← C23×#33388 #19181014
Subgroups — C23×#33388 #19181014
Order 5888.
1128 conjugacy classes of subgroups.
The subgroup lattice has 252 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C23×#3584 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C23×#3407 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C23×#2912 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23×#2145 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×C23×#360 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C23×#585 | — | 2 | 1 | no | — |
| 7 | C23×#578 | C4 | 1 | 4 | yes | complemented |
| 8 | C23×#577 | — | 2 | 4 | no | — |
| 9 | C23×#576 | C4 | 1 | 4 | yes | complemented |
| 10 | C23×#575 | — | 2 | 3 | no | — |
| 11 | C2×C23×#135 | C22 | 1 | 6 | yes | — |
| 12 | C2×C23×#131 | C22 | 1 | 1 | yes | characteristic |
| 13 | C23×#520 | — | 2 | 1 | no | — |
| 14 | C23×#504 | — | 2 | 4 | no | — |
| 15 | C2×C23×M32 | C22 | 1 | 6 | yes | complemented |
| 16 | C23×#428 | — | 2 | 3 | no | — |
| 17 | C23×#427 | C22 | 1 | 3 | yes | complemented |
| 18 | C2×C23×#106 | C22 | 1 | 3 | yes | complemented |
| 19 | C23×#360 | C22 | 1 | 12 | yes | complemented |
| 20 | C23×#360 | C22 | 1 | 4 | yes | complemented |
| 21 | C23×2−1+4 | — | 2 | 4 | no | — |
| 22 | C23×2−1+4 | — | 4 | 1 | no | — |
| 23 | C23×H(5,𝔽2) | — | 2 | 4 | no | — |
| 24 | C23×H(5,𝔽2) | — | 4 | 3 | no | — |
| 25 | C2×#41×C23 | — | 2 | 6 | no | — |
| 26 | C2×#41×C23 | — | 4 | 4 | no | — |
| 27 | C2×#41×C23 | D4 | 1 | 1 | yes | characteristic |
| 28 | C23×#137 | — | 2 | 12 | no | — |
| 29 | C23×AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 30 | C23×#135 | — | 2 | 6 | no | — |
| 31 | C23×#135 | — | 2 | 6 | no | — |
| 32 | C23×#135 | C2×C4 | 1 | 12 | yes | complemented |
| 33 | C23×#135 | C23 | 1 | 4 | yes | — |
| 34 | C2×Q16×C23 | C2×C4 | 1 | 6 | yes | complemented |
| 35 | C2×SD16×C23 | — | 2 | 6 | no | — |
| 36 | C2×D8×C23 | C2×C4 | 1 | 6 | yes | complemented |
| 37 | C23×#131 | C2×C4 | 1 | 4 | yes | — |
| 38 | C23×#131 | D4 | 1 | 4 | yes | — |
| 39 | C2×M16×C23 | D4 | 1 | 3 | yes | — |
| 40 | C22×C8×C23 | C23 | 1 | 3 | yes | — |
| 41 | C4×Q8×C23 | — | 4 | 1 | no | — |
| 42 | C4×D4×C23 | — | 4 | 3 | no | — |
| 43 | C23×#117 | — | 2 | 3 | no | — |
| 44 | C2×C4⋊C4×C23 | — | 2 | 3 | no | — |
| 45 | C23×M32 | — | 2 | 6 | no | — |
| 46 | C23×M32 | C23 | 1 | 4 | yes | — |
| 47 | C2×C16×C23 | — | 2 | 6 | no | — |
| 48 | C23×#107 | — | 2 | 3 | no | — |
| 49 | C23×#106 | — | 2 | 3 | no | — |
| 50 | C23×#106 | C23 | 1 | 3 | yes | — |
| 51 | C23×#106 | C23 | 1 | 1 | yes | characteristic |
| 52 | #41×C23 | — | 2 | 12 | no | — |
| 53 | #41×C23 | — | 2 | 4 | no | — |
| 54 | #41×C23 | — | 4 | 4 | no | — |
| 55 | #41×C23 | — | 4 | 12 | no | — |
| 56 | #41×C23 | — | 4 | 6 | no | — |
| 57 | #41×C23 | — | 4 | 6 | no | — |
| 58 | #41×C23 | C22⋊C4 | 1 | 4 | yes | — |
