Finite Group Databasebeta

← #20285683

Subgroups — #20285683

Order 6784.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811159C211yesmaximal, characteristic, complemented
2C2×#13133204C211yesmaximal, characteristic, complemented
3#16811004C212yesmaximal, complemented
4#16810976C212yesmaximal, complemented
5#16810843C212yesmaximal, complemented
6#16810837C212yesmaximal, complemented
7#16810778C211yesmaximal, characteristic, complemented
8#16810764C211yesmaximal, characteristic, complemented
9#16810757C211yesmaximal, characteristic
10#16810673C211yesmaximal, characteristic, complemented
11C53×#450C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C2×#11719925C2212yescomplemented
14C2×#11719913C2212yescomplemented
15C2×#11719922C2212yes—
16#13133212—21no—
17#13133208C2211yescharacteristic, complemented
18#13133206C2211yescharacteristic, complemented
19#13133204—21no—
20#13133204—21no—
21#13133204C2212yes—
22#13133204C2212yes—
23#13133201—21no—
24#13133191C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic, complemented
26#13133179C2211yescharacteristic, complemented
27#13133177—21no—
28#13133175—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53C2211yescharacteristic
32#13133146C2212yescomplemented
33#13133142C2212yescomplemented
34#13133140C2212yescomplemented
35#13133138C2212yes—
36C22⋊#88322C2211yescharacteristic
37C8×Dic53C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×SD16×C53C2212yes—
40C22⋊Q8×C53C2211yescharacteristic
41#102×C53C2212yes—
42C22⋊C8×C53C2211yescharacteristic
43C24×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—42no—
47C2×D4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—21no—
50C22×Dic53—21no—
51C22×Dic53D411yescharacteristic
52C22×Dic53D411yescharacteristic
53C2×D212C2311yescharacteristic
54C2×C4×D53—21no—
55C2×Dic106—21no—
56C2×Dic106—42no—
57C2×Dic106C2312yes—
58SD16×C53—42no—
59C2×C8×C53C2312yes—
60C4⋊C4×C53—21no—
61C4⋊C4×C53C2311yescharacteristic
62C22⋊C4×C53—21no—
63C22⋊Dic53—22no—
64C22⋊Dic53—22no—
65C22⋊Dic53—22no—
66C22⋊Dic53—22no—
67#11719925—42no—
68#11719922—42no—
69#11719921—21no—
70#11719921—21no—
71#11719921—21no—
72#11719921—21no—
73C4⋊Dic53—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77C4×Dic53—21no—
78C4×Dic53D411yescharacteristic
79C4×Dic53D411yescharacteristic
80C4×Dic53C2311yescharacteristic
81C4×Dic53C2311yescharacteristic
82C2×#88322C2312yes—
83#11719913—42no—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53C2×D411yescharacteristic
89C22×D53—21no—
90Q8×C53—22no—
91D4×C53—22no—
92D4×C53—22no—
93D4×C53—42no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2411yescharacteristic
99C22⋊D53—42no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53C2×D411yescharacteristic
106C2×Dic53C2×D411yescharacteristic
107C2×Dic53C2×D411yescharacteristic
108C2×Dic53C2×D411yescharacteristic
109D212—22no—
110C4×D53—42no—
111Dic106—22no—
112Dic106—22no—
113Dic106—42no—
114C8×C53—22no—
115#88322—22no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—84no—
122C22×C53#13512yes—
123C22×C53C22×D411yescharacteristic
124C2×D53—41no—
125C2×D53—41no—
126C2×D53—41no—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—41no—
130C4×C53H(5,𝔽2)12yes—
131Dic53—21no—
132Dic53—21no—
133Dic53—21no—
134Dic53—21no—
135Dic53—42no—
136Dic53—41no—
137#3211—531nomaximal, Sylow
