Finite Group Databasebeta

← #20285086

Subgroups — #20285086

Order 6784.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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#127×D53C211yesmaximal, characteristic
3#16810986C214yesmaximal, complemented
4#16810971C211yesmaximal, characteristic, complemented
5#16810970C211yesmaximal, characteristic, complemented
6#16810969C211yesmaximal, characteristic, complemented
7#16810626C214yesmaximal, complemented
8C4×#11719912C211yesmaximal, characteristic, complemented
9C53×#486C211yesmaximal, characteristic, complemented
10C2×D4×D53—21no—
11C2×D4×D53—21no—
12C2×D4×D53C2212yes—
13C2×#11719912C2212yes—
14#13133217C2211yescharacteristic, complemented
15#13133214C2211yescharacteristic
16#13133213—21no—
17#13133213—21no—
18#13133213C2212yes—
19#13133208C2212yescomplemented
20#13133197—21no—
21#13133193—21no—
22Q8×Dic53—21no—
23#13133182—21no—
24#13133175C2212yes—
25C42×D53C2211yescharacteristic
26#13133139C2214yescomplemented
27#13133138C2214yescomplemented
28#13133137C2214yescomplemented
29#13133127C2212yescomplemented
30C4×#88322C2211yescharacteristic, complemented
31C4⋊Q8×C53C2211yescharacteristic, complemented
32#127×C53C2211yescharacteristic
33#102×C53C2214yescomplemented
34C4×C8×C53C2211yescharacteristic, complemented
35C23×D53—22no—
36C2×Q8×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53—21no—
39C2×D4×C53C2312yes—
40C2×C22⋊D53—22no—
41C2×C22⋊D53—22no—
42C2×C22⋊D53—22no—
43D4×D53—28no—
44D4×D53—44no—
45D4×D53—44no—
46C2×D212—21no—
47C2×D212—21no—
48C2×D212C2312yes—
49C2×C4×D53D411yescharacteristic
50C2×C4×D53D411yescharacteristic
51C2×C4×D53C2311yescharacteristic
52C2×Dic106—21no—
53C2×C8×C53C2312yes—
54C4⋊C4×C53—21no—
55C4⋊C4×C53C2312yes—
56C42×C53C2311yescharacteristic
57#11719921—22no—
58C4⋊Dic53—21no—
59C4⋊Dic53—22no—
60C4⋊Dic53C2312yes—
61#11719919—22no—
62C4×Dic53—22no—
63C4×Dic53D411yescharacteristic
64C4×Dic53D411yescharacteristic
65C4×Dic53C2311yescharacteristic
66C2×#88322C2312yes—
67#11719912—24no—
68C23×C53—22no—
69C22×D53—22no—
70C22×D53—44no—
71C22×D53—44no—
72C22×D53—44no—
73C22×D53C2×D411yescharacteristic
74Q8×C53—42no—
75D4×C53—24no—
76D4×C53—42no—
77D4×C53—42no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C22⋊D53—44no—
83C22⋊D53—44no—
84C22⋊D53—44no—
85C2×Dic53—22no—
86C2×Dic53C2×D411yescharacteristic
87C2×Dic53C2×D411yescharacteristic
88C2×Dic53C2×D411yescharacteristic
89D212—24no—
90D212—42no—
91D212—42no—
92C4×D53—22no—
93C4×D53—22no—
94C4×D53#4114yes—
95Dic106—42no—
96C8×C53—22no—
97#88322—22no—
98C22×C53—42no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53C22×D411yescharacteristic
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—88no—
108C2×D53#12412yes—
109C4×C53—42no—
110C4×C53AGL(1,ℤ/8)12yes—
111C4×C53AGL(1,ℤ/8)12yes—
112C4×C53C2×#4112yes—
113Dic53—21no—
114Dic53—21no—
115Dic53—42no—
116Dic53#12412yes—
117#3138—531nomaximal, Sylow
118C2×C53—82no—
119C2×C53C2×#12411yescharacteristic
