 | #709 | — | 6 | 6 | no | — |
| 93 | C2×D20 | — | 3 | 1 | no | — |
| 94 | C2×C4×D5 | — | 6 | 3 | no | — |
| 95 | C2×Dic10 | — | 3 | 12 | no | — |
| 96 | C2×Dic10 | — | 3 | 3 | no | — |
| 97 | C2×Dic10 | — | 6 | 6 | no | — |
| 98 | C2×Dic10 | — | 6 | 3 | no | — |
| 99 | #689 | — | 3 | 6 | no | — |
| 100 | #689 | — | 3 | 6 | no | — |
| 101 | #688 | — | 3 | 2 | no | — |
| 102 | #688 | — | 3 | 2 | no | — |
| 103 | #687 | — | 3 | 12 | no | — |
| 104 | #687 | — | 3 | 24 | no | — |
| 105 | #686 | — | 3 | 12 | no | — |
| 106 | #681 | — | 3 | 12 | no | — |
| 107 | C2×#168 | — | 3 | 12 | no | — |
| 108 | C2×#168 | — | 3 | 1 | no | — |
| 109 | C2×#168 | — | 3 | 3 | no | — |
| 110 | C2×2−1+4 | — | 15 | 1 | no | — |
| 111 | C2×H(5,𝔽2) | D15 | 1 | 1 | yes | characteristic, complemented |
| 112 | #577 | — | 15 | 8 | no | — |
| 113 | #577 | — | 15 | 8 | no | — |
| 114 | C2×#137 | — | 15 | 3 | no | — |
| 115 | C2×AGL(1,ℤ/8) | — | 15 | 3 | no | — |
| 116 | C2×#135 | — | 15 | 3 | no | — |
| 117 | C22×SD16 | — | 15 | 3 | no | — |
| 118 | C2×#131 | — | 15 | 1 | no | — |
| 119 | C22×C3×C5 | — | 4 | 1 | no | — |
| 120 | C2×D15 | — | 4 | 2 | no | — |
| 121 | C2×D15 | — | 4 | 4 | no | — |
| 122 | C3×C4×C5 | — | 4 | 6 | no | — |
| 123 | Dic15 | — | 4 | 6 | no | — |
| 124 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 125 | D3×Q8 | — | 20 | 8 | no | — |
| 126 | C2×C4×D3 | — | 20 | 3 | no | — |
| 127 | C2×Dic6 | — | 20 | 3 | no | — |
| 128 | #41⋊C3 | C22×D5 | 1 | 6 | yes | — |
| 129 | C2×SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic |
| 130 | GL(2,𝔽3) | — | 5 | 2 | no | — |
| 131 | #226 | — | 5 | 6 | no | — |
| 132 | C23×C5 | — | 3 | 3 | no | — |
| 133 | C23×C5 | — | 6 | 4 | no | — |
| 134 | C23×C5 | — | 24 | 2 | no | — |
| 135 | C22×D5 | — | 6 | 1 | no | — |
| 136 | C5×Q8 | — | 3 | 1 | no | — |
| 137 | C5×Q8 | C2×S4 | 1 | 4 | yes | — |
| 138 | C5×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 139 | C5×D4 | — | 3 | 6 | no | — |
| 140 | C5×D4 | — | 3 | 12 | no | — |
| 141 | C5×D4 | — | 6 | 3 | no | — |
| 142 | C2×C4×C5 | — | 3 | 1 | no | — |
| 143 | C2×C4×C5 | — | 3 | 6 | no | — |
| 144 | C2×C4×C5 | — | 3 | 6 | no | — |
| 145 | C2×C4×C5 | C2×S4 | 1 | 3 | yes | — |
| 146 | C22⋊D5 | — | 6 | 12 | no | — |
| 147 | C2×Dic5 | — | 6 | 12 | no | — |
| 148 | C2×Dic5 | — | 6 | 3 | no | — |
| 149 | D20 | — | 3 | 2 | no | — |
| 150 | D20 | — | 6 | 1 | no | — |
| 151 | C4×D5 | — | 6 | 12 | no | — |
| 152 | Dic10 | — | 3 | 6 | no | — |
| 153 | Dic10 | — | 6 | 3 | no | — |
| 154 | Dic10 | — | 6 | 12 | no | — |
| 155 | #168 | — | 3 | 2 | no | — |
| 156 | #168 | — | 3 | 6 | no | — |
| 157 | 2−1+4 | — | 15 | 8 | no | — |
| 158 | 2−1+4 | — | 30 | 4 | no | — |
| 159 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 160 | H(5,𝔽2) | C2×D15 | 1 | 4 | yes | complemented |
| 161 | C2×#41 | — | 3 | 1 | no | — |
| 162 | C2×#41 | — | 15 | 3 | no | — |
| 163 | C2×#41 | — | 30 | 3 | no | — |
| 164 | C2×#41 | C2×D15 | 1 | 3 | yes | — |
| 165 | C22×Q8 | — | 15 | 3 | no | — |
