Finite Group Databasebeta

← #20285588

Subgroups — #20285588

Order 6784.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811194C211yesmaximal, characteristic, complemented
2#16811187C211yesmaximal, characteristic
3#16810965C212yesmaximal, complemented
4#16810948C212yesmaximal, complemented
5#16810869C212yesmaximal, complemented
6#16810852C212yesmaximal, complemented
7#16810776C212yesmaximal, complemented
8#16810764C212yesmaximal, complemented
9C53×#408C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic, complemented
11C22×D212C2211yescharacteristic
12C2×#11719913C2212yes—
13C2×#11719913C2212yes—
14#13133215—21no—
15#13133213—21no—
16#13133206—21no—
17#13133204—21no—
18#13133198—21no—
19#13133198—21no—
20#13133196C2212yes—
21#13133196C2212yescomplemented
22#13133191C2212yes—
23#13133191C2212yes—
24#13133185—21no—
25#13133185—21no—
26#13133176C2211yescharacteristic
27#13133176C2211yescharacteristic, complemented
28#13133151C2212yescomplemented
29#13133149C2212yescomplemented
30#13133142C2212yescomplemented
31#13133142C2212yescomplemented
32#13133139C2212yes—
33#13133139C2212yes—
34C2×#41×C53C2211yescharacteristic
35C2×M16×C53C2211yescharacteristic, complemented
36C22×C8×C53C2211yescharacteristic, complemented
37C22⋊C8×C53C2212yes—
38C22⋊C8×C53C2212yescomplemented
39C23×D53—21no—
40#41×C53—41no—
41#41×C53—41no—
42C2×Q8×C53D411yescharacteristic
43C2×D4×C53D411yescharacteristic
44C2×D4×C53D411yescharacteristic
45C2×D4×C53D411yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C22×Dic53—21no—
52C2×D212—21no—
53C2×D212—41no—
54C2×D212—41no—
55C2×D212C2312yes—
56C2×Dic106—21no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106C2312yes—
60M16×C53—41no—
61C2×C8×C53—41no—
62C2×C8×C53C2312yes—
63C2×C8×C53C2312yes—
64C22⋊Dic53—21no—
65C22⋊Dic53—21no—
66C22⋊Dic53—22no—
67C22⋊Dic53—22no—
68#11719921—22no—
69#11719921—22no—
70C4⋊Dic53C2312yes—
71C4⋊Dic53C2312yes—
72#11719919—22no—
73#11719919—22no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76#11719913—42no—
77#11719913—42no—
78C23×C53C2×D411yescharacteristic
79C23×C53C2×D411yescharacteristic
80C23×C53C2×D411yescharacteristic
81C22×D53—22no—
82C22×D53—42no—
83C22×D53—82no—
84Q8×C53—41no—
85D4×C53—41no—
86D4×C53—41no—
87D4×C53—41no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—82no—
99C22⋊D53—82no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—42no—
104D212—22no—
105D212—22no—
106D212—42no—
107Dic106—22no—
108Dic106—22no—
109Dic106—42no—
110C8×C53—22no—
111C8×C53—22no—
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—22no—
118C22×C53C22×D411yescharacteristic
119C2×D53—42no—
120C2×D53—42no—
121C2×D53—42no—
122C2×D53—84no—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53H(5,𝔽2)11yescharacteristic
127C4×C53H(5,𝔽2)11yescharacteristic
128Dic53—42no—
129Dic53—42no—
130Dic53—42no—
131#3200—531nomaximal, Sylow
132C2×C53—41no—
133C2×C53—41no—
134C2×C53—41no—
135C2×C53#53411yescharacteristic
136C2×C53#57712yes—
137D53—82no—
138#536—531no—
139#535—531no—
140#485—532no—
141#482—532no—
142#468—532no—
143#465—532no—
144#450—532no—
145#448—532no—
146#408D5311yescharacteristic, complemented
147C53#320011yescharacteristic, complemented, Sylow
148C2×#41C2×D5311yescharacteristic
149C22×Q8—531no—
150C22×D4—531no—
151C2×SD16—532no—
152C2×SD16—532no—
153C2×M16C2×D5311yescharacteristic, complemented
154C22×C8C2×D5311yescharacteristic, complemented
155C4⋊Q8—1061no—
156#127—1061no—
157#124—1061no—
158#124—1061no—
159#123—1061no—
160#123—1061no—
161C22⋊Q8—532no—
162C22⋊Q8—532no—
163#121—532no—
164#121—532no—
165#120—1061no—
166#120—1061no—
167#117—531no—
168#117—531no—
169#107—532no—
170#106—532no—
171#103—532no—
172#103—532no—
173#102—532no—
174#102—532no—
175C22⋊C8C2×D5312yes—
176C22⋊C8C2×D5312yescomplemented
177C24—1061no—
178#41—41no—
179#41—41no—
180C2×Q8—532no—
181C2×Q8—1061no—
182C2×Q8—2121no—
183C2×Q8—2121no—
184C2×Q8D21211yescharacteristic
185C2×D4—532no—
186C2×D4—1062no—
187C2×D4—1062no—
188C2×D4—1061no—
189C2×D4—2121no—
190C2×D4—2121no—
191C2×D4D21211yescharacteristic
192C2×D4D21211yescharacteristic
193C2×D4D21211yescharacteristic
194C22×C4—1061no—
195C22×C4C22×D5311yescharacteristic
196C22×C4C22×D5311yescharacteristic
197C22×C4C22×D5311yescharacteristic
198SD16—2122no—
199SD16—2122no—
200M16—41no—
201C2×C8—41no—
202C2×C8C22×D5312yes—
203C2×C8C22×D5312yes—
204C4⋊C4—532no—
205C4⋊C4—532no—
206C4⋊C4—1062no—
207C4⋊C4—1062no—
208C22⋊C4—1061no—
209C22⋊C4—1061no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C42—1061no—
215C42—1061no—
216C23—1062no—
217C23—4242no—
218C23—2122no—
219C23C2×D21211yescharacteristic
220C23C2×D21211yescharacteristic
221C23C2×D21211yescharacteristic
222Q8—41no—
223Q8—1062no—
224Q8—1062no—
225Q8—2122no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—1062no—
230D4—1062no—
231D4—4242no—
232D4—4242no—
233D4—2122no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—2122no—
244C2×C4C2×D21211yescharacteristic
245C2×C4C2×D21211yescharacteristic
246C2×C4C2×D21211yescharacteristic
247C2×C4C23×D5311yescharacteristic
248C8—22no—
249C8—22no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—4244no—
257C22—2122no—
258C22—2122no—
259C22—2122no—
260C22C22×D21211yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—2122no—
265C4—2122no—
266C4—2122no—
267C4#1313326211yescharacteristic
268C4#1313326211yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—4242no—
273C2#1681118511yescharacteristic
274C2#1681143512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3200
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16811194C2semidirect
3. #16810965C2semidirect
4. #16810948C2semidirect
5. #16810869C2semidirect
6. #16810852C2semidirect
7. #16810776C2semidirect
8. #16810764C2semidirect
9. C53×#408C2semidirect
10. C22×Dic106C22semidirect
21. #13133196C22semidirect
27. #13133176C22semidirect
28. #13133151C22semidirect
29. #13133149C22semidirect
30. #13133142C22semidirect
31. #13133142C22semidirect
35. C2×M16×C53C22semidirect
36. C22×C8×C53C22semidirect
38. C22⋊C8×C53C22semidirect
146. #408D53semidirect
147. C53#3200semidirect
153. C2×M16C2×D53semidirect
154. C22×C8C2×D53semidirect
176. C22⋊C8C2×D53semidirect