Finite Group DatabasebetaFinite Group DatabasebetaOrder 7552.
444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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 | #16838218 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #16838217 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #16837781 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16837676 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #16837675 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #16837586 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #16837584 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #16837516 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C59×#456 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #14273775 | C22 | 1 | 1 | yes | characteristic |
| 11 | #14273774 | C22 | 1 | 2 | yes | — |
| 12 | #14273760 | C22 | 1 | 2 | yes | — |
| 13 | #14273759 | — | 2 | 2 | no | — |
| 14 | #14273758 | — | 2 | 2 | no | — |
| 15 | #14273758 | C22 | 1 | 2 | yes | — |
| 16 | #14273757 | — | 2 | 2 | no | — |
| 17 | #14273757 | C22 | 1 | 2 | yes | — |
| 18 | #14273756 | — | 2 | 2 | no | — |
| 19 | #14273748 | — | 2 | 2 | no | — |
| 20 | #14273747 | — | 2 | 2 | no | — |
| 21 | #14273747 | C22 | 1 | 2 | yes | — |
| 22 | C2×#11745672 | — | 2 | 2 | no | — |
| 23 | C2×#11745672 | C22 | 1 | 2 | yes | — |
| 24 | C2×#11745672 | C22 | 1 | 2 | yes | — |
| 25 | Q16×D59 | — | 2 | 2 | no | — |
| 26 | #14273735 | C22 | 1 | 2 | yes | — |
| 27 | #14273734 | C22 | 1 | 2 | yes | — |
| 28 | #14273727 | C22 | 1 | 1 | yes | characteristic |
| 29 | #14273662 | C22 | 1 | 2 | yes | — |
| 30 | #14273655 | C22 | 1 | 2 | yes | — |
| 31 | #14273646 | C22 | 1 | 1 | yes | characteristic |
| 32 | #14273645 | C22 | 1 | 2 | yes | — |
| 33 | C4×C59⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 34 | 2−1+4×C59 | C22 | 1 | 1 | yes | characteristic |
| 35 | #137×C59 | C22 | 1 | 2 | yes | — |
| 36 | C4⋊Q8×C59 | C22 | 1 | 1 | yes | characteristic |
| 37 | #104×C59 | C22 | 1 | 2 | yes | — |
| 38 | #101×C59 | C22 | 1 | 1 | yes | characteristic |
| 39 | #41×C59 | — | 2 | 2 | no | — |
| 40 | #41×C59 | — | 4 | 1 | no | — |
| 41 | #41×C59 | C23 | 1 | 2 | yes | — |
| 42 | C2×Q8×C59 | — | 2 | 2 | no | — |
| 43 | C2×Q8×C59 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×Q8×C59 | C23 | 1 | 1 | yes | characteristic |
| 45 | #11745689 | — | 2 | 2 | no | — |
| 46 | Q8×D59 | — | 2 | 2 | no | — |
| 47 | Q8×D59 | — | 2 | 2 | no | — |
| 48 | #11745687 | — | 2 | 2 | no | — |
| 49 | #11745687 | — | 4 | 2 | no | — |
| 50 | #11745685 | — | 2 | 2 | no | — |
| 51 | #11745685 | — | 4 | 1 | no | — |
| 52 | #11745685 | C23 | 1 | 2 | yes | — |
| 53 | C2×Dic118 | — | 2 | 2 | no | — |
| 54 | C2×Dic118 | C23 | 1 | 2 | yes | — |
| 55 | Q16×C59 | — | 2 | 2 | no | — |
| 56 | SD16×C59 | — | 2 | 2 | no | — |
| 57 | M16×C59 | C23 | 1 | 2 | yes | — |
