Finite Group Databasebeta

← #20281759

Subgroups — #20281759

Order 6784.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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#16810552C212yesmaximal, complemented
2#16810548C211yesmaximal, characteristic, complemented
3#16810534C212yesmaximal, complemented
4#16810517C212yesmaximal, complemented
5#16810515C212yesmaximal, complemented
6#16810512C212yesmaximal, complemented
7#16810478C212yesmaximal
8#16810478C211yesmaximal, characteristic
9C53×#393C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic, complemented
12C2×#11719919C2212yescomplemented
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2211yescharacteristic
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×C4×Dic53C2212yescomplemented
22#13133124C2212yescomplemented
23#13133124C2212yescomplemented
24#13133124C2212yescomplemented
25#13133124C2211yescharacteristic, complemented
26C22×Q8×C53C2211yescharacteristic, complemented
27C2×C22⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29#95×C53C2211yescharacteristic, complemented
30#95×C53C2212yescomplemented
31C24×C53C2311yescharacteristic
32C23×D53—22no—
33C23×D53—24no—
34C23×D53—41no—
35C23×D53—42no—
36C23×D53—41no—
37C23×D53C2312yes—
38C2×Q8×C53—42no—
39C2×Q8×C53—41no—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22⋊C4×C53—42no—
48C22⋊C4×C53—41no—
49C22⋊Dic53—24no—
50C22⋊Dic53—41no—
51#11719921—24no—
52#11719921—24no—
53#11719921—42no—
54#11719921—42no—
55#11719921—42no—
56#11719921—42no—
57C4⋊Dic53—41no—
58#11719919—42no—
59C4×Dic53—42no—
60C23×C53—22no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—22no—
64C23×C53#4112yes—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—44no—
73C22×D53—48no—
74C22×D53—48no—
75C22×D53—44no—
76C22×D53—44no—
77C22×D53#4114yes—
78Q8×C53—44no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C22×C53—22no—
99C22×C53—24no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53C2×#4112yes—
104C22×C53C2×#4111yescharacteristic
105C22×C53H(5,𝔽2)12yes—
106C22×C53H(5,𝔽2)11yescharacteristic
107C22×C53H(5,𝔽2)11yescharacteristic
108C2×D53—24no—
109C2×D53—24no—
110C2×D53—24no—
111C2×D53—416no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C4×C53—42no—
117C4×C53—41no—
118C4×C53—42no—
119C4×C53—41no—
120Dic53—42no—
121Dic53—42no—
122Dic53—42no—
123#2736—531nomaximal, Sylow
124C2×C53—42no—
125C2×C53#53812yes—
126C2×C53#53811yescharacteristic
127C2×C53#55112yes—
128C2×C53#55111yescharacteristic
129C2×C53H(3,𝔽4)11yescharacteristic
130D53—44no—
131#397—531no—
132#397—532no—
133#393—532no—
134#393D5311yescharacteristic, complemented
135#388—532no—
136#388—532no—
137#388—532no—
138#379—532no—
139#379—531no—
140C53#273611yescharacteristic, complemented, Sylow
141C25—531no—
142C22×Q8C2×D5311yescharacteristic, complemented
143C2×C4⋊C4—532no—
144C2×C4⋊C4—531no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—531no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4C2×D5312yes—
154C2×C22⋊C4C2×D5311yescharacteristic
155C2×C42—532no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95—531no—
160#95C2×D5311yescharacteristic, complemented
161#95C2×D5312yescomplemented
162C24—532no—
163C24—1064no—
164C24—1062no—
165C24—2122no—
166C24—2121no—
167C24—2121no—
168C24C22×D5311yescharacteristic
169C2×Q8—42no—
170C2×Q8—41no—
171C22×C4—532no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4C22×D5312yes—
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177C22×C4C22×D5312yes—
178C4⋊C4—2122no—
179C4⋊C4—2121no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C22⋊C4—1064no—
183C22⋊C4—1064no—
184C22⋊C4—1064no—
185C22⋊C4—2122no—
186C22⋊C4—2122no—
187C22⋊C4—2122no—
188C22⋊C4—2122no—
189C22⋊C4—2121no—
190C42—2122no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23—534no—
196C23—1062no—
197C23—1062no—
198C23—1062no—
199C23—1062no—
200C23—1062no—
201C23—1062no—
202C23—2128no—
203C23—2124no—
204C23—2124no—
205C23—2128no—
206C23—2124no—
207C23#1171994912yes—
208C23C23×D5311yescharacteristic
209Q8—44no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C22—24no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—1064no—
235C22—1064no—
236C22—1064no—
237C22—21216no—
238C22—2122no—
239C22—2122no—
240C22—2122no—
241C22—2122no—
242C22#41×D5312yes—
243C22C2×#1171994911yescharacteristic
244C22#1313326212yes—
245C22#1313326211yescharacteristic
246C22#1313326311yescharacteristic
247C4—41no—
248C4—42no—
249C4—41no—
250C4—42no—
251C4—2122no—
252C4—2122no—
253C4—2122no—
254C2—42no—
255C2—2124no—
256C2#1681120712yes—
257C2#1681121311yescharacteristic
258C2#1681130012yes—
259C2#1681130311yescharacteristic
260C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2736
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #16810552C2semidirect
2. #16810548C2semidirect
3. #16810534C2semidirect
4. #16810517C2semidirect
5. #16810515C2semidirect
6. #16810512C2semidirect
9. C53×#393C2semidirect
11. C2×C4⋊Dic53C22semidirect
12. C2×#11719919C22semidirect
21. C2×C4×Dic53C22semidirect
22. #13133124C22semidirect
23. #13133124C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
26. C22×Q8×C53C22semidirect
29. #95×C53C22semidirect
30. #95×C53C22semidirect
134. #393D53semidirect
140. C53#2736semidirect
142. C22×Q8C2×D53semidirect
160. #95C2×D53semidirect
161. #95C2×D53semidirect