| 59 | #41×C23 | SD16 | 1 | 4 | yes | — |
| 60 | C2×Q8×C23 | — | 2 | 6 | no | — |
| 61 | C2×Q8×C23 | — | 4 | 4 | no | — |
| 62 | C2×Q8×C23 | D8 | 1 | 1 | yes | characteristic |
| 63 | C2×D4×C23 | — | 2 | 6 | no | — |
| 64 | C2×D4×C23 | — | 4 | 6 | no | — |
| 65 | C2×D4×C23 | — | 4 | 12 | no | — |
| 66 | C2×D4×C23 | D8 | 1 | 3 | yes | — |
| 67 | C22×C4×C23 | — | 4 | 6 | no | — |
| 68 | C22×C4×C23 | — | 4 | 3 | no | — |
| 69 | C22×C4×C23 | C2×D4 | 1 | 3 | yes | — |
| 70 | Q16×C23 | — | 2 | 6 | no | — |
| 71 | Q16×C23 | C22×C4 | 1 | 4 | yes | — |
| 72 | SD16×C23 | — | 2 | 12 | no | — |
| 73 | SD16×C23 | — | 2 | 4 | no | — |
| 74 | D8×C23 | — | 2 | 6 | no | — |
| 75 | D8×C23 | C22×C4 | 1 | 4 | yes | — |
| 76 | M16×C23 | C22⋊C4 | 1 | 12 | yes | — |
| 77 | C2×C8×C23 | C22×C4 | 1 | 6 | yes | — |
| 78 | C2×C8×C23 | C2×D4 | 1 | 3 | yes | — |
| 79 | C2×C8×C23 | C2×D4 | 1 | 6 | yes | — |
| 80 | C2×C8×C23 | C24 | 1 | 1 | yes | characteristic |
| 81 | C4⋊C4×C23 | — | 2 | 3 | no | — |
| 82 | C4⋊C4×C23 | — | 2 | 1 | no | — |
| 83 | C4⋊C4×C23 | — | 4 | 3 | no | — |
| 84 | C22⋊C4×C23 | — | 4 | 3 | no | — |
| 85 | C42×C23 | — | 4 | 3 | no | — |
| 86 | C16×C23 | — | 2 | 4 | no | — |
| 87 | #33388 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 88 | C23×C23 | — | 4 | 6 | no | — |
| 89 | C23×C23 | C2×D8 | 1 | 3 | yes | — |
| 90 | Q8×C23 | — | 2 | 4 | no | — |
| 91 | Q8×C23 | — | 4 | 6 | no | — |
| 92 | Q8×C23 | #102 | 1 | 4 | yes | — |
| 93 | D4×C23 | — | 2 | 4 | no | — |
| 94 | D4×C23 | — | 4 | 4 | no | — |
| 95 | D4×C23 | — | 4 | 12 | no | — |
| 96 | D4×C23 | — | 4 | 6 | no | — |
| 97 | D4×C23 | #102 | 1 | 12 | yes | — |
| 98 | C2×C4×C23 | — | 4 | 6 | no | — |
| 99 | C2×C4×C23 | — | 4 | 4 | no | — |
| 100 | C2×C4×C23 | — | 4 | 12 | no | — |
| 101 | C2×C4×C23 | — | 4 | 1 | no | — |
| 102 | C2×C4×C23 | — | 4 | 3 | no | — |
| 103 | C2×C4×C23 | — | 16 | 3 | no | — |
| 104 | C2×C4×C23 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 105 | C2×C4×C23 | C2×D8 | 1 | 3 | yes | — |
| 106 | C2×C4×C23 | C2×SD16 | 1 | 6 | yes | — |
| 107 | C2×C4×C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 108 | C8×C23 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 109 | C8×C23 | C23×C4 | 1 | 1 | yes | characteristic |
| 110 | C8×C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 111 | #3584 | C2×C23 | 1 | 1 | yes | characteristic |
| 112 | #3407 | C2×C23 | 1 | 1 | yes | characteristic, complemented |
| 113 | #2912 | C2×C23 | 1 | 1 | yes | characteristic, complemented |
| 114 | #2145 | C2×C23 | 1 | 6 | yes | complemented |
| 115 | C2×#360 | C2×C23 | 1 | 6 | yes | complemented |
| 116 | C22×C23 | — | 4 | 3 | no | — |
| 117 | C22×C23 | — | 4 | 4 | no | — |
| 118 | C22×C23 | — | 16 | 6 | no | — |
| 119 | C22×C23 | C2×#102 | 1 | 6 | yes | — |