| 58 | C4⋊C4×C59 | — | 4 | 1 | no | — |
| 59 | C42×C59 | C23 | 1 | 1 | yes | characteristic |
| 60 | #11745672 | — | 2 | 2 | no | — |
| 61 | #11745672 | — | 2 | 2 | no | — |
| 62 | #11745672 | — | 2 | 2 | no | — |
| 63 | #11745672 | — | 2 | 2 | no | — |
| 64 | #11745672 | — | 2 | 2 | no | — |
| 65 | #11745672 | — | 2 | 2 | no | — |
| 66 | #11745672 | D4 | 1 | 2 | yes | — |
| 67 | #11745671 | — | 2 | 2 | no | — |
| 68 | #11745671 | D4 | 1 | 2 | yes | — |
| 69 | #11745670 | — | 2 | 2 | no | — |
| 70 | #11745670 | D4 | 1 | 2 | yes | — |
| 71 | #11745669 | D4 | 1 | 2 | yes | — |
| 72 | C4⋊Dic59 | — | 4 | 1 | no | — |
| 73 | #11745664 | — | 2 | 2 | no | — |
| 74 | #11745664 | C23 | 1 | 2 | yes | — |
| 75 | C2×C59⋊C8 | — | 2 | 2 | no | — |
| 76 | C2×C59⋊C8 | C23 | 1 | 2 | yes | — |
| 77 | Dic236 | — | 2 | 2 | no | — |
| 78 | #11745660 | — | 2 | 2 | no | — |
| 79 | #11745659 | — | 2 | 2 | no | — |
| 80 | C8×D59 | — | 2 | 2 | no | — |
| 81 | Q8×C59 | — | 2 | 2 | no | — |
| 82 | Q8×C59 | — | 2 | 2 | no | — |
| 83 | Q8×C59 | — | 4 | 1 | no | — |
| 84 | Q8×C59 | C2×D4 | 1 | 2 | yes | — |
| 85 | D4×C59 | — | 2 | 2 | no | — |
| 86 | D4×C59 | — | 4 | 1 | no | — |
| 87 | D4×C59 | C2×D4 | 1 | 2 | yes | — |
| 88 | C2×C4×C59 | — | 2 | 1 | no | — |
| 89 | C2×C4×C59 | — | 2 | 2 | no | — |
| 90 | C2×C4×C59 | — | 2 | 1 | no | — |
| 91 | C2×C4×C59 | — | 2 | 1 | no | — |
| 92 | C2×C4×C59 | — | 4 | 2 | no | — |
| 93 | C2×C4×C59 | C24 | 1 | 1 | yes | characteristic |
| 94 | C22⋊D59 | — | 2 | 2 | no | — |
| 95 | C22⋊D59 | — | 4 | 2 | no | — |
| 96 | C2×Dic59 | — | 2 | 2 | no | — |
| 97 | C2×Dic59 | — | 4 | 2 | no | — |
| 98 | D236 | — | 4 | 1 | no | — |
| 99 | D236 | C2×D4 | 1 | 2 | yes | — |
| 100 | C4×D59 | — | 2 | 2 | no | — |
| 101 | C4×D59 | — | 4 | 2 | no | — |
| 102 | C4×D59 | — | 4 | 2 | no | — |
| 103 | Dic118 | — | 2 | 2 | no | — |
| 104 | Dic118 | — | 2 | 2 | no | — |
| 105 | Dic118 | — | 4 | 1 | no | — |
| 106 | Dic118 | — | 4 | 2 | no | — |
| 107 | Dic118 | C2×D4 | 1 | 2 | yes | — |
| 108 | C8×C59 | — | 2 | 2 | no | — |
| 109 | C59⋊C8 | — | 2 | 2 | no | — |
| 110 | C59⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 111 | C59⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 112 | C22×C59 | — | 2 | 2 | no | — |
| 113 | C22×C59 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 114 | C2×D59 | — | 2 | 2 | no | — |
| 115 | C4×C59 | — | 2 | 2 | no | — |
| 116 | C4×C59 | — | 2 | 2 | no | — |
| 117 | C4×C59 | — | 4 | 1 | no | — |
| 118 | C4×C59 | — | 4 | 1 | no | — |
| 119 | C4×C59 | C22×D4 | 1 | 2 | yes | — |
| 120 | Dic59 | — | 2 | 2 | no | — |
| 121 | Dic59 | — | 4 | 2 | no | — |
| 122 | #3293 | — | 59 | 1 | no | maximal, Sylow |
| 123 | C2×C59 | — | 2 | 1 | no | — |
| 124 | C2×C59 | — | 4 | 2 | no | — |
| 125 | C2×C59 | D42 | 1 | 1 | yes | characteristic |