138C2×C53—22no—
139C2×C53—82no—
140C2×C53#53511yescharacteristic
141C2×C53C2×#13511yescharacteristic
142C2×C53#57511yescharacteristic
143D53—81no—
144#529—531no—
145C2×#124—531no—
146#495—532no—
147#488—532no—
148#464—532no—
149#463—532no—
150#451—531no—
151#450D5311yescharacteristic, complemented
152#448—531no—
153#447—531no—
154#431—531no—
155C53#321111yescharacteristic, complemented, Sylow
156C22×Q8—531no—
157C22×D4C2×D5311yescharacteristic
158C2×Q16—532no—
159C2×SD16—532no—
160C2×SD16C2×D5312yes—
161C2×D8—532no—
162#126—1061no—
163#125—531no—
164#124—531no—
165#124—532no—
166#124—532no—
167#124—1061no—
168#124—1061no—
169#123—1061no—
170C22⋊Q8C2×D5311yescharacteristic
171#121—531no—
172C4×Q8—531no—
173C4×D4—531no—
174C4×D4—1061no—
175#117—1061no—
176C2×C22⋊C4—1061no—
177C2×C22⋊C4—1061no—
178C2×C42—531no—
179C4⋊C8—532no—
180#103—532no—
181#103—532no—
182#102—532no—
183#102C2×D5312yes—
184C22⋊C8—531no—
185C22⋊C8C2×D5311yescharacteristic
186C4×C8—532no—
187C24—21no—
188C2×Q8—532no—
189C2×Q8—1061no—
190C2×Q8—2122no—
191C2×Q8C22×D5311yescharacteristic
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—531no—
195C2×D4—1061no—
196C2×D4C22×D5312yes—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4C22×D5311yescharacteristic
202Q16—2122no—
203SD16—42no—
204SD16—2122no—
205D8—2122no—
206C2×C8—532no—
207C2×C8C22×D5312yes—
208C4⋊C4—21no—
209C4⋊C4—531no—
210C4⋊C4—1061no—
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C4⋊C4C22×D5311yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1061no—
216C22⋊C4—1061no—
217C22⋊C4—1061no—
218C22⋊C4—1062no—
219C22⋊C4—1062no—
220C22⋊C4—1062no—
221C22⋊C4—1062no—
222C22⋊C4—1061no—
223C42—531no—
224C42—531no—
225C42—531no—
226C42—531no—
227C42—1061no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—1061no—
233C23D4×D5311yescharacteristic
234Q8—22no—
235Q8—1062no—
236Q8—1062no—
237Q8—2122no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—1062no—
242D4—2122no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—531no—
247C2×C4—531no—
248C2×C4—531no—
249C2×C4—531no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—2122no—
254C2×C4—2121no—
255C2×C4—2122no—
256C2×C4D4×D5311yescharacteristic
257C2×C4C23×D5311yescharacteristic
258C8—22no—
259C8—1062no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—2121no—
267C22—2121no—
268C22—2121no—
269C22#1313323812yes—
270C22C2×D4×D5311yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—1061no—
275C4—1061no—
276C4—1061no—
277C4—1061no—
278C4—2121no—
279C4—2122no—
280C4#1313326212yes—
281C2—22no—
282C2—82no—
283C2—4241no—
284C2#1681118811yescharacteristic
285C2C2×#1313323811yescharacteristic
286C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3211
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811159C2semidirect
2. C2×#13133204C2semidirect
3. #16811004C2semidirect
4. #16810976C2semidirect
5. #16810843C2semidirect
6. #16810837C2semidirect
7. #16810778C2semidirect
8. #16810764C2semidirect
10. #16810673C2semidirect
11. C53×#450C2semidirect
13. C2×#11719925C22semidirect
14. C2×#11719913C22semidirect
17. #13133208C22semidirect
18. #13133206C22semidirect
25. Q8×Dic53C22semidirect
26. #13133179C22semidirect
32. #13133146C22semidirect
33. #13133142C22semidirect
34. #13133140C22semidirect
37. C8×Dic53C22semidirect
151. #450D53semidirect
155. C53#3211semidirect