Finite Group Databasebeta

← #20286414

Subgroups — #20286414

Order 6784.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×SD16×D53C212yesmaximal, complemented
2#16811288C211yesmaximal, characteristic, complemented
3D4×D212C211yesmaximal, characteristic
4#16810997C211yesmaximal, characteristic, complemented
5#16810855C212yesmaximal, complemented
6#16810824C211yesmaximal, characteristic, complemented
7#16810772C212yesmaximal, complemented
8#16810763C212yesmaximal, complemented
9C4×#11719923C211yesmaximal, characteristic, complemented
10#16810688C211yesmaximal, characteristic, complemented
11C53×#461C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2212yescomplemented
13C22×D212—21no—
14C2×D4×D53C2212yes—
15C2×#11719913C2212yescomplemented
16C2×#11719923C2211yescharacteristic
17C2×#11719924C2212yescomplemented
18C2×#11719924C2212yescomplemented
19SD16×D53—28no—
20C2×C8×D53C2212yescomplemented
21#13133216—21no—
22#13133214C2211yescharacteristic
23#13133195—22no—
24#13133193C2212yescomplemented
25#13133192C2212yes—
26#13133191—21no—
27#13133187—22no—
28C4×Dic106C2211yescharacteristic, complemented
29C4×D212C2211yescharacteristic
30#13133179—21no—
31#13133148C2211yescharacteristic, complemented
32#13133140C2211yescharacteristic, complemented
33#13133139C2212yescomplemented
34#13133138C2211yescharacteristic
35#13133129C2212yescomplemented
36C4×#88322C2211yescharacteristic, complemented
37C2×SD16×C53C2212yes—
38C4⋊Q8×C53C2211yescharacteristic, complemented
39C4×D4×C53C2211yescharacteristic
40C4⋊C8×C53C2211yescharacteristic, complemented
41#103×C53C2212yescomplemented
42C23×D53—22no—
43C2×Q8×C53C2312yes—
44C2×D4×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C2×C22⋊D53—22no—
47Q8×D53—24no—
48Q8×D53—42no—
49D4×D53—24no—
50D4×D53—42no—
51C2×D212—22no—
52C2×D212—21no—
53C2×D212—21no—
54C2×D212—44no—
55C2×D212D411yescharacteristic
56C2×D212C2312yes—
57C2×C4×D53—22no—
58C2×C4×D53C2312yes—
59C2×Dic106—22no—
60C2×Dic106C2311yescharacteristic
61SD16×C53—24no—
62C2×C8×C53C2312yes—
63C4⋊C4×C53—21no—
64C4⋊C4×C53—21no—
65C4⋊C4×C53C2311yescharacteristic
66C22⋊C4×C53—21no—
67C42×C53C2311yescharacteristic
68#11719924—24no—
69#11719924—42no—
70#11719923D414yescomplemented
71#11719921—22no—
72#11719921—22no—
73#11719921D412yes—
74C4⋊Dic53D411yescharacteristic
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77#11719919—22no—
78C4×Dic53—21no—
79C2×#88322C2311yescharacteristic
80C2×#88322C2311yescharacteristic
81#11719913—24no—
82C8×D53—24no—
83C23×C53—21no—
84C22×D53—22no—
85C22×D53—44no—
86C22×D53—44no—
87C22×D53—82no—
88C22×D53C2×D412yes—
89Q8×C53—22no—
90Q8×C53—22no—
91D4×C53—21no—
92D4×C53C2×D412yes—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2411yescharacteristic
99C22⋊D53—44no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53C2×D412yes—
103D212—22no—
104D212—22no—
105D212—44no—
106D212—42no—
107D212—81no—
108D212SD1614yescomplemented
109C4×D53—22no—
110C4×D53—22no—
111C4×D53—44no—
112Dic106—21no—
113Dic106—44no—
114Dic106C2×D412yescomplemented
115C8×C53—22no—
116#88322—21no—
117#88322C2×D412yescomplemented
118C22×C53—22no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53C22×D411yescharacteristic
122C2×D53—22no—
123C2×D53—22no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—88no—
128C2×D53C2×SD1614yescomplemented
129C4×C53—21no—
130C4×C53—42no—
131C4×C53—41no—
132C4×C53C2×SD1612yescomplemented
133C4×C53C22×D411yescharacteristic
134C4×C53H(5,𝔽2)11yescharacteristic
135Dic53—22no—
136Dic53—22no—
137Dic53—41no—
138D4×SD16—531nomaximal, Sylow
139C2×C53—42no—
140C2×C53D4211yescharacteristic
141C2×C53C22×SD1611yescharacteristic
142C2×C53#57711yescharacteristic
143D53—24no—
144D53—82no—
145C22×SD16—532no—
146D4×Q8—531no—
147D42—531no—
148#492—531no—
149#465—532no—
