Finite Group Databasebeta

← #20285577

Subgroups — #20285577

Order 6784.

442 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811189C212yesmaximal, complemented
2#16810949C212yesmaximal, complemented
3#16810946C212yesmaximal, complemented
4#16810876C211yesmaximal, characteristic, complemented
5#16810857C212yesmaximal, complemented
6#16810758C212yesmaximal, complemented
7#16810755C212yesmaximal, complemented
8#16810591C211yesmaximal, characteristic, complemented
9C53×#469C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic
11C2×D4×D53C2212yescomplemented
12C2×D424C2211yescharacteristic
13C2×#11719922C2212yes—
14C22×#88322C2211yescharacteristic, complemented
15#13133213—22no—
16#13133203—22no—
17#13133192C2212yes—
18#13133191C2212yescomplemented
19#13133188C2212yes—
20#13133187—22no—
21#13133185—22no—
22#13133175C2212yescomplemented
23#13133150C2212yescomplemented
24#13133149C2211yescharacteristic, complemented
25#13133139C2212yescomplemented
26#13133138C2212yescomplemented
27#13133137C2212yescomplemented
28#13133129C2212yescomplemented
29#13133129C2212yescomplemented
30C2×D8×C53C2211yescharacteristic
31C2×M16×C53C2211yescharacteristic, complemented
32#121×C53C2212yescomplemented
33#106×C53C2211yescharacteristic, complemented
34#102×C53C2212yescomplemented
35C23×D53—22no—
36C2×D4×C53—22no—
37C2×D4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—22no—
40C2×C22⋊D53—22no—
41C2×C22⋊D53—22no—
42C2×C22⋊D53D412yes—
43D4×D53—42no—
44D4×D53—42no—
45#11719945—41no—
46#11719945—41no—
47C2×D212—22no—
48C2×D212D411yescharacteristic
49C2×D212C2312yes—
50C2×C4×D53C2312yes—
51C2×Dic106D411yescharacteristic
52D8×C53—41no—
53M16×C53—41no—
54C2×C8×C53C2311yescharacteristic
55C2×C8×C53C2311yescharacteristic
56C4⋊C4×C53C2312yes—
57C22⋊C4×C53—22no—
58C22⋊Dic53—22no—
59#11719922—42no—
60#11719921—22no—
61#11719921—22no—
62#11719921—22no—
63C4⋊Dic53C2312yes—
64C4×Dic53—22no—
65C2×#88322—41no—
66C2×#88322C2312yes—
67D424—41no—
68C23×C53—22no—
69C23×C53C2×D411yescharacteristic
70C22×D53—22no—
71C22×D53—44no—
72C22×D53—82no—
73C22×D53—82no—
74C22×D53C2×D412yes—
75D4×C53—22no—
76D4×C53—22no—
77D4×C53—82no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2411yescharacteristic
83C22⋊D53—42no—
84C22⋊D53—44no—
85C22⋊D53—82no—
86C22⋊D53—82no—
87C2×Dic53—22no—
88C2×Dic53C2×D412yes—
89D212—22no—
90D212—22no—
91D212—41no—
92D212—82no—
93C4×D53—22no—
94C4×D53—22no—
95Dic106—41no—
96C8×C53—21no—
97C8×C53—21no—
98#88322—22no—
99C22×C53—21no—
100C22×C53—22no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53C22×D411yescharacteristic
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—42no—
111C2×D53—88no—
112C4×C53—21no—
113C4×C53—42no—
114C4×C53H(5,𝔽2)11yescharacteristic
115C4×C53H(5,𝔽2)11yescharacteristic
116Dic53—22no—
117Dic53—42no—
118#3199—531nomaximal, Sylow
119C2×C53—41no—
120C2×C53—82no—
121C2×C53#53411yescharacteristic
122C2×C53#57612yes—
123D53—42no—
124D53—82no—
125#535—532no—
126#482—532no—
127#482—532no—
128#469—531no—
129#469D5311yescharacteristic, complemented