| 166 | C22×Q8 | — | 30 | 1 | no | — |
| 167 | C22×D4 | — | 3 | 3 | no | — |
| 168 | #137 | — | 15 | 12 | no | — |
| 169 | #137 | — | 15 | 12 | no | — |
| 170 | AGL(1,ℤ/8) | — | 15 | 24 | no | — |
| 171 | #135 | — | 15 | 12 | no | — |
| 172 | #135 | — | 15 | 12 | no | — |
| 173 | C2×Q16 | — | 15 | 3 | no | — |
| 174 | C2×SD16 | — | 15 | 3 | no | — |
| 175 | C2×SD16 | — | 15 | 6 | no | — |
| 176 | C2×SD16 | — | 15 | 24 | no | — |
| 177 | C2×SD16 | — | 15 | 1 | no | — |
| 178 | C2×D8 | — | 15 | 3 | no | — |
| 179 | #131 | — | 15 | 8 | no | — |
| 180 | C2×M16 | — | 15 | 3 | no | — |
| 181 | C22×C8 | — | 15 | 3 | no | — |
| 182 | C2×C3×C5 | — | 4 | 2 | no | — |
| 183 | C2×C3×C5 | — | 4 | 1 | no | — |
| 184 | D15 | — | 4 | 4 | no | — |
| 185 | C22×D3 | — | 20 | 1 | no | — |
| 186 | C3×Q8 | — | 4 | 4 | no | — |
| 187 | C2×C3×C4 | — | 4 | 3 | no | — |
| 188 | C2×Dic3 | — | 20 | 3 | no | — |
| 189 | C4×D3 | — | 20 | 12 | no | — |
| 190 | Dic6 | — | 20 | 12 | no | — |
| 191 | SL(2,𝔽3) | C23×D5 | 1 | 1 | yes | characteristic |
| 192 | C22×C5 | — | 3 | 6 | no | — |
| 193 | C22×C5 | — | 6 | 6 | no | — |
| 194 | C22×C5 | — | 24 | 4 | no | — |
| 195 | C22×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 196 | C2×D5 | — | 6 | 2 | no | — |
| 197 | C2×D5 | — | 12 | 2 | no | — |
| 198 | C4×C5 | — | 3 | 1 | no | — |
| 199 | C4×C5 | — | 3 | 1 | no | — |
| 200 | C4×C5 | C22×S4 | 1 | 6 | yes | — |
| 201 | Dic5 | — | 6 | 6 | no | — |
| 202 | C24 | — | 6 | 1 | no | — |
| 203 | #41 | — | 3 | 4 | no | — |
| 204 | #41 | — | 3 | 4 | no | — |
| 205 | #41 | — | 3 | 6 | no | — |
| 206 | #41 | — | 15 | 12 | no | — |
| 207 | #41 | — | 30 | 6 | no | — |
| 208 | #41 | — | 30 | 24 | no | — |
| 209 | #41 | C22×D15 | 1 | 6 | yes | — |
| 210 | C2×Q8 | — | 15 | 3 | no | — |
| 211 | C2×Q8 | — | 15 | 12 | no | — |
| 212 | C2×Q8 | — | 30 | 6 | no | — |
| 213 | C2×Q8 | — | 30 | 8 | no | — |
| 214 | C2×Q8 | — | 30 | 3 | no | — |
| 215 | C2×Q8 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 216 | C2×Q8 | C22×D15 | 1 | 1 | yes | characteristic |
| 217 | C2×D4 | — | 3 | 12 | no | — |
| 218 | C2×D4 | — | 3 | 3 | no | — |
| 219 | C2×D4 | — | 3 | 3 | no | — |
| 220 | C2×D4 | — | 6 | 6 | no | — |
| 221 | C2×D4 | — | 15 | 1 | no | — |
| 222 | C2×D4 | — | 30 | 3 | no | — |
| 223 | C22×C4 | — | 3 | 3 | no | — |
| 224 | C22×C4 | — | 30 | 3 | no | — |
| 225 | C22×C4 | — | 30 | 3 | no | — |
| 226 | Q16 | — | 15 | 6 | no | — |
| 227 | Q16 | — | 15 | 6 | no | — |
| 228 | SD16 | — | 15 | 12 | no | — |
| 229 | SD16 | — | 15 | 24 | no | — |
| 230 | SD16 | — | 15 | 2 | no | — |
| 231 | SD16 | — | 15 | 2 | no | — |
| 232 | D8 | — | 15 | 12 | no | — |
| 233 | M16 | — | 15 | 12 | no | — |
| 234 | C2×C8 | — | 15 | 1 | no | — |
| 235 | C2×C8 | — | 15 | 3 | no | — |
| 236 | C2×C8 | — | 15 | 12 | no | — |
| 237 | C3×C5 | — | 4 | 1 | no | Hall |
| 238 | C22×C3 | — | 4 | 1 | no | — |
| 239 | C2×D3 | — | 20 | 2 | no | — |
| 240 | C2×D3 | — | 20 | 4 | no | — |
| 241 | C3×C4 | — | 4 | 6 | no | — |
| 242 | Dic3 | — | 20 | 6 | no | — |