| 126 | D59 | — | 4 | 2 | no | — |
| 127 | #578 | — | 59 | 2 | no | — |
| 128 | #578 | — | 59 | 2 | no | — |
| 129 | #494 | — | 59 | 1 | no | — |
| 130 | #473 | — | 59 | 2 | no | — |
| 131 | #473 | — | 59 | 2 | no | — |
| 132 | #456 | — | 59 | 1 | no | — |
| 133 | #456 | — | 59 | 2 | no | — |
| 134 | #456 | D59 | 1 | 1 | yes | characteristic, complemented |
| 135 | #443 | — | 59 | 2 | no | — |
| 136 | C59 | #3293 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 137 | 2−1+4 | — | 59 | 2 | no | — |
| 138 | 2−1+4 | — | 59 | 1 | no | — |
| 139 | 2−1+4 | C2×D59 | 1 | 1 | yes | characteristic |
| 140 | #137 | — | 59 | 2 | no | — |
| 141 | #137 | — | 59 | 2 | no | — |
| 142 | #137 | — | 59 | 2 | no | — |
| 143 | #137 | — | 118 | 2 | no | — |
| 144 | #137 | — | 118 | 2 | no | — |
| 145 | #137 | — | 118 | 2 | no | — |
| 146 | #137 | — | 118 | 2 | no | — |
| 147 | #137 | C2×D59 | 1 | 2 | yes | — |
| 148 | #135 | — | 59 | 2 | no | — |
| 149 | #135 | — | 118 | 2 | no | — |
| 150 | #135 | — | 118 | 2 | no | — |
| 151 | C2×Q16 | — | 59 | 2 | no | — |
| 152 | C2×Q16 | — | 59 | 2 | no | — |
| 153 | C2×Q16 | — | 118 | 2 | no | — |
| 154 | C2×Q16 | — | 118 | 2 | no | — |
| 155 | #131 | — | 59 | 2 | no | — |
| 156 | #131 | — | 59 | 2 | no | — |
| 157 | C4⋊Q8 | — | 59 | 1 | no | — |
| 158 | C4⋊Q8 | C2×D59 | 1 | 1 | yes | characteristic |
| 159 | #108 | — | 59 | 2 | no | — |
| 160 | #104 | — | 59 | 2 | no | — |
| 161 | #104 | C2×D59 | 1 | 2 | yes | — |
| 162 | #101 | — | 59 | 1 | no | — |
| 163 | #101 | — | 59 | 2 | no | — |
| 164 | #101 | C2×D59 | 1 | 1 | yes | characteristic |
| 165 | C4×C8 | — | 59 | 1 | no | — |
| 166 | #41 | — | 2 | 2 | no | — |
| 167 | #41 | — | 4 | 1 | no | — |
| 168 | #41 | — | 59 | 2 | no | — |
| 169 | #41 | — | 118 | 2 | no | — |
| 170 | #41 | — | 118 | 2 | no | — |
| 171 | #41 | — | 118 | 2 | no | — |
| 172 | #41 | — | 236 | 1 | no | — |
| 173 | #41 | — | 236 | 2 | no | — |
| 174 | #41 | C22×D59 | 1 | 2 | yes | — |
| 175 | C2×Q8 | — | 2 | 2 | no | — |
| 176 | C2×Q8 | — | 59 | 2 | no | — |
| 177 | C2×Q8 | — | 118 | 2 | no | — |
| 178 | C2×Q8 | — | 118 | 2 | no | — |
| 179 | C2×Q8 | — | 118 | 2 | no | — |
| 180 | C2×Q8 | C22×D59 | 1 | 1 | yes | characteristic |
| 181 | C2×Q8 | C22×D59 | 1 | 1 | yes | characteristic |
| 182 | Q16 | — | 2 | 2 | no | — |
| 183 | Q16 | — | 59 | 2 | no | — |
| 184 | Q16 | — | 118 | 2 | no | — |
| 185 | Q16 | — | 118 | 2 | no | — |
| 186 | Q16 | — | 118 | 2 | no | — |
| 187 | Q16 | — | 118 | 2 | no | — |
| 188 | Q16 | — | 118 | 2 | no | — |
| 189 | Q16 | — | 118 | 2 | no | — |
| 190 | Q16 | — | 118 | 2 | no | — |
| 191 | SD16 | — | 2 | 2 | no | — |
| 192 | SD16 | — | 59 | 2 | no | — |
| 193 | SD16 | — | 59 | 2 | no | — |
| 194 | SD16 | — | 118 | 2 | no | — |
| 195 | SD16 | — | 118 | 2 | no | — |
| 196 | SD16 | — | 118 | 2 | no | — |
| 197 | D8 | — | 59 | 2 | no | — |
