Finite Group Databasebeta

← #20285713

Subgroups — #20285713

Order 6784.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#16811178C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic
3#16811015C211yesmaximal, characteristic, complemented
4#16810983C211yesmaximal, characteristic, complemented
5#16810982C211yesmaximal, characteristic, complemented
6#16810847C212yesmaximal, complemented
7#16810843C212yesmaximal, complemented
8#16810777C212yesmaximal, complemented
9#16810772C212yesmaximal
10#16810678C211yesmaximal, characteristic, complemented
11C53×#497C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×#11719925C2212yescomplemented
15C2×#11719924C2212yes—
16#13133217C2211yescharacteristic, complemented
17#13133208—21no—
18#13133206C2211yescharacteristic
19#13133205C2211yescharacteristic
20#13133204C2211yescharacteristic
21#13133203—21no—
22#13133203—21no—
23#13133197—21no—
24#13133193C2212yes—
25Q8×Dic53C2212yescomplemented
26#13133182—21no—
27#13133175—21no—
28C22⋊C4×D53—21no—
29C22⋊C4×D53—21no—
30C42×D53C2211yescharacteristic
31#13133146C2212yescomplemented
32#13133142C2212yescomplemented
33#13133141C2212yescomplemented
34#13133139C2212yes—
35#13133138C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic, complemented
38C2×Q16×C53C2212yescomplemented
39C2×SD16×C53C2212yes—
40C4⋊Q8×C53C2211yescharacteristic, complemented
41#124×C53C2211yescharacteristic
42#97×C53C2211yescharacteristic, complemented
43C23×D53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×Q8×C53C2312yes—
46C2×D4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48Q8×D53—42no—
49D4×D53—42no—
50C2×D212C2311yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53D411yescharacteristic
53C2×C4×D53D411yescharacteristic
54C2×C4×D53C2311yescharacteristic
55C2×Dic106—21no—
56C2×Dic106—21no—
57C2×Dic106C2311yescharacteristic
58Q16×C53—42no—
59SD16×C53—42no—
60C2×C8×C53C2312yes—
61C4⋊C4×C53—21no—
62C4⋊C4×C53—21no—
63C22⋊C4×C53—21no—
64C22⋊C4×C53—21no—
65C42×C53C2311yescharacteristic
66C22⋊Dic53—21no—
67C22⋊Dic53—21no—
68#11719925—42no—
69#11719924—42no—
70#11719921—22no—
71#11719921—21no—
72#11719921—21no—
73#11719921—21no—
74#11719921—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53—22no—
77C4⋊Dic53C2312yes—
78#11719919—22no—
79C4×Dic53—22no—
80C4×Dic53D411yescharacteristic
81C4×Dic53D411yescharacteristic
82C4×Dic53C2311yescharacteristic
83C2×#88322C2312yes—
84C23×C53—21no—
85C22×D53—21no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53—42no—
89C22×D53C2×D411yescharacteristic
90Q8×C53—22no—
91Q8×C53—22no—
92Q8×C53—22no—
93D4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2411yescharacteristic
99C22⋊D53—42no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D411yescharacteristic
104C2×Dic53C2×D411yescharacteristic
105D212—22no—
106C4×D53—22no—
107C4×D53—22no—
108C4×D53—41no—
109C4×D53—42no—
110Dic106—22no—
111Dic106—42no—
112Dic106—42no—
113C8×C53—22no—
114#88322—22no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53—41no—
118C22×C53C22×D411yescharacteristic
119C2×D53—21no—
120C2×D53—21no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—84no—
125C2×D53#13712yes—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—42no—
130C4×C53H(5,𝔽2)12yes—
131Dic53—21no—
132Dic53—21no—
133Dic53—21no—
134Dic53—41no—
135Dic53—42no—
136#3215—531nomaximal, Sylow
137C2×C53—81no—
138C2×C53#53511yescharacteristic
