Finite Group Databasebeta

← #20285634

Subgroups — #20285634

Order 6784.

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

normal subgroups only

This group has 532 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#16811174C211yesmaximal, characteristic, complemented
2#127×D53C211yesmaximal, characteristic, complemented
3#16811000C211yesmaximal, characteristic, complemented
4#16810999C211yesmaximal, characteristic, complemented
5#16810971C211yesmaximal, characteristic, complemented
6#16810817C214yesmaximal, complemented
7#16810758C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#486C211yesmaximal, characteristic, complemented
10C2×#11719949—21no—
11C2×D4×D53—21no—
12C2×D4×D53—21no—
13C2×D4×D53C2212yes—
14C2×D424C2214yescomplemented
15C2×#11719922C2214yescomplemented
16#13133214C2211yescharacteristic
17#13133214C2211yescharacteristic, complemented
18#13133213—21no—
19#13133213—21no—
20#13133213C2212yes—
21#13133206—21no—
22#13133192—21no—
23#13133192C2212yes—
24#13133179—21no—
25#13133179C2212yescomplemented
26C42×D53C2211yescharacteristic
27#13133146C2212yescomplemented
28#13133137C2214yescomplemented
29#13133129C2212yes—
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—
43#11719949—44no—
44D4×D53—44no—
45D4×D53—44no—
46D4×D53—44no—
47C2×D212—22no—
48C2×D212—21no—
49C2×D212—21no—
50C2×D212—21no—
51C2×D212—21no—
52C2×D212C2312yes—
53C2×D212C2312yes—
54C2×C4×D53—22no—
55C2×C4×D53D411yescharacteristic
56C2×C4×D53D411yescharacteristic
57C2×C4×D53C2311yescharacteristic
58C2×C8×C53C2312yes—
59C4⋊C4×C53—21no—
60C4⋊C4×C53C2312yes—
61C42×C53C2311yescharacteristic
62#11719922—44no—
63#11719921—22no—
64#11719921—22no—
65C4×Dic53D411yescharacteristic
66C4×Dic53D411yescharacteristic
67C4×Dic53C2311yescharacteristic
68C2×#88322C2312yes—
69D424—44no—
70C23×C53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—44no—
74C22×D53—44no—
75C22×D53—44no—
76C22×D53C2×D411yescharacteristic
77Q8×C53—42no—
78D4×C53—24no—
79D4×C53—42no—
80D4×C53—42no—
81C2×C4×C53—22no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C22⋊D53—44no—
86C22⋊D53—44no—
87C22⋊D53—44no—
88C2×Dic53C2×D411yescharacteristic
89C2×Dic53C2×D411yescharacteristic
90C2×Dic53C2×D411yescharacteristic
91D212—24no—
92D212—24no—
93D212—44no—
94D212—42no—
95D212—42no—
96D212—42no—
97D212—42no—
98C4×D53—22no—
99C4×D53—22no—
100C4×D53—44no—
101C4×D53D814yescomplemented
102C8×C53—22no—
103#88322—22no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53C22×D411yescharacteristic
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—88no—
117C2×D53C2×D812yes—
118C4×C53—42no—
119C4×C53C2×D812yes—
120C4×C53AGL(1,ℤ/8)12yes—
121C4×C53H(5,𝔽2)12yes—
122Dic53—21no—
123Dic53—21no—
124Dic53C2×D812yes—
125#3205—531nomaximal, Sylow
126C2×C53—82no—
127C2×C53#53511yescharacteristic
128C2×C53C22×D811yescharacteristic
129C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
130D53—22no—
131D53—82no—
132D53—82no—
133#532—531no—
134C2×#127—531no—
135#493—531no—
136#493—531no—
137#486—531no—
138#486D5311yescharacteristic, complemented