| 243 | C2×C5 | — | 6 | 6 | no | — |
| 244 | C2×C5 | #5924 | 1 | 2 | yes | complemented |
| 245 | C2×C5 | C23×S4 | 1 | 1 | yes | characteristic |
| 246 | D5 | — | 12 | 2 | no | — |
| 247 | C23 | — | 3 | 3 | no | — |
| 248 | C23 | — | 6 | 4 | no | — |
| 249 | C23 | — | 24 | 2 | no | — |
| 250 | C23 | — | 30 | 1 | no | — |
| 251 | Q8 | — | 3 | 1 | no | — |
| 252 | Q8 | — | 15 | 6 | no | — |
| 253 | Q8 | — | 30 | 3 | no | — |
| 254 | Q8 | — | 30 | 12 | no | — |
| 255 | Q8 | C2×C5⋊S4 | 1 | 4 | yes | — |
| 256 | Q8 | C23×D15 | 1 | 1 | yes | characteristic |
| 257 | D4 | — | 3 | 6 | no | — |
| 258 | D4 | — | 3 | 12 | no | — |
| 259 | D4 | — | 6 | 3 | no | — |
| 260 | D4 | — | 15 | 2 | no | — |
| 261 | D4 | — | 30 | 1 | no | — |
| 262 | D4 | — | 30 | 12 | no | — |
| 263 | C2×C4 | — | 3 | 1 | no | — |
| 264 | C2×C4 | — | 3 | 6 | no | — |
| 265 | C2×C4 | — | 3 | 6 | no | — |
| 266 | C2×C4 | — | 30 | 12 | no | — |
| 267 | C2×C4 | — | 30 | 3 | no | — |
| 268 | C2×C4 | — | 30 | 12 | no | — |
| 269 | C2×C4 | C2×C5⋊S4 | 1 | 3 | yes | — |
| 270 | C8 | — | 15 | 2 | no | — |
| 271 | C8 | — | 15 | 6 | no | — |
| 272 | C2×C3 | — | 4 | 2 | no | — |
| 273 | C2×C3 | — | 4 | 1 | no | — |
| 274 | D3 | — | 20 | 4 | no | — |
| 275 | C5 | C2×#5924 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 276 | C22 | — | 3 | 6 | no | — |
| 277 | C22 | — | 6 | 6 | no | — |
| 278 | C22 | — | 24 | 4 | no | — |
| 279 | C22 | — | 30 | 2 | no | — |
| 280 | C22 | — | 60 | 2 | no | — |
| 281 | C22 | C22×C5⋊S4 | 1 | 1 | yes | characteristic |
| 282 | C4 | — | 3 | 1 | no | — |
| 283 | C4 | — | 3 | 1 | no | — |
| 284 | C4 | — | 30 | 6 | no | — |
| 285 | C4 | C22×C5⋊S4 | 1 | 6 | yes | — |
| 286 | C3 | — | 4 | 1 | no | Sylow |
| 287 | C2 | — | 6 | 6 | no | — |
| 288 | C2 | — | 60 | 2 | no | — |
| 289 | C2 | #11756908 | 1 | 2 | yes | direct product factor |
| 290 | C2 | C23×C5⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C5⋊S4 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×C5⋊S4 |
| Fitting | C2×C5×H(5,𝔽2) | D3 |
| socle | C22×C5 | C22×S4 |
| hypercenter | C2×Q8 | C5⋊S4 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×#994 | C2 | semidirect |
| 2. #11756908 | C2 | direct (N × H) |
| 3. C2×#91955 | C2 | semidirect |
| 4. C2×#91954 | C2 | semidirect |
| 7. C5×#994 | C22 | semidirect |
| 9. #91955 | C22 | semidirect |
| 10. #91954 | C22 | semidirect |
| 11. C2×C5⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C2×C5⋊#226 | C22 | semidirect |
| 14. C2×C5×H(5,𝔽2) | D3 | semidirect |
| 29. C5⋊GL(2,𝔽3) | C23 | semidirect |
| 30. C5⋊#226 | C23 | semidirect |
| 31. C2×#994 | D5 | semidirect |
| 36. C5×H(5,𝔽2) | C2×D3 | semidirect |
| 69. #994 | C2×D5 | semidirect |
| 111. C2×H(5,𝔽2) | D15 | semidirect |
| 160. H(5,𝔽2) | C2×D15 | semidirect |
| 244. C2×C5 | #5924 | semidirect |
| 275. C5 | C2×#5924 | semidirect |
| 289. C2 | #11756908 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#994 | 2 |
| #11756908 | 2 |
| C2×#91955 | 2 |
| C2×#91954 | 2 |
| C2×#66170 | 3 |
| C2×Q8×D15 | 4 |
| C2×#5924 | 5 |