← C4×Q8×#1043 #72223258
Subgroups — C4×Q8×#1043 #72223258
Order 3200.
2736 conjugacy classes of subgroups.
The subgroup lattice has 234 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×Q8×#1043 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×Q8×C5⋊D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C4⋊C4×#1043 | C2 | 1 | 6 | yes | maximal |
| 4 | C4×C4⋊#1043 | C2 | 1 | 3 | yes | maximal |
| 5 | C42×#1043 | C2 | 1 | 3 | yes | maximal |
| 6 | C2×Q8×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | Q8×#1043 | C4 | 1 | 16 | yes | direct product factor |
| 8 | C2×C4⋊#1043 | C22 | 1 | 6 | yes | — |
| 9 | C2×C4×#1043 | C22 | 1 | 3 | yes | — |
| 10 | C2×C4×#1043 | C22 | 1 | 6 | yes | — |
| 11 | C2×C4×#1043 | C22 | 1 | 6 | yes | — |
| 12 | C2×C4×#1043 | C22 | 1 | 1 | yes | characteristic |
| 13 | C4×C52×Q8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | Q8×#1039 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | C4⋊C4×C5⋊D5 | C22 | 1 | 3 | yes | — |
| 16 | #11717207 | C4 | 1 | 3 | yes | complemented |
| 17 | C42×C5⋊D5 | C22 | 1 | 3 | yes | — |
| 18 | C4×#6045 | C4 | 1 | 3 | yes | complemented |
| 19 | #11716874 | C22 | 1 | 6 | yes | — |
| 20 | C4×Q8×AGL(1,𝔽5) | — | 5 | 6 | no | maximal |
| 21 | C22×#1043 | C23 | 1 | 6 | yes | — |
| 22 | C22×#1043 | C23 | 1 | 2 | yes | — |
| 23 | C2×C52×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | Q8×C5⋊D5 | C2×C4 | 1 | 8 | yes | — |
| 25 | C2×C4×C5⋊D5 | C23 | 1 | 3 | yes | — |
| 26 | C2×C4×C5⋊D5 | C23 | 1 | 3 | yes | — |
| 27 | C2×C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×#6045 | C2×C4 | 1 | 3 | yes | — |
| 29 | C4⋊#1043 | C2×C4 | 1 | 24 | yes | — |
| 30 | C4×#1043 | — | 2 | 12 | no | — |
| 31 | C4×#1043 | — | 2 | 6 | no | — |
| 32 | C4×#1043 | C2×C4 | 1 | 24 | yes | — |
| 33 | C4×#1043 | Q8 | 1 | 4 | yes | direct product factor |
| 34 | C52×C4⋊C4 | C2×C4 | 1 | 3 | yes | — |
| 35 | C42×C52 | C2×C4 | 1 | 3 | yes | — |
| 36 | C4⋊#1039 | C2×C4 | 1 | 3 | yes | — |
| 37 | #87806 | C2×C4 | 1 | 6 | yes | — |
| 38 | C4×#1039 | C2×C4 | 1 | 3 | yes | — |
| 39 | C4×#1039 | C2×C4 | 1 | 3 | yes | — |
| 40 | C4×#1039 | C2×C4 | 1 | 3 | yes | — |
| 41 | C2×Q8×AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 42 | C4×Q8×D5 | — | 5 | 6 | no | — |
| 43 | C4⋊C4×AGL(1,𝔽5) | — | 5 | 36 | no | — |
| 44 | C4×C4⋊AGL(1,𝔽5) | — | 5 | 18 | no | — |
| 45 | C42×AGL(1,𝔽5) | — | 5 | 18 | no | — |
| 46 | C22×C5⋊D5 | C24 | 1 | 1 | yes | characteristic |
| 47 | C2×#1043 | — | 2 | 6 | no | — |
| 48 | C2×#1043 | — | 2 | 6 | no | — |
| 49 | C2×#1043 | C22×C4 | 1 | 12 | yes | — |
| 50 | C2×#1043 | C22×C4 | 1 | 4 | yes | — |
| 51 | C2×#1043 | C2×Q8 | 1 | 4 | yes | — |
| 52 | C2×#1043 | #41 | 1 | 4 | yes | — |