120C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
122D53—22no—
123D53—82no—
124#533—531no—
125C2×#127—531no—
126#490—534no—
127#486—531no—
128#486—531no—
129#486—531no—
130#486D5311yescharacteristic, complemented
131#418—534no—
132C4×M16—531no—
133C53#313811yescharacteristic, complemented, Sylow
134C22×D4—532no—
135C22×D4—1061no—
136C22×D4—1061no—
137C2×M16—532no—
138C4⋊Q8—531no—
139C4⋊Q8C2×D5311yescharacteristic, complemented
140#127—532no—
141#127—531no—
142#127—1061no—
143#127—1061no—
144#127C2×D5311yescharacteristic
145#125—532no—
146C22⋊Q8—1061no—
147C22⋊Q8—1061no—
148C4×Q8—1061no—
149C4×Q8—1061no—
150#117—532no—
151C2×C42—531no—
152#102—534no—
153#102—534no—
154#102—534no—
155#102C2×D5314yescomplemented
156#97—532no—
157C4×C8—531no—
158C4×C8C2×D5311yescharacteristic, complemented
159C24—1062no—
160C2×Q8—21no—
161C2×Q8—1061no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—532no—
165C2×D4—1068no—
166C2×D4—1062no—
167C2×D4—1062no—
168C2×D4—1062no—
169C2×D4—1061no—
170C2×D4—1061no—
171C2×D4—2124no—
172C2×D4—2124no—
173C2×D4C22×D5312yes—
174C22×C4—531no—
175C22×C4—531no—
176C22×C4—531no—
177M16—1064no—
178C2×C8—532no—
179C2×C8C22×D5312yes—
180C4⋊C4—21no—
181C4⋊C4—532no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C4⋊C4—1061no—
185C4⋊C4C22×D5312yes—
186C22⋊C4—1062no—
187C42—531no—
188C42—531no—
189C42—531no—
190C42—1062no—
191C42C22×D5311yescharacteristic
192C23—22no—
193C23—531no—
194C23—1062no—
195C23—2124no—
196C23—2124no—
197C23—2124no—
198Q8—42no—
199Q8—2122no—
200D4—24no—
201D4—42no—
202D4—42no—
203D4—1064no—
204D4—2124no—
205D4—2124no—
206D4—2124no—
207D4—2122no—
208D4—2122no—
209C2×C4—22no—
210C2×C4—531no—
211C2×C4—534no—
212C2×C4—531no—
213C2×C4—531no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4D4×D5311yescharacteristic
218C2×C4D4×D5311yescharacteristic
219C2×C4C23×D5311yescharacteristic
220C8—22no—
221C8—1062no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—532no—
226C22—1061no—
227C22—1061no—
228C22—4248no—
229C22—2122no—
230C22—2122no—
231C22—2122no—
232C22C2×D4×D5311yescharacteristic
233C4—42no—
234C4—532no—
235C4—1061no—
236C4—1061no—
237C4—2122no—
238C4#1313324112yes—
239C4#1313324412yes—
240C4#41×D5312yes—
241C2—82no—
242C2—1062no—
243C2—4242no—
244C2#124×D5311yescharacteristic
245C2C2×#1313324111yescharacteristic
246C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3138
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16810986C2semidirect
4. #16810971C2semidirect
5. #16810970C2semidirect
6. #16810969C2semidirect
7. #16810626C2semidirect
8. C4×#11719912C2semidirect
9. C53×#486C2semidirect
14. #13133217C22semidirect
19. #13133208C22semidirect
26. #13133139C22semidirect
27. #13133138C22semidirect
28. #13133137C22semidirect
29. #13133127C22semidirect
30. C4×#88322C22semidirect
31. C4⋊Q8×C53C22semidirect
33. #102×C53C22semidirect
34. C4×C8×C53C22semidirect
130. #486D53semidirect
133. C53#3138semidirect
139. C4⋊Q8C2×D53semidirect
155. #102C2×D53semidirect
158. C4×C8C2×D53semidirect