| 198 | M16 | — | 59 | 2 | no | — |
| 199 | M16 | — | 118 | 2 | no | — |
| 200 | M16 | — | 118 | 2 | no | — |
| 201 | M16 | C22×D59 | 1 | 2 | yes | — |
| 202 | C2×C8 | — | 59 | 2 | no | — |
| 203 | C2×C8 | — | 118 | 2 | no | — |
| 204 | C2×C8 | — | 118 | 2 | no | — |
| 205 | C4⋊C4 | — | 4 | 1 | no | — |
| 206 | C4⋊C4 | — | 236 | 1 | no | — |
| 207 | C42 | C22×D59 | 1 | 1 | yes | characteristic |
| 208 | Q8 | — | 2 | 2 | no | — |
| 209 | Q8 | — | 2 | 2 | no | — |
| 210 | Q8 | — | 4 | 1 | no | — |
| 211 | Q8 | — | 59 | 2 | no | — |
| 212 | Q8 | — | 118 | 2 | no | — |
| 213 | Q8 | — | 118 | 2 | no | — |
| 214 | Q8 | — | 236 | 2 | no | — |
| 215 | Q8 | — | 236 | 1 | no | — |
| 216 | Q8 | D4×D59 | 1 | 2 | yes | — |
| 217 | D4 | — | 2 | 2 | no | — |
| 218 | D4 | — | 4 | 1 | no | — |
| 219 | D4 | — | 59 | 2 | no | — |
| 220 | D4 | — | 118 | 2 | no | — |
| 221 | D4 | — | 236 | 1 | no | — |
| 222 | D4 | — | 236 | 2 | no | — |
| 223 | D4 | D4×D59 | 1 | 2 | yes | — |
| 224 | C2×C4 | — | 2 | 1 | no | — |
| 225 | C2×C4 | — | 2 | 2 | no | — |
| 226 | C2×C4 | — | 2 | 1 | no | — |
| 227 | C2×C4 | — | 2 | 1 | no | — |
| 228 | C2×C4 | — | 4 | 2 | no | — |
| 229 | C2×C4 | — | 118 | 2 | no | — |
| 230 | C2×C4 | — | 118 | 2 | no | — |
| 231 | C2×C4 | — | 236 | 2 | no | — |
| 232 | C2×C4 | — | 236 | 2 | no | — |
| 233 | C2×C4 | — | 236 | 2 | no | — |
| 234 | C2×C4 | C23×D59 | 1 | 1 | yes | characteristic |
| 235 | C8 | — | 2 | 2 | no | — |
| 236 | C8 | — | 59 | 2 | no | — |
| 237 | C8 | — | 59 | 2 | no | — |
| 238 | C8 | — | 118 | 2 | no | — |
| 239 | C22 | — | 2 | 2 | no | — |
| 240 | C22 | — | 118 | 2 | no | — |
| 241 | C22 | #14273772 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 2 | 2 | no | — |
| 243 | C4 | — | 2 | 2 | no | — |
| 244 | C4 | — | 4 | 1 | no | — |
| 245 | C4 | — | 4 | 1 | no | — |
| 246 | C4 | — | 118 | 2 | no | — |
| 247 | C4 | — | 236 | 2 | no | — |
| 248 | C4 | C2×D4×D59 | 1 | 2 | yes | — |
| 249 | C2 | — | 2 | 1 | no | — |
| 250 | C2 | — | 4 | 2 | no | — |
| 251 | C2 | — | 236 | 2 | no | — |
| 252 | C2 | #16838034 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16838034 |
| commutator | C2×C4×C59 | C24 |
| Frattini | C2×C4 | C23×D59 |
| Fitting | C59×#456 | C2 |
| socle | C2×C59 | D42 |
| hypercenter | #456 | D59 |
| hypocenter | C59 | #3293 |
| LCS | G ▷ C2×C4×C59 ▷ C2×C59 ▷ C59 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #456 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16838218 | C2 | semidirect |
| 2. #16838217 | C2 | semidirect |
| 3. #16837781 | C2 | semidirect |
| 4. #16837676 | C2 | semidirect |
| 5. #16837675 | C2 | semidirect |
| 6. #16837586 | C2 | semidirect |
| 7. #16837584 | C2 | semidirect |
| 8. #16837516 | C2 | semidirect |
| 9. C59×#456 | C2 | semidirect |
| 134. #456 | D59 | semidirect |
| 136. C59 | #3293 | semidirect |