Finite Group Databasebeta

← #22592006

Subgroups — #22592006

Order 9088.

780 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×#14556105C212yesmaximal, complemented
2C22×C22⋊Dic71C212yesmaximal
3#17985935C211yesmaximal, characteristic, complemented
4#17985922C212yesmaximal, complemented
5#17985911C211yesmaximal, characteristic, complemented
6#17985874C212yesmaximal, complemented
7#17985871C212yesmaximal, complemented
8#17985857C212yesmaximal, complemented
9#379×C71C211yesmaximal, characteristic
10C22×Dic142C2211yescharacteristic, complemented
11C23×Dic71C2212yes—
12C23×Dic71C2212yes—
13#14556105—28no—
14C2×C4⋊Dic71C2212yescomplemented
15C2×C4⋊Dic71C2211yescharacteristic, complemented
16C2×Dic71⋊C4C2212yescomplemented
17C2×Dic71⋊C4C2212yescomplemented
18C2×Dic71⋊C4C2212yescomplemented
19C2×C22⋊Dic71—28no—
20C2×C22⋊Dic71C2212yes—
21C2×C22⋊Dic71C2212yes—
22C2×C22⋊Dic71C2212yes—
23C2×C22⋊Dic71C2212yes—
24C71⋊#95C2211yescharacteristic, complemented
25C71⋊#95C2212yescomplemented
26C71⋊#95C2211yescharacteristic, complemented
27C71⋊#95C2212yescomplemented
28C25×C71C2211yescharacteristic
29C2×C22⋊C4×C71C2212yes—
30C2×C22⋊C4×C71C2212yes—
31C2×C22⋊C4×C71C2211yescharacteristic
32C2×C22⋊C4×C71C2211yescharacteristic
33C2×Dic142—24no—
34C2×Dic142—41no—
35C22×Dic71—24no—
36C22×Dic71—24no—
37C22×Dic71—24no—
38C22×Dic71—24no—
39C22×Dic71—42no—
40C22×Dic71—42no—
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C22×Dic71C2312yes—
44C22×Dic71C2312yes—
45C4⋊Dic71—24no—
46C4⋊Dic71—41no—
47Dic71⋊C4—24no—
48Dic71⋊C4—24no—
49Dic71⋊C4—42no—
50C22⋊Dic71—24no—
51C22⋊Dic71—24no—
52C22⋊Dic71—24no—
53C22⋊Dic71D418yes—
54C24×C71—24no—
55C24×C71—24no—
56C24×C71—41no—
57C24×C71—41no—
58C24×C71Q814yescomplemented
59C24×C71C2311yescharacteristic
60C24×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2312yes—
64C22⋊C4×C71—24no—
65C22⋊C4×C71—24no—
66C22⋊C4×C71—41no—
67C22⋊C4×C71—41no—
68C23×C71—22no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71—216no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—21no—
76C23×C71—21no—
77C23×C71—44no—
78C23×C71—44no—
79C23×C71—48no—
80C23×C71—44no—
81C23×C71C2×D414yes—
82C23×C71C2×Q812yes—
83C23×C71C2×Q814yes—
84C23×C71#4114yes—
85C23×C71C2411yescharacteristic
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—48no—
104C2×Dic71—48no—
105C2×Dic71C2×D414yes—
106C2×Dic71C2×D414yes—
107Dic142—44no—
108C22×C71—24no—
109C22×C71—24no—
110C22×C71—24no—
111C22×C71—24no—
112C22×C71—24no—
113C22×C71—24no—
114C22×C71—24no—
115C22×C71—22no—
116C22×C71—416no—
117C22×C71—41no—
118C22×C71—42no—
119C22×C71—41no—
120C22×C71C22⋊Q818yescomplemented
121C22×C71C22×D412yes—
122C22×C71C22×Q811yescharacteristic
123C22×C71C2×#4112yes—
124C22×C71H(5,𝔽2)11yescharacteristic
125C22×C71H(5,𝔽2)11yescharacteristic
126C4×C71—41no—
127C4×C71—42no—
128C4×C71—41no—
129Dic71—24no—
130Dic71—24no—
131Dic71—42no—
132Dic71—42no—
133C2×C71—28no—
134C2×C71—42no—
135C2×C71C2×C22⋊Q812yes—
136C2×C71#54311yescharacteristic
137C2×C71D4211yescharacteristic
138C2×C71#54612yes—
139C2×C71#55111yescharacteristic
140#2437—711nomaximal, Sylow
141C71#243711yescharacteristic, complemented, Sylow