| 120 | C22×C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 121 | C4×C23 | — | 4 | 4 | no | — |
| 122 | C4×C23 | — | 8 | 1 | no | — |
| 123 | C4×C23 | — | 8 | 3 | no | — |
| 124 | C4×C23 | C2×#102 | 1 | 6 | yes | — |
| 125 | C4×C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C4×C23 | C22×SD16 | 1 | 1 | yes | characteristic |
| 127 | #585 | — | 2 | 1 | no | — |
| 128 | #578 | C4×C23 | 1 | 4 | yes | complemented |
| 129 | #577 | — | 2 | 4 | no | — |
| 130 | #576 | C4×C23 | 1 | 4 | yes | complemented |
| 131 | #575 | — | 2 | 3 | no | — |
| 132 | C2×#135 | C22×C23 | 1 | 6 | yes | — |
| 133 | C2×#131 | C22×C23 | 1 | 1 | yes | characteristic |
| 134 | #520 | — | 2 | 1 | no | — |
| 135 | #504 | — | 2 | 4 | no | — |
| 136 | C2×M32 | C22×C23 | 1 | 6 | yes | complemented |
| 137 | #428 | — | 2 | 3 | no | — |
| 138 | #427 | C22×C23 | 1 | 3 | yes | complemented |
| 139 | C2×#106 | C22×C23 | 1 | 3 | yes | complemented |
| 140 | #360 | C22×C23 | 1 | 12 | yes | complemented |
| 141 | #360 | C22×C23 | 1 | 4 | yes | complemented |
| 142 | C2×C23 | — | 2 | 1 | no | — |
| 143 | C2×C23 | — | 2 | 6 | no | — |
| 144 | C2×C23 | — | 8 | 4 | no | — |
| 145 | C2×C23 | C22×#102 | 1 | 1 | yes | characteristic |
| 146 | 2−1+4 | — | 2 | 4 | no | — |
| 147 | 2−1+4 | — | 4 | 1 | no | — |
| 148 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 149 | H(5,𝔽2) | — | 4 | 3 | no | — |
| 150 | C2×#41 | — | 2 | 6 | no | — |
| 151 | C2×#41 | — | 4 | 4 | no | — |
| 152 | C2×#41 | D4×C23 | 1 | 1 | yes | characteristic |
| 153 | #137 | — | 2 | 12 | no | — |
| 154 | AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 155 | #135 | — | 2 | 6 | no | — |
| 156 | #135 | — | 2 | 6 | no | — |
| 157 | #135 | C2×C4×C23 | 1 | 12 | yes | complemented |
| 158 | #135 | C23×C23 | 1 | 4 | yes | — |
| 159 | C2×Q16 | C2×C4×C23 | 1 | 6 | yes | complemented |
| 160 | C2×SD16 | — | 2 | 6 | no | — |
| 161 | C2×D8 | C2×C4×C23 | 1 | 6 | yes | complemented |
| 162 | #131 | C2×C4×C23 | 1 | 4 | yes | — |
| 163 | #131 | D4×C23 | 1 | 4 | yes | — |
| 164 | C2×M16 | D4×C23 | 1 | 3 | yes | — |
| 165 | C22×C8 | C23×C23 | 1 | 3 | yes | — |
| 166 | C4×Q8 | — | 4 | 1 | no | — |
| 167 | C4×D4 | — | 4 | 3 | no | — |
| 168 | #117 | — | 2 | 3 | no | — |
| 169 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 170 | M32 | — | 2 | 6 | no | — |
| 171 | M32 | C23×C23 | 1 | 4 | yes | — |
| 172 | C2×C16 | — | 2 | 6 | no | — |
| 173 | #107 | — | 2 | 3 | no | — |
| 174 | #106 | — | 2 | 3 | no | — |
| 175 | #106 | C23×C23 | 1 | 3 | yes | — |
| 176 | #106 | C23×C23 | 1 | 1 | yes | characteristic |
| 177 | C23 | #33388 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 178 | #41 | — | 2 | 12 | no | — |
| 179 | #41 | — | 2 | 4 | no | — |
| 180 | #41 | — | 4 | 4 | no | — |
| 181 | #41 | — | 4 | 12 | no | — |
| 182 | #41 | — | 4 | 6 | no | — |
| 183 | #41 | — | 4 | 6 | no | — |