139#459—534no—
140#447—534no—
141#431—531no—
142C53#320511yescharacteristic, complemented, Sylow
143C2×#41—1061no—
144C22×D4—532no—
145C22×D4—1061no—
146C22×D4—1061no—
147C2×D8—534no—
148C2×D8—534no—
149C4⋊Q8C2×D5311yescharacteristic, complemented
150#127—532no—
151#127—531no—
152#127—531no—
153#127—1061no—
154#127—1061no—
155#127C2×D5311yescharacteristic
156#124—1061no—
157#121—532no—
158#121—1061no—
159C4×D4—532no—
160C4×D4—1061no—
161C2×C42—531no—
162C4⋊C8—532no—
163#102—534no—
164#102C2×D5314yescomplemented
165C22⋊C8—532no—
166C4×C8—531no—
167C4×C8C2×D5311yescharacteristic, complemented
168C24—1062no—
169#41—2124no—
170C2×Q8—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—532no—
174C2×D4—532no—
175C2×D4—1062no—
176C2×D4—1062no—
177C2×D4—1062no—
178C2×D4—1062no—
179C2×D4—1061no—
180C2×D4—1061no—
181C2×D4—1061no—
182C2×D4—1061no—
183C2×D4—2124no—
184C2×D4—2124no—
185C2×D4—2124no—
186C2×D4C22×D5312yes—
187C22×C4—531no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—1062no—
191D8—2124no—
192D8—2124no—
193C2×C8—532no—
194C2×C8C22×D5312yes—
195C4⋊C4—21no—
196C4⋊C4C22×D5312yes—
197C22⋊C4—1062no—
198C22⋊C4—1062no—
199C42—531no—
200C42—531no—
201C42—531no—
202C42C22×D5311yescharacteristic
203C23—22no—
204C23—531no—
205C23—1062no—
206C23—1062no—
207C23—2124no—
208C23—2124no—
209C23—2124no—
210Q8—42no—
211D4—24no—
212D4—42no—
213D4—42no—
214D4—1064no—
215D4—1064no—
216D4—2124no—
217D4—2124no—
218D4—2124no—
219D4—2124no—
220D4—2122no—
221D4—2122no—
222D4—2122no—
223D4—2122no—
224C2×C4—22no—
225C2×C4—531no—
226C2×C4—534no—
227C2×C4—531no—
228C2×C4—531no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—2124no—
232C2×C4D4×D5311yescharacteristic
233C2×C4D4×D5311yescharacteristic
234C2×C4C23×D5311yescharacteristic
235C8—22no—
236C8—1062no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—532no—
241C22—1061no—
242C22—1061no—
243C22—4248no—
244C22—2122no—
245C22—2122no—
246C22—2122no—
247C22—2122no—
248C22—2122no—
249C22—2122no—
250C22C2×D4×D5311yescharacteristic
251C4—42no—
252C4—532no—
253C4—1061no—
254C4—1061no—
255C4D8×D5312yes—
256C4#1313324412yes—
257C4#1313326312yes—
258C2—82no—
259C2—1062no—
260C2—4242no—
261C2—4242no—
262C2#1681119111yescharacteristic
263C2C2×D8×D5311yescharacteristic
264C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3205
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. #16811174C2semidirect
2. #127×D53C2semidirect
3. #16811000C2semidirect
4. #16810999C2semidirect
5. #16810971C2semidirect
6. #16810817C2semidirect
7. #16810758C2semidirect
8. #16810674C2semidirect
9. C53×#486C2semidirect
14. C2×D424C22semidirect
15. C2×#11719922C22semidirect
17. #13133214C22semidirect
25. #13133179C22semidirect
27. #13133146C22semidirect
28. #13133137C22semidirect
30. C4×#88322C22semidirect
31. C4⋊Q8×C53C22semidirect
33. #102×C53C22semidirect
34. C4×C8×C53C22semidirect
101. C4×D53D8semidirect
138. #486D53semidirect
142. C53#3205semidirect
149. C4⋊Q8C2×D53semidirect
164. #102C2×D53semidirect
167. C4×C8C2×D53semidirect