| 53 | C52×Q8 | C42 | 1 | 4 | yes | complemented |
| 54 | C2×C4×C52 | C22×C4 | 1 | 3 | yes | — |
| 55 | C2×C4×C52 | C22×C4 | 1 | 3 | yes | — |
| 56 | C2×C4×C52 | C22×C4 | 1 | 1 | yes | characteristic |
| 57 | C2×#1039 | C22×C4 | 1 | 3 | yes | — |
| 58 | C2×#1039 | C22×C4 | 1 | 3 | yes | — |
| 59 | C2×#1039 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | C4×C5⋊D5 | — | 2 | 6 | no | — |
| 61 | C4×C5⋊D5 | C22×C4 | 1 | 12 | yes | — |
| 62 | C4×C5⋊D5 | C2×Q8 | 1 | 2 | yes | — |
| 63 | C4×C5⋊D5 | #41 | 1 | 2 | yes | — |
| 64 | #6045 | C42 | 1 | 12 | yes | complemented |
| 65 | C2×Q8×D5 | — | 5 | 6 | no | — |
| 66 | Q8×AGL(1,𝔽5) | — | 5 | 96 | no | — |
| 67 | C2×C4⋊AGL(1,𝔽5) | — | 5 | 36 | no | — |
| 68 | C2×C4×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 69 | C2×C4×AGL(1,𝔽5) | — | 5 | 36 | no | — |
| 70 | C2×C4×AGL(1,𝔽5) | — | 5 | 18 | no | — |
| 71 | C2×C4×AGL(1,𝔽5) | — | 5 | 36 | no | — |
| 72 | C4×C5×Q8 | AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 73 | Q8×Dic5 | — | 5 | 6 | no | — |
| 74 | D5×C4⋊C4 | — | 5 | 18 | no | — |
| 75 | #4033 | — | 5 | 18 | no | — |
| 76 | C42×D5 | — | 5 | 18 | no | — |
| 77 | C4×Dic10 | — | 5 | 18 | no | — |
| 78 | (C2×AGL(1,𝔽5))⋊C4 | — | 5 | 36 | no | — |
| 79 | C42×Q8 | — | 25 | 1 | no | Sylow |
| 80 | C22×C52 | C23×C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C5⋊D5 | C23×C4 | 1 | 2 | yes | — |
| 82 | C2×C5⋊D5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 83 | C2×C5⋊D5 | C2×#41 | 1 | 2 | yes | — |
| 84 | C2×C5⋊D5 | C2×#41 | 1 | 1 | yes | characteristic |
| 85 | #1043 | — | 2 | 12 | no | — |
| 86 | #1043 | C4×Q8 | 1 | 8 | yes | direct product factor |
| 87 | C4×C52 | — | 2 | 3 | no | — |
| 88 | C4×C52 | C2×C42 | 1 | 6 | yes | — |
| 89 | C4×C52 | C4×Q8 | 1 | 2 | yes | complemented |
| 90 | #1039 | — | 2 | 3 | no | — |
| 91 | #1039 | C2×C42 | 1 | 6 | yes | — |
| 92 | #1039 | C4×Q8 | 1 | 2 | yes | complemented |
| 93 | C22×AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 94 | C22×AGL(1,𝔽5) | — | 5 | 36 | no | — |
| 95 | C2×C5×Q8 | C2×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 96 | Q8×D5 | — | 5 | 48 | no | — |
| 97 | C2×C4×D5 | — | 5 | 6 | no | — |
| 98 | C2×C4×D5 | — | 5 | 18 | no | — |
| 99 | C2×C4×D5 | — | 5 | 18 | no | — |
| 100 | C2×Dic10 | — | 5 | 18 | no | — |
| 101 | C4⋊AGL(1,𝔽5) | — | 5 | 144 | no | — |
| 102 | C4×AGL(1,𝔽5) | — | 5 | 144 | no | — |
| 103 | C4×AGL(1,𝔽5) | — | 5 | 24 | no | — |
| 104 | C4×AGL(1,𝔽5) | — | 10 | 36 | no | — |
| 105 | C4×AGL(1,𝔽5) | — | 10 | 72 | no | — |
| 106 | C5×C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 107 | C42×C5 | C2×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 108 | C4⋊Dic5 | — | 5 | 18 | no | — |