150#461D5311yescharacteristic, complemented
151#460—531no—
152#450—532no—
153#448—532no—
154C4×SD16—531no—
155C8×D4—531no—
156C53D4×SD1611yescharacteristic, complemented, Sylow
157C22×Q8—532no—
158C22×D4—532no—
159C22×D4—1061no—
160C2×SD16—531no—
161C2×SD16—532no—
162C2×SD16—532no—
163C2×SD16—532no—
164C2×SD16—1068no—
165C2×SD16C2×D5312yes—
166C22×C8—532no—
167C4⋊Q8—1061no—
168C4⋊Q8C2×D5311yescharacteristic, complemented
169#127—531no—
170C22⋊Q8—532no—
171C22⋊Q8—1062no—
172#121—532no—
173#121—1061no—
174#120—1062no—
175C4×Q8—531no—
176C4×D4—531no—
177C4×D4—1061no—
178C4×D4C2×D5311yescharacteristic
179#106—531no—
180C4⋊C8C2×D5311yescharacteristic, complemented
181#103—531no—
182#103C2×D5312yescomplemented
183#102—531no—
184#102—532no—
185C22⋊C8—532no—
186C4×C8—531no—
187C24—1062no—
188C2×Q8—531no—
189C2×Q8—1062no—
190C2×Q8—1064no—
191C2×Q8—2122no—
192C2×Q8C22×D5312yes—
193C2×D4—531no—
194C2×D4—532no—
195C2×D4—1062no—
196C2×D4—1062no—
197C2×D4—1064no—
198C2×D4—1061no—
199C2×D4—1061no—
200C2×D4—2124no—
201C2×D4—2122no—
202C2×D4C22×D5311yescharacteristic
203C22×C4—21no—
204C22×C4—532no—
205C22×C4—1062no—
206SD16—24no—
207SD16—534no—
208SD16—1064no—
209SD16—1064no—
210SD16—2122no—
211C2×C8—531no—
212C2×C8—531no—
213C2×C8—1064no—
214C2×C8C22×D5312yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—531no—
218C4⋊C4—531no—
219C4⋊C4—1062no—
220C4⋊C4—1061no—
221C4⋊C4C22×D5311yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—532no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C42—1061no—
227C42C22×D5311yescharacteristic
228C23—21no—
229C23—532no—
230C23—1062no—
231C23—4242no—
232C23—2124no—
233C23—2124no—
234Q8—22no—
235Q8—22no—
236Q8—532no—
237Q8—1061no—
238Q8—2124no—
239D4—21no—
240D4—534no—
241D4—1062no—
242D4—1062no—
243D4—4241no—
244D4—2124no—
245D4—2124no—
246D4—2122no—
247D4D4×D5312yes—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—532no—
252C2×C4—1061no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1061no—
256C2×C4—2124no—
257C2×C4D4×D5311yescharacteristic
258C2×C4D4×D5311yescharacteristic
259C2×C4C23×D5311yescharacteristic
260C8—22no—
261C8—532no—
262C8—1061no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—534no—
267C22—1062no—
268C22—1062no—
269C22—4248no—
270C22—2122no—
271C22—2122no—
272C22—2122no—
273C22C2×D4×D5311yescharacteristic
274C4—21no—
275C4—41no—
276C4—42no—
277C4—1062no—
278C4—1062no—
279C4—2121no—
280C4SD16×D5312yescomplemented
281C4C2×D4×D5311yescharacteristic
282C4#1313326311yescharacteristic
283C2—42no—
284C2—1064no—
285C2—4242no—
286C2#1681125711yescharacteristic
287C2C2×SD16×D5311yescharacteristic
288C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#461C2
socleC22×C53C22×D4
hypercenter#461D53
hypocenterC53D4×SD16
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D53C2semidirect
2. #16811288C2semidirect
4. #16810997C2semidirect
5. #16810855C2semidirect
6. #16810824C2semidirect
7. #16810772C2semidirect
8. #16810763C2semidirect
9. C4×#11719923C2semidirect
10. #16810688C2semidirect
11. C53×#461C2semidirect
12. C2×Q8×D53C22semidirect
15. C2×#11719913C22semidirect
17. C2×#11719924C22semidirect
18. C2×#11719924C22semidirect
20. C2×C8×D53C22semidirect
24. #13133193C22semidirect
28. C4×Dic106C22semidirect
31. #13133148C22semidirect
32. #13133140C22semidirect
33. #13133139C22semidirect
35. #13133129C22semidirect
36. C4×#88322C22semidirect
38. C4⋊Q8×C53C22semidirect
40. C4⋊C8×C53C22semidirect
41. #103×C53C22semidirect
70. #11719923D4semidirect
108. D212SD16semidirect
114. Dic106C2×D4semidirect
117. #88322C2×D4semidirect
128. C2×D53C2×SD16semidirect
132. C4×C53C2×SD16semidirect
150. #461D53semidirect
156. C53D4×SD16semidirect
168. C4⋊Q8C2×D53semidirect
180. C4⋊C8C2×D53semidirect
182. #103C2×D53semidirect
280. C4SD16×D53semidirect