Finite Group Databasebeta

← #20285575

Subgroups — #20285575

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#16810950C212yesmaximal, complemented
3#16810945C212yesmaximal, complemented
4#16810874C212yesmaximal, complemented
5#16810859C211yesmaximal, characteristic, complemented
6#16810758C212yesmaximal, complemented
7#16810755C212yesmaximal, complemented
8#16810589C211yesmaximal, characteristic, complemented
9C53×#466C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic
11C2×D4×D53C2212yescomplemented
12C2×D424C2211yescharacteristic
13C2×#11719922C2212yes—
14C2×#11719917C2211yescharacteristic, complemented
15#13133213—22no—
16#13133203—22no—
17#13133192C2212yes—
18#13133191C2212yescomplemented
19#13133188C2212yes—
20#13133187—22no—
21#13133185—22no—
22#13133175C2212yescomplemented
23#13133151C2211yescharacteristic, complemented
24#13133148C2212yescomplemented
25#13133139C2212yescomplemented
26#13133138C2212yescomplemented
27#13133137C2212yescomplemented
28#13133129C2212yescomplemented
29#13133129C2212yescomplemented
30C2×D8×C53C2211yescharacteristic
31C22×C8×C53C2211yescharacteristic, complemented
32#121×C53C2212yescomplemented
33#107×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—
53C2×C8×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—
65#11719917—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—21no—
101C22×C53—21no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53C22×D411yescharacteristic
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—22no—
109C2×D53—42no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—88no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53H(5,𝔽2)11yescharacteristic
116C4×C53H(5,𝔽2)11yescharacteristic
117Dic53—22no—
118Dic53—42no—
119#3199—531nomaximal, Sylow
120C2×C53—41no—
121C2×C53—82no—
122C2×C53#53411yescharacteristic
123C2×C53#57611yescharacteristic
124C2×C53#57611yescharacteristic
125D53—42no—
126D53—82no—
127#535—532no—
128#482—532no—
129#482—532no—
130#469—532no—
131#466—531no—
132#466D5311yescharacteristic, complemented
133#447—532no—
134#447—532no—
135#408—531no—
136C53#319911yescharacteristic, complemented, Sylow
137C2×#41—531no—
138C22×D4—532no—
139C2×D8—532no—
140C2×D8—531no—
141C2×D8C2×D5311yescharacteristic
142C2×M16—531no—
143C22×C8C2×D5311yescharacteristic, complemented
144#127—1062no—
145#124—1062no—
146#121—532no—
147#121—532no—
148#121—532no—
149#121C2×D5312yescomplemented
150#120—1062no—
151#120—1062no—
152#117—532no—
153#107—531no—
154#107C2×D5311yescharacteristic, complemented
155#106—532no—
156#102—532no—
157#102—532no—
158#102—532no—
159#102C2×D5312yescomplemented
160C22⋊C8—532no—
161C22⋊C8—532no—
162C24—1062no—
163#41—2121no—
164#41—2121no—
165C2×Q8—531no—
166C2×D4—22no—
167C2×D4—532no—
168C2×D4—531no—
169C2×D4—532no—
170C2×D4—1062no—
171C2×D4—1062no—
172C2×D4—1062no—
173C2×D4—1062no—
174C2×D4—2122no—
175C2×D4—2122no—
176C2×D4C22×D5312yes—
177C22×C4—532no—
178C22×C4C22×D5311yescharacteristic
179D8—41no—
180D8—2121no—
181D8—2122no—
182M16—2121no—
183C2×C8—41no—
184C2×C8—532no—
185C2×C8C22×D5311yescharacteristic
186C2×C8C22×D5311yescharacteristic
187C4⋊C4—532no—
188C4⋊C4C22×D5312yes—
189C22⋊C4—22no—
190C22⋊C4—1062no—
191C22⋊C4—1062no—
192C22⋊C4—1062no—
193C22⋊C4—1062no—
194C42—1062no—
195C23—22no—
196C23—532no—
197C23—1062no—
198C23—4242no—
199C23—4242no—
200C23—2124no—
201C23D4×D5311yescharacteristic
202Q8—2121no—
203D4—22no—
204D4—22no—
205D4—82no—
206D4—1062no—
207D4—1062no—
208D4—4242no—
209D4—4242no—
210D4—4242no—
211D4—2121no—
212D4—2122no—
213D4—2124no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—532no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4D4×D5311yescharacteristic
222C2×C4C23×D5311yescharacteristic
223C8—21no—
224C8—21no—
225C8—1062no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—1062no—
233C22—1062no—
234C22—1062no—
235C22—4248no—
236C22—2122no—
237C22—2122no—
238C22—2122no—
239C22C2×D4×D5311yescharacteristic
240C4—21no—
241C4—42no—
242C4—1062no—
243C4—2122no—
244C4#1313326211yescharacteristic
245C4#1313326311yescharacteristic
246C2—41no—
247C2—82no—
248C2—4242no—
249C2—2122no—
250C2#1681118111yescharacteristic
251C2#1681142411yescharacteristic
252C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#466C2
socleC22×C53C22×D4
hypercenter#466D53
hypocenterC53#3199
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #16811189C2semidirect
2. #16810950C2semidirect
3. #16810945C2semidirect
4. #16810874C2semidirect
5. #16810859C2semidirect
6. #16810758C2semidirect
7. #16810755C2semidirect
8. #16810589C2semidirect
9. C53×#466C2semidirect
11. C2×D4×D53C22semidirect
14. C2×#11719917C22semidirect
18. #13133191C22semidirect
22. #13133175C22semidirect
23. #13133151C22semidirect
24. #13133148C22semidirect
25. #13133139C22semidirect
26. #13133138C22semidirect
27. #13133137C22semidirect
28. #13133129C22semidirect
29. #13133129C22semidirect
31. C22×C8×C53C22semidirect
32. #121×C53C22semidirect
33. #107×C53C22semidirect
34. #102×C53C22semidirect
132. #466D53semidirect
136. C53#3199semidirect
143. C22×C8C2×D53semidirect
149. #121C2×D53semidirect
154. #107C2×D53semidirect
159. #102C2×D53semidirect