| 109 | #683 | — | 5 | 36 | no | — |
| 110 | C4×Dic5 | — | 5 | 18 | no | — |
| 111 | C4×Dic5 | — | 5 | 18 | no | — |
| 112 | C4×Dic5 | — | 5 | 18 | no | — |
| 113 | C2×C4×Q8 | — | 25 | 2 | no | — |
| 114 | C2×C4×Q8 | — | 25 | 1 | no | — |
| 115 | C4×C4⋊C4 | — | 25 | 3 | no | — |
| 116 | C4×C4⋊C4 | — | 25 | 6 | no | — |
| 117 | C43 | — | 25 | 3 | no | — |
| 118 | C2×C52 | C22×C42 | 1 | 1 | yes | characteristic |
| 119 | C2×C52 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 120 | C2×C52 | C4×#41 | 1 | 1 | yes | characteristic |
| 121 | C5⋊D5 | C2×C4×Q8 | 1 | 2 | yes | — |
| 122 | C5⋊D5 | C4×#41 | 1 | 2 | yes | — |
| 123 | C22×D5 | — | 5 | 6 | no | — |
| 124 | C2×AGL(1,𝔽5) | — | 5 | 24 | no | — |
| 125 | C2×AGL(1,𝔽5) | — | 5 | 24 | no | — |
| 126 | C2×AGL(1,𝔽5) | — | 5 | 24 | no | — |
| 127 | C2×AGL(1,𝔽5) | — | 5 | 72 | no | — |
| 128 | C2×AGL(1,𝔽5) | — | 10 | 36 | no | — |
| 129 | C2×AGL(1,𝔽5) | — | 10 | 36 | no | — |
| 130 | C5×Q8 | C4×AGL(1,𝔽5) | 1 | 24 | yes | complemented |
| 131 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 132 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 133 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 134 | C2×Dic5 | — | 5 | 6 | no | — |
| 135 | C2×Dic5 | — | 5 | 18 | no | — |
| 136 | C2×Dic5 | — | 5 | 18 | no | — |
| 137 | C4×D5 | — | 5 | 72 | no | — |
| 138 | C4×D5 | — | 5 | 12 | no | — |
| 139 | C4×D5 | — | 5 | 12 | no | — |
| 140 | C4×D5 | — | 10 | 36 | no | — |
| 141 | Dic10 | — | 5 | 72 | no | — |
| 142 | C22×Q8 | — | 25 | 1 | no | — |
| 143 | C4×Q8 | — | 25 | 16 | no | — |
| 144 | C4×Q8 | — | 25 | 1 | no | — |
| 145 | C4×Q8 | — | 25 | 3 | no | — |
| 146 | C4×Q8 | — | 25 | 3 | no | — |
| 147 | C4×Q8 | #1043 | 1 | 1 | yes | characteristic, direct product factor |
| 148 | C2×C4⋊C4 | — | 25 | 6 | no | — |
| 149 | C2×C4⋊C4 | — | 25 | 3 | no | — |
| 150 | C2×C42 | — | 25 | 6 | no | — |
| 151 | C2×C42 | — | 25 | 1 | no | — |
| 152 | C2×C42 | — | 25 | 3 | no | — |
| 153 | C2×C42 | — | 25 | 6 | no | — |
| 154 | C2×C42 | — | 25 | 3 | no | — |
| 155 | #95 | — | 25 | 6 | no | — |
| 156 | C52 | C42×Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 157 | C22×C5 | C23×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 158 | C2×D5 | — | 5 | 12 | no | — |
| 159 | C2×D5 | — | 5 | 6 | no | — |
| 160 | C2×D5 | — | 5 | 6 | no | — |
| 161 | C2×D5 | — | 5 | 12 | no | — |
| 162 | AGL(1,𝔽5) | — | 5 | 48 | no | — |
| 163 | AGL(1,𝔽5) | — | 10 | 72 | no | — |
| 164 | C4×C5 | — | 2 | 18 | no | — |
| 165 | C4×C5 | C2×C4×AGL(1,𝔽5) | 1 | 36 | yes | — |
| 166 | C4×C5 | Q8×AGL(1,𝔽5) | 1 | 12 | yes | complemented |
| 167 | Dic5 | — | 5 | 36 | no | — |
| 168 | Dic5 | — | 5 | 12 | no | — |
| 169 | Dic5 | — | 10 | 18 | no | — |