130#466—532no—
131#447—532no—
132#447—532no—
133#408—531no—
134C53#319911yescharacteristic, complemented, Sylow
135C2×#41—531no—
136C22×D4—532no—
137C2×D8—532no—
138C2×D8—531no—
139C2×D8C2×D5311yescharacteristic
140C2×M16C2×D5311yescharacteristic, complemented
141C22×C8—531no—
142#127—1062no—
143#124—1062no—
144#121—532no—
145#121—532no—
146#121—532no—
147#121C2×D5312yescomplemented
148#120—1062no—
149#120—1062no—
150#117—532no—
151#107—532no—
152#106—531no—
153#106C2×D5311yescharacteristic, complemented
154#102—532no—
155#102—532no—
156#102—532no—
157#102C2×D5312yescomplemented
158C22⋊C8—532no—
159C22⋊C8—532no—
160C24—1062no—
161#41—2121no—
162#41—2121no—
163C2×Q8—531no—
164C2×D4—22no—
165C2×D4—532no—
166C2×D4—531no—
167C2×D4—532no—
168C2×D4—1062no—
169C2×D4—1062no—
170C2×D4—1062no—
171C2×D4—1062no—
172C2×D4—2122no—
173C2×D4—2122no—
174C2×D4C22×D5312yes—
175C22×C4—532no—
176C22×C4C22×D5311yescharacteristic
177D8—41no—
178D8—2121no—
179D8—2122no—
180M16—41no—
181C2×C8—532no—
182C2×C8—2121no—
183C2×C8C22×D5311yescharacteristic
184C2×C8C22×D5311yescharacteristic
185C4⋊C4—532no—
186C4⋊C4C22×D5312yes—
187C22⋊C4—22no—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4—1062no—
192C42—1062no—
193C23—22no—
194C23—532no—
195C23—1062no—
196C23—4242no—
197C23—4242no—
198C23—2124no—
199C23D4×D5311yescharacteristic
200Q8—2121no—
201D4—22no—
202D4—22no—
203D4—82no—
204D4—1062no—
205D4—1062no—
206D4—4242no—
207D4—4242no—
208D4—4242no—
209D4—2121no—
210D4—2122no—
211D4—2124no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—532no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4D4×D5311yescharacteristic
220C2×C4C23×D5311yescharacteristic
221C8—21no—
222C8—21no—
223C8—1062no—
224C22—22no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—1062no—
230C22—1062no—
231C22—1062no—
232C22—4248no—
233C22—2122no—
234C22—2122no—
235C22—2122no—
236C22C2×D4×D5311yescharacteristic
237C4—21no—
238C4—42no—
239C4—1062no—
240C4—2122no—
241C4#1313326211yescharacteristic
242C4#1313326311yescharacteristic
243C2—41no—
244C2—82no—
245C2—4242no—
246C2—2122no—
247C2#1681118111yescharacteristic
248C2#1681142712yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3199
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16811189C2semidirect
2. #16810949C2semidirect
3. #16810946C2semidirect
4. #16810876C2semidirect
5. #16810857C2semidirect
6. #16810758C2semidirect
7. #16810755C2semidirect
8. #16810591C2semidirect
9. C53×#469C2semidirect
11. C2×D4×D53C22semidirect
14. C22×#88322C22semidirect
18. #13133191C22semidirect
22. #13133175C22semidirect
23. #13133150C22semidirect
24. #13133149C22semidirect
25. #13133139C22semidirect
26. #13133138C22semidirect
27. #13133137C22semidirect
28. #13133129C22semidirect
29. #13133129C22semidirect
31. C2×M16×C53C22semidirect
32. #121×C53C22semidirect
33. #106×C53C22semidirect
34. #102×C53C22semidirect
129. #469D53semidirect
134. C53#3199semidirect
140. C2×M16C2×D53semidirect
147. #121C2×D53semidirect
153. #106C2×D53semidirect
157. #102C2×D53semidirect