Finite Group Databasebeta

← #20285582

Subgroups — #20285582

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#16811187C211yesmaximal, characteristic
2#16810961C212yesmaximal, complemented
3#16810947C212yesmaximal, complemented
4#16810871C212yesmaximal, complemented
5#16810854C212yesmaximal, complemented
6#16810773C212yesmaximal, complemented
7#16810762C212yesmaximal, complemented
8#16810590C211yesmaximal, characteristic, complemented
9C53×#536C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C2×#11719924C2212yes—
12C2×#11719924C2212yes—
13C2×#11719917C2211yescharacteristic, complemented
14C22×#88322C2211yescharacteristic, complemented
15#13133213—21no—
16#13133206—21no—
17#13133191C2212yes—
18#13133191C2212yes—
19#13133185—21no—
20#13133185—21no—
21#13133176C2211yescharacteristic
22#13133150C2212yescomplemented
23#13133148C2212yescomplemented
24#13133141C2212yescomplemented
25#13133141C2212yescomplemented
26#13133137C2212yes—
27#13133137C2212yes—
28C22⋊#88322C2212yes—
29C22⋊#88322C2212yescomplemented
30C2×#41×C53C2211yescharacteristic
31C22×Q8×C53C2211yescharacteristic, complemented
32C4⋊Q8×C53—21no—
33#124×C53—21no—
34#123×C53—21no—
35#123×C53—21no—
36C22⋊Q8×C53C2212yes—
37C22⋊Q8×C53C2212yescomplemented
38#117×C53C2211yescharacteristic, complemented
39C23×D53—21no—
40#41×C53—41no—
41#41×C53—41no—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53D411yescharacteristic
46C2×Q8×C53C2312yes—
47C2×D4×C53D411yescharacteristic
48C2×D4×C53D411yescharacteristic
49C2×D4×C53D411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53C2311yescharacteristic
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C2×D212—21no—
57C2×D212—41no—
58C2×D212—41no—
59C2×D212C2312yes—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C4⋊C4×C53C2312yes—
63C22⋊C4×C53—21no—
64C22⋊C4×C53—22no—
65C22⋊C4×C53—22no—
66C42×C53—21no—
67C22⋊Dic53—21no—
68#11719924—42no—
69#11719924—42no—
70#11719921—22no—
71#11719921—22no—
72C4⋊Dic53C2312yes—
73C4×Dic53—21no—
74#11719917—41no—
75C2×#88322—41no—
76C2×#88322C2312yes—
77C2×#88322C2312yes—
78C23×C53C2×D411yescharacteristic
79C23×C53C2×D411yescharacteristic
80C23×C53C2×D411yescharacteristic
81C22×D53—22no—
82C22×D53—42no—
83C22×D53—82no—
84Q8×C53—22no—
85Q8×C53—22no—
86Q8×C53—41no—
87Q8×C53—42no—
88D4×C53—41no—
89D4×C53—41no—
90D4×C53—41no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—42no—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—82no—
105C22⋊D53—82no—
106C2×Dic53—22no—
107D212—22no—
108D212—22no—
109D212—42no—
110#88322—22no—
111#88322—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×C53—42no—
127C4×C53—42no—
128C4×C53H(5,𝔽2)11yescharacteristic
129C4×C53H(5,𝔽2)11yescharacteristic
130Dic53—42no—
131#3200—531nomaximal, Sylow
132C2×C53—41no—
133C2×C53—41no—
134C2×C53—41no—
135C2×C53#53411yescharacteristic
136C2×C53#57712yes—
137D53—82no—
138#536D5311yescharacteristic, complemented
139#535—531no—
140#485—532no—
141#482—532no—
142#468—532no—
143#465—532no—
144#450—532no—
145#448—532no—
146#408—531no—
147C53#320011yescharacteristic, complemented, Sylow
148C2×#41C2×D5311yescharacteristic
149C22×Q8C2×D5311yescharacteristic, complemented
150C22×D4—531no—
151C2×SD16—532no—
152C2×SD16—532no—
153C2×M16—531no—
154C22×C8—531no—
155C4⋊Q8—21no—
156#127—1061no—
157#124—21no—
158#124—1061no—
159#123—21no—
160#123—21no—
161C22⋊Q8C2×D5312yes—
162C22⋊Q8C2×D5312yescomplemented
163#121—532no—
164#121—532no—
165#120—1061no—
166#120—1061no—
167#117—531no—
168#117C2×D5311yescharacteristic, complemented
169#107—532no—
170#106—532no—
171#103—532no—
172#103—532no—
173#102—532no—
174#102—532no—
175C22⋊C8—532no—
176C22⋊C8—532no—
177C24—1061no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊D5311yescharacteristic
184C2×Q8C22×D5312yes—
185C2×D4—532no—
186C2×D4—1062no—
187C2×D4—1062no—
188C2×D4—1061no—
189C2×D4—2121no—
190C2×D4—2121no—
191C2×D4C22⋊D5311yescharacteristic
192C2×D4C22⋊D5311yescharacteristic
193C2×D4C22⋊D5311yescharacteristic
194C22×C4—21no—
195C22×C4C22×D5311yescharacteristic
196C22×C4C22×D5311yescharacteristic
197C22×C4C22×D5311yescharacteristic
198SD16—2122no—
199SD16—2122no—
200M16—2121no—
201C2×C8—532no—
202C2×C8—532no—
203C2×C8—2121no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—532no—
207C4⋊C4C22×D5312yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—1061no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C42—21no—
215C42—1061no—
216C23—1062no—
217C23—4242no—
218C23—2122no—
219C23C2×C22⋊D5311yescharacteristic
220C23C2×C22⋊D5311yescharacteristic
221C23C2×C22⋊D5311yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—42no—
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—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—1062no—
244C2×C4C2×C22⋊D5311yescharacteristic
245C2×C4C2×C22⋊D5311yescharacteristic
246C2×C4C2×C22⋊D5311yescharacteristic
247C2×C4C23×D5311yescharacteristic
248C8—1062no—
249C8—1062no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—4244no—
257C22—2122no—
258C22—2122no—
259C22—2122no—
260C22C22×C22⋊D5311yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—2122no—
267C4#1313326211yescharacteristic
268C4#1313326211yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—4242no—
273C2#1681118211yescharacteristic
274C2#1681143112yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#536C2
socleC22×C53C22×D4
hypercenter#536D53
hypocenterC53#3200
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #16810961C2semidirect
3. #16810947C2semidirect
4. #16810871C2semidirect
5. #16810854C2semidirect
6. #16810773C2semidirect
7. #16810762C2semidirect
8. #16810590C2semidirect
9. C53×#536C2semidirect
13. C2×#11719917C22semidirect
14. C22×#88322C22semidirect
22. #13133150C22semidirect
23. #13133148C22semidirect
24. #13133141C22semidirect
25. #13133141C22semidirect
29. C22⋊#88322C22semidirect
31. C22×Q8×C53C22semidirect
37. C22⋊Q8×C53C22semidirect
38. #117×C53C22semidirect
138. #536D53semidirect
147. C53#3200semidirect
149. C22×Q8C2×D53semidirect
162. C22⋊Q8C2×D53semidirect
168. #117C2×D53semidirect