142C2×C22⋊Q8—712no—
143C22×C22⋊C4—712no—
144#399—711no—
145#396—712no—
146#393—711no—
147#385—712no—
148#385—712no—
149#380—712no—
150#379D7111yescharacteristic
151C25C2×D7111yescharacteristic
152C22×Q8—711no—
153C23×C4—712no—
154C23×C4—712no—
155C22⋊Q8—1428no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—711no—
158C2×C4⋊C4—712no—
159C2×C4⋊C4—712no—
160C2×C4⋊C4—712no—
161C2×C22⋊C4—712no—
162C2×C22⋊C4—712no—
163C2×C22⋊C4—712no—
164C2×C22⋊C4—712no—
165C2×C22⋊C4—1428no—
166C2×C22⋊C4C2×D7112yes—
167C2×C22⋊C4C2×D7112yes—
168C2×C22⋊C4C2×D7111yescharacteristic
169C2×C22⋊C4C2×D7111yescharacteristic
170#95—711no—
171#95—712no—
172#95—712no—
173#95—711no—
174C24—24no—
175C24—24no—
176C24—41no—
177C24—41no—
178C24Dic14214yescomplemented
179C24C22×D7111yescharacteristic
180C24C22×D7112yes—
181C2×Q8—2841no—
182C2×Q8—1424no—
183C22×C4—2842no—
184C22×C4—2842no—
185C22×C4—712no—
186C22×C4—712no—
187C22×C4—712no—
188C22×C4—712no—
189C22×C4—1424no—
190C22×C4—1424no—
191C22×C4—1424no—
192C22×C4—1424no—
193C22×C4C22×D7111yescharacteristic
194C22×C4C22×D7112yes—
195C22×C4C22×D7111yescharacteristic
196C4⋊C4—2842no—
197C4⋊C4—2841no—
198C4⋊C4—1424no—
199C4⋊C4—1424no—
200C4⋊C4—1424no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—718no—
206C22⋊C4—1424no—
207C22⋊C4—1424no—
208C22⋊C4—1424no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—216no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—44no—
222C23D4×D7114yes—
223C23C2×Dic14212yes—
224C23C2×Dic14214yes—
225C23#1176586414yes—
226C23C23×D7111yescharacteristic
227Q8—2844no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—2848no—
236C2×C4—2848no—
237C2×C4—714no—
238C2×C4—714no—
239C2×C4—1422no—
240C2×C4—1422no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4—1422no—
249C22—24no—
250C22—24no—
251C22—24no—
252C22—24no—
253C22—24no—
254C22—24no—
255C22—22no—
256C22—24no—
257C22—416no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22#1455610518yescomplemented
262C22C2×D4×D7112yes—
263C22C22×Dic14211yescharacteristic
264C22C2×#1176586412yes—
265C22#1455617311yescharacteristic
266C22#1455617311yescharacteristic
267C4—41no—
268C4—42no—
269C4—41no—
270C4—2842no—
271C4—2842no—
272C4—1424no—
273C4—1424no—
274C2—28no—
275C2—42no—
276C2C2×#1455610512yes—
277C2#1798662511yescharacteristic
278C2#1798663111yescharacteristic
279C2#1798663612yes—
280C2#1798668011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#379×C71C2
hypercenter#379D71
hypocenterC71#2437
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#14556105C2semidirect
3. #17985935C2semidirect
4. #17985922C2semidirect
5. #17985911C2semidirect
6. #17985874C2semidirect
7. #17985871C2semidirect
8. #17985857C2semidirect
10. C22×Dic142C22semidirect
14. C2×C4⋊Dic71C22semidirect
15. C2×C4⋊Dic71C22semidirect
16. C2×Dic71⋊C4C22semidirect
17. C2×Dic71⋊C4C22semidirect
18. C2×Dic71⋊C4C22semidirect
24. C71⋊#95C22semidirect
25. C71⋊#95C22semidirect
26. C71⋊#95C22semidirect
27. C71⋊#95C22semidirect
58. C24×C71Q8semidirect
120. C22×C71C22⋊Q8semidirect
141. C71#2437semidirect
178. C24Dic142semidirect
261. C22#14556105semidirect