| 170 | C2×Q8 | — | 25 | 8 | no | — |
| 171 | C2×Q8 | — | 25 | 3 | no | — |
| 172 | C2×Q8 | C2×#1043 | 1 | 1 | yes | characteristic |
| 173 | C22×C4 | — | 25 | 2 | no | — |
| 174 | C22×C4 | — | 25 | 3 | no | — |
| 175 | C22×C4 | — | 25 | 1 | no | — |
| 176 | C22×C4 | — | 25 | 6 | no | — |
| 177 | C22×C4 | — | 25 | 3 | no | — |
| 178 | C4⋊C4 | — | 25 | 24 | no | — |
| 179 | C4⋊C4 | — | 25 | 6 | no | — |
| 180 | C4⋊C4 | — | 25 | 3 | no | — |
| 181 | C4⋊C4 | C2×#1043 | 1 | 3 | yes | — |
| 182 | C42 | — | 25 | 24 | no | — |
| 183 | C42 | — | 25 | 4 | no | — |
| 184 | C42 | — | 25 | 3 | no | — |
| 185 | C42 | — | 25 | 3 | no | — |
| 186 | C42 | — | 25 | 3 | no | — |
| 187 | C42 | — | 50 | 12 | no | — |
| 188 | C42 | — | 50 | 6 | no | — |
| 189 | C42 | C2×#1043 | 1 | 3 | yes | — |
| 190 | C2×C5 | C22×C4×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 191 | C2×C5 | C2×Q8×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 192 | C2×C5 | #41×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 193 | D5 | — | 5 | 12 | no | — |
| 194 | D5 | — | 5 | 12 | no | — |
| 195 | C23 | — | 25 | 1 | no | — |
| 196 | Q8 | — | 25 | 12 | no | — |
| 197 | Q8 | C4×#1043 | 1 | 4 | yes | direct product factor |
| 198 | C2×C4 | — | 25 | 4 | no | — |
| 199 | C2×C4 | — | 25 | 4 | no | — |
| 200 | C2×C4 | — | 25 | 4 | no | — |
| 201 | C2×C4 | — | 25 | 1 | no | — |
| 202 | C2×C4 | — | 25 | 3 | no | — |
| 203 | C2×C4 | — | 25 | 12 | no | — |
| 204 | C2×C4 | — | 25 | 2 | no | — |
| 205 | C2×C4 | — | 25 | 2 | no | — |
| 206 | C2×C4 | — | 25 | 12 | no | — |
| 207 | C2×C4 | — | 25 | 3 | no | — |
| 208 | C2×C4 | — | 50 | 6 | no | — |
| 209 | C2×C4 | — | 50 | 6 | no | — |
| 210 | C2×C4 | — | 50 | 6 | no | — |
| 211 | C2×C4 | C22×#1043 | 1 | 3 | yes | — |
| 212 | C2×C4 | C22×#1043 | 1 | 3 | yes | — |
| 213 | C2×C4 | C22×#1043 | 1 | 1 | yes | characteristic |
| 214 | C5 | C4×Q8×AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 215 | C22 | — | 25 | 2 | no | — |
| 216 | C22 | — | 25 | 1 | no | — |
| 217 | C22 | — | 25 | 1 | no | — |
| 218 | C22 | — | 25 | 2 | no | — |
| 219 | C22 | C23×#1043 | 1 | 1 | yes | characteristic |
| 220 | C4 | — | 2 | 3 | no | — |
| 221 | C4 | — | 25 | 8 | no | — |
| 222 | C4 | — | 25 | 6 | no | — |
| 223 | C4 | — | 25 | 2 | no | — |
| 224 | C4 | — | 50 | 12 | no | — |
| 225 | C4 | — | 50 | 3 | no | — |
| 226 | C4 | C2×C4×#1043 | 1 | 6 | yes | — |
| 227 | C4 | Q8×#1043 | 1 | 2 | yes | direct product factor |
| 228 | C2 | — | 25 | 2 | no | — |
| 229 | C2 | — | 25 | 2 | no | — |
| 230 | C2 | C22×C4×#1043 | 1 | 1 | yes | characteristic |
| 231 | C2 | C2×Q8×#1043 | 1 | 1 | yes | characteristic |
| 232 | C2 | #41×#1043 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta