Finite Group Databasebeta

← #20283956

Subgroups — #20283956

Order 6784.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719949C211yesmaximal, characteristic, complemented
2C2×#13133242C211yesmaximal, characteristic, complemented
3#16811130C211yesmaximal, characteristic, complemented
4#16810784C214yesmaximal, complemented
5#16810767C214yesmaximal, complemented
6#16810632C211yesmaximal, characteristic, complemented
7#16810629C211yesmaximal, characteristic, complemented
8#16810586C211yesmaximal, characteristic, complemented
9C2×#137×C53C211yesmaximal, characteristic, complemented
10C2×#11719949—24no—
11C2×#11719949—24no—
12C2×#11719949—22no—
13C2×#11719949C2214yes—
14C22×D212—21no—
15C22×D212C2211yescharacteristic
16C22×C4×D53—21no—
17C22×C4×D53C2211yescharacteristic
18#13133242—24no—
19C2×#11719924C2212yescomplemented
20C2×#11719922C2212yescomplemented
21C2×#11719917C2211yescharacteristic
22#13133198—21no—
23#13133193C2212yescomplemented
24#13133188C2212yescomplemented
25#13133185—21no—
26C4×C22⋊D53—22no—
27#13133176C2211yescharacteristic
28#13133141C2214yescomplemented
29#13133139C2214yescomplemented
30#13133129C2214yescomplemented
31C2×#41×C53C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic
33#137×C53—24no—
34C2×Q16×C53C2212yescomplemented
35C2×SD16×C53C2212yescomplemented
36C2×M16×C53C2211yescharacteristic
37C23×D53—21no—
38C23×D53D411yescharacteristic
39#41×C53—22no—
40#41×C53—22no—
41C2×Q8×C53—22no—
42C2×Q8×C53—21no—
43C2×Q8×C53D414yescomplemented
44C2×Q8×C53C2311yescharacteristic
45C2×Q8×C53C2312yes—
46C2×D4×C53—21no—
47C2×D4×C53C2311yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—22no—
52C22×Dic53D411yescharacteristic
53#11719949—216no—
54#11719949—28no—
55#11719949—28no—
56C2×D212—28no—
57C2×D212—24no—
58C2×D212—21no—
59C2×D212—22no—
60C2×D212D414yescomplemented
61C2×D212C2312yes—
62C2×C4×D53—28no—
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×C4×D53—24no—
66C2×C4×D53D412yes—
67C2×C4×D53C2312yes—
68Q16×C53—24no—
69SD16×C53—24no—
70M16×C53—22no—
71C2×C8×C53C2312yes—
72C22⋊Dic53—21no—
73C22⋊Dic53—22no—
74#11719924—24no—
75#11719922—24no—
76#11719921—22no—
77C4⋊Dic53C2312yes—
78#11719919—22no—
79C4×Dic53—21no—
80#11719917—22no—
81C2×#88322C2312yes—
82C23×C53—21no—
83C23×C53C2×D411yescharacteristic
84C22×D53—22no—
85C22×D53—24no—
86C22×D53—22no—
87C22×D53—42no—
88C22×D53—44no—
89C22×D53C2×D412yes—
90Q8×C53—24no—
91Q8×C53—22no—
92Q8×C53—21no—
93Q8×C53—21no—
94Q8×C53C2×D414yes—
95D4×C53—21no—
96D4×C53—21no—
97D4×C53—42no—
98C2×C4×C53—24no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2411yescharacteristic
106C22⋊D53—44no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53C2×D412yes—
110D212—216no—
111D212—24no—
112D212—22no—
113D212C2×D414yes—
114C4×D53—216no—
115C4×D53—24no—
116C4×D53—22no—
117C4×D53—22no—
118C8×C53—22no—
119#88322—22no—
120C22×C53—22no—
121C22×C53—41no—
122C22×C53—42no—
123C22×C53#12012yes—
124C22×C53C22×D411yescharacteristic
125C2×D53—22no—
126C2×D53—24no—
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—44no—
130C2×D53—48no—
131C2×D53—44no—
132C4×C53—24no—
133C4×C53—41no—
134C4×C53#12012yes—
135C4×C53C22×D411yescharacteristic
136C4×C53C22×D411yescharacteristic
137Dic53—22no—
138Dic53—42no—
139#3005—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53—81no—
142C2×C53C2×#12011yescharacteristic
143C2×C53#57512yes—
144D53—42no—
145D53—44no—
146C22×#41—531no—
147C2×#137D5311yescharacteristic, complemented
148C2×AGL(1,ℤ/8)—531no—
149#525—531no—
150#452—534no—
151#449—534no—
152#419—531no—
153#418—531no—
154#407—531no—
155C53#300511yescharacteristic, complemented, Sylow
156C2×#41—534no—
157C2×#41—1062no—
158C2×#41—1064no—
159C2×#41—1064no—
160C2×#41C2×D5311yescharacteristic
161C22×Q8C2×D5311yescharacteristic
162C22×D4—531no—
163C22×D4—1061no—
164C23×C4—531no—
165C23×C4—1061no—
166#137—24no—
167AGL(1,ℤ/8)—1064no—
168C2×Q16C2×D5312yescomplemented
169C2×SD16—532no—
170C2×SD16C2×D5312yescomplemented
171C2×D8—532no—
172C2×M16—531no—
173C2×M16C2×D5311yescharacteristic
174#123—1061no—
175C22⋊Q8—532no—
176#121—532no—
177#120—1061no—
178C4×D4—1062no—
179#117—531no—
180#103—534no—
181#102—534no—
182C22⋊C8—534no—
183C24—531no—
184C24—1061no—
185#41—22no—
186#41—22no—
187#41—1068no—
188#41—10616no—
189#41—1068no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8C22⋊D5314yescomplemented
193C2×Q8C22×D5311yescharacteristic
194C2×Q8C22×D5312yes—
195C2×D4—21no—
196C2×D4—534no—
197C2×D4—532no—
198C2×D4—1064no—
199C2×D4—1068no—
200C2×D4—1062no—
201C2×D4—1061no—
202C2×D4—1062no—
203C2×D4C22×D5311yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—531no—
207C22×C4—532no—
208C22×C4—532no—
209C22×C4—1064no—
210C22×C4—1068no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4C22×D5311yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—1064no—
217D8—1064no—
218M16—22no—
219M16—1062no—
220C2×C8—532no—
221C2×C8C22×D5312yes—
222C4⋊C4—532no—
223C4⋊C4—1062no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C22⋊C4—1061no—
227C42—1061no—
228C23—21no—
229C23—532no—
230C23—1064no—
231C23—1062no—
232C23—1062no—
233C23—2122no—
234C23—2124no—
235C23D4×D5311yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D5314yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—534no—
245D4—10616no—
246D4—1064no—
247D4—1062no—
248D4—2124no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—532no—
254C2×C4—10616no—
255C2×C4—1062no—
256C2×C4—1064no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4D4×D5311yescharacteristic
261C2×C4D4×D5312yes—
262C2×C4C2×C22⋊D5312yes—
263C2×C4C23×D5311yescharacteristic
264C8—22no—
265C8—1062no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—1064no—
270C22—1062no—
271C22—1062no—
272C22—1062no—
273C22—2124no—
274C22—2128no—
275C22—2124no—
276C22#1313318512yes—
277C22C2×D4×D5311yescharacteristic
278C4—24no—
279C4—41no—
280C4—1062no—
281C4—2122no—
282C4#1313318512yes—
283C4C2×D4×D5311yescharacteristic
284C4C22×C22⋊D5311yescharacteristic
285C2—22no—
286C2—81no—
287C2—2122no—
288C2—2124no—
289C2C2×#1313318511yescharacteristic
290C2#1681142212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#137×C53C2
socleC22×C53C22×D4
hypercenterC2×#137D53
hypocenterC53#3005
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#11719949C2semidirect
2. C2×#13133242C2semidirect
3. #16811130C2semidirect
4. #16810784C2semidirect
5. #16810767C2semidirect
6. #16810632C2semidirect
7. #16810629C2semidirect
8. #16810586C2semidirect
9. C2×#137×C53C2semidirect
19. C2×#11719924C22semidirect
20. C2×#11719922C22semidirect
23. #13133193C22semidirect
24. #13133188C22semidirect
28. #13133141C22semidirect
29. #13133139C22semidirect
30. #13133129C22semidirect
34. C2×Q16×C53C22semidirect
35. C2×SD16×C53C22semidirect
43. C2×Q8×C53D4semidirect
60. C2×D212D4semidirect
147. C2×#137D53semidirect
155. C53#3005semidirect
168. C2×Q16C2×D53semidirect
170. C2×SD16C2×D53semidirect
192. C2×Q8C22⋊D53semidirect