139C2×C53C2×#13711yescharacteristic
140C2×C53#57511yescharacteristic
141D53—22no—
142D53—81no—
143#533—531no—
144C2×#124—531no—
145#497—531no—
146#497D5311yescharacteristic, complemented
147#489—531no—
148#489—531no—
149#464—532no—
150#464—532no—
151#451—532no—
152#450—532no—
153#432—531no—
154C53#321511yescharacteristic, complemented, Sylow
155C22×Q8—531no—
156C22×D4—531no—
157C2×Q16—532no—
158C2×Q16C2×D5312yescomplemented
159C2×SD16—532no—
160C2×SD16C2×D5312yes—
161C4⋊Q8—531no—
162C4⋊Q8C2×D5311yescharacteristic, complemented
163#125—1061no—
164#124—531no—
165#124—531no—
166#124—531no—
167#124—1061no—
168#124—1061no—
169#124C2×D5311yescharacteristic
170C22⋊Q8—532no—
171C22⋊Q8—1061no—
172C4×Q8—532no—
173C4×Q8—1061no—
174#117—1061no—
175C2×C22⋊C4—1061no—
176C2×C22⋊C4—1061no—
177C2×C42—531no—
178C4⋊C8—532no—
179#103—532no—
180#103—532no—
181#102—532no—
182#102—532no—
183C22⋊C8—532no—
184#97—531no—
185#97C2×D5311yescharacteristic, complemented
186C24—1061no—
187C2×Q8—531no—
188C2×Q8—1061no—
189C2×Q8—1061no—
190C2×Q8—2122no—
191C2×Q8C22×D5311yescharacteristic
192C2×Q8C22×D5312yes—
193C2×D4—531no—
194C2×D4—1061no—
195C2×D4—2122no—
196C2×D4C22×D5311yescharacteristic
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—1061no—
201Q16—42no—
202Q16—2122no—
203SD16—42no—
204SD16—2122no—
205C2×C8—532no—
206C2×C8C22×D5312yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—532no—
210C4⋊C4—1062no—
211C4⋊C4—1062no—
212C4⋊C4—1061no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1061no—
216C22⋊C4—1061no—
217C22⋊C4—1062no—
218C22⋊C4—1061no—
219C22⋊C4—1061no—
220C22⋊C4—1061no—
221C22⋊C4—1061no—
222C42—531no—
223C42—531no—
224C42—531no—
225C42—1062no—
226C42C22×D5311yescharacteristic
227C23—21no—
228C23—531no—
229C23—1061no—
230C23—2122no—
231C23—2122no—
232C23—2122no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—1062no—
237Q8—2122no—
238Q8—2122no—
239D4—22no—
240D4—1062no—
241D4—2122no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—531no—
245C2×C4—531no—
246C2×C4—531no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—2121no—
252C2×C4—2122no—
253C2×C4D4×D5311yescharacteristic
254C2×C4D4×D5311yescharacteristic
255C2×C4C23×D5311yescharacteristic
256C8—22no—
257C8—1062no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—532no—
262C22—1061no—
263C22—1061no—
264C22—4244no—
265C22—2121no—
266C22—2121no—
267C22—2121no—
268C22C2×D4×D5311yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—1061no—
274C4—1061no—
275C4—1061no—
276C4—2122no—
277C4—2121no—
278C4#1313326212yes—
279C2—81no—
280C2—1062no—
281C2—4241no—
282C2#1681118611yescharacteristic
283C2#137×D5311yescharacteristic
284C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#497C2
socleC22×C53C22×D4
hypercenter#497D53
hypocenterC53#3215
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811015C2semidirect
4. #16810983C2semidirect
5. #16810982C2semidirect
6. #16810847C2semidirect
7. #16810843C2semidirect
8. #16810777C2semidirect
10. #16810678C2semidirect
11. C53×#497C2semidirect
14. C2×#11719925C22semidirect
16. #13133217C22semidirect
25. Q8×Dic53C22semidirect
31. #13133146C22semidirect
32. #13133142C22semidirect
33. #13133141C22semidirect
37. #13133128C22semidirect
38. C2×Q16×C53C22semidirect
40. C4⋊Q8×C53C22semidirect
42. #97×C53C22semidirect
146. #497D53semidirect
154. C53#3215semidirect
158. C2×Q16C2×D53semidirect
162. C4⋊Q8C2×D53semidirect
185. #97C2×D53semidirect