| 184 | #41 | C22⋊C4×C23 | 1 | 4 | yes | — |
| 185 | #41 | SD16×C23 | 1 | 4 | yes | — |
| 186 | C2×Q8 | — | 2 | 6 | no | — |
| 187 | C2×Q8 | — | 4 | 4 | no | — |
| 188 | C2×Q8 | D8×C23 | 1 | 1 | yes | characteristic |
| 189 | C2×D4 | — | 2 | 6 | no | — |
| 190 | C2×D4 | — | 4 | 6 | no | — |
| 191 | C2×D4 | — | 4 | 12 | no | — |
| 192 | C2×D4 | D8×C23 | 1 | 3 | yes | — |
| 193 | C22×C4 | — | 4 | 6 | no | — |
| 194 | C22×C4 | — | 4 | 3 | no | — |
| 195 | C22×C4 | C2×D4×C23 | 1 | 3 | yes | — |
| 196 | Q16 | — | 2 | 6 | no | — |
| 197 | Q16 | C22×C4×C23 | 1 | 4 | yes | — |
| 198 | SD16 | — | 2 | 12 | no | — |
| 199 | SD16 | — | 2 | 4 | no | — |
| 200 | D8 | — | 2 | 6 | no | — |
| 201 | D8 | C22×C4×C23 | 1 | 4 | yes | — |
| 202 | M16 | C22⋊C4×C23 | 1 | 12 | yes | — |
| 203 | C2×C8 | C22×C4×C23 | 1 | 6 | yes | — |
| 204 | C2×C8 | C2×D4×C23 | 1 | 3 | yes | — |
| 205 | C2×C8 | C2×D4×C23 | 1 | 6 | yes | — |
| 206 | C2×C8 | C24×C23 | 1 | 1 | yes | characteristic |
| 207 | C4⋊C4 | — | 2 | 3 | no | — |
| 208 | C4⋊C4 | — | 2 | 1 | no | — |
| 209 | C4⋊C4 | — | 4 | 3 | no | — |
| 210 | C22⋊C4 | — | 4 | 3 | no | — |
| 211 | C42 | — | 4 | 3 | no | — |
| 212 | C16 | — | 2 | 4 | no | — |
| 213 | C23 | — | 4 | 6 | no | — |
| 214 | C23 | C2×D8×C23 | 1 | 3 | yes | — |
| 215 | Q8 | — | 2 | 4 | no | — |
| 216 | Q8 | — | 4 | 6 | no | — |
| 217 | Q8 | C23×#102 | 1 | 4 | yes | — |
| 218 | D4 | — | 2 | 4 | no | — |
| 219 | D4 | — | 4 | 4 | no | — |
| 220 | D4 | — | 4 | 12 | no | — |
| 221 | D4 | — | 4 | 6 | no | — |
| 222 | D4 | C23×#102 | 1 | 12 | yes | — |
| 223 | C2×C4 | — | 4 | 6 | no | — |
| 224 | C2×C4 | — | 4 | 4 | no | — |
| 225 | C2×C4 | — | 4 | 12 | no | — |
| 226 | C2×C4 | — | 4 | 1 | no | — |
| 227 | C2×C4 | — | 4 | 3 | no | — |
| 228 | C2×C4 | — | 16 | 3 | no | — |
| 229 | C2×C4 | C2×C22⋊C4×C23 | 1 | 6 | yes | — |
| 230 | C2×C4 | C2×D8×C23 | 1 | 3 | yes | — |
| 231 | C2×C4 | C2×SD16×C23 | 1 | 6 | yes | — |
| 232 | C2×C4 | C22×D4×C23 | 1 | 1 | yes | characteristic |
| 233 | C8 | C2×C22⋊C4×C23 | 1 | 6 | yes | — |
| 234 | C8 | C23×C4×C23 | 1 | 1 | yes | characteristic |
| 235 | C8 | C22×D4×C23 | 1 | 1 | yes | characteristic |
| 236 | C22 | — | 4 | 3 | no | — |
| 237 | C22 | — | 4 | 4 | no | — |
| 238 | C22 | — | 16 | 6 | no | — |
| 239 | C22 | C2×C23×#102 | 1 | 6 | yes | — |
| 240 | C22 | C22×D8×C23 | 1 | 1 | yes | characteristic |
| 241 | C4 | — | 4 | 4 | no | — |
| 242 | C4 | — | 8 | 3 | no | — |
| 243 | C4 | — | 8 | 1 | no | — |
| 244 | C4 | C2×C23×#102 | 1 | 6 | yes | — |
| 245 | C4 | C22×C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 246 | C4 | C22×SD16×C23 | 1 | 1 | yes | characteristic |
| 247 | C2 | — | 2 | 1 | no | — |
| 248 | C2 | — | 2 | 6 | no | — |
| 249 | C2 | — | 8 | 4 | no | — |
| 250 | C2 | C22×C23×#102 | 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