Finite Group Databasebeta

← #22594542

Subgroups — #22594542

Order 9088.

816 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556100C212yesmaximal, complemented
2C2×#14556096C212yesmaximal, complemented
3#17985935C211yesmaximal, characteristic, complemented
4#17985928C212yesmaximal, complemented
5#17985915C212yesmaximal, complemented
6#17985879C212yesmaximal, complemented
7#17985876C212yesmaximal, complemented
8#17985854C211yesmaximal, characteristic, complemented
9#392×C71C211yesmaximal, characteristic, complemented
10C24×D71C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C22×C22⋊D71C2212yes—
13C23×Dic71C2212yescomplemented
14#14556100—28no—
15#14556096—28no—
16C2×C4⋊Dic71C2211yescharacteristic, complemented
17C2×Dic71⋊C4C2212yescomplemented
18C2×C22⋊Dic71C2212yes—
19C2×C22⋊Dic71C2211yescharacteristic
20C2×C22⋊Dic71C2212yescomplemented
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C71⋊#95C2211yescharacteristic, complemented
26C71⋊#95C2212yescomplemented
27C71⋊#95C2212yescomplemented
28C22×D4×C71C2212yes—
29C22×D4×C71C2211yescharacteristic, complemented
30C2×C22⋊C4×C71C2211yescharacteristic
31C2×C22⋊C4×C71C2212yescomplemented
32#95×C71C2211yescharacteristic, complemented
33C23×D71—22no—
34C23×D71—24no—
35C23×D71—42no—
36C23×D71—41no—
37C23×D71—41no—
38C23×D71C2312yes—
39C2×C22⋊D71—24no—
40C2×C22⋊D71—24no—
41C2×C22⋊D71—42no—
42C2×C22⋊D71—42no—
43C2×C22⋊D71—42no—
44C2×C22⋊D71D418yescomplemented
45C22×Dic71—24no—
46C22×Dic71—42no—
47C22×Dic71—42no—
48C22×Dic71C2312yes—
49C22×Dic71C2312yes—
50C22×Dic71C2312yes—
51C4⋊Dic71—41no—
52Dic71⋊C4—24no—
53C22⋊Dic71—24no—
54C22⋊Dic71—24no—
55C22⋊Dic71—41no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—42no—
60C24×C71C2312yes—
61C24×C71C2311yescharacteristic
62C2×D4×C71—24no—
63C2×D4×C71—24no—
64C2×D4×C71—24no—
65C2×D4×C71—42no—
66C2×D4×C71—41no—
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2312yes—
69C22×C4×C71C2311yescharacteristic
70C22⋊C4×C71—41no—
71C22⋊C4×C71—42no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—21no—
77C23×C71—22no—
78C23×C71—22no—
79C23×C71—22no—
80C23×C71—21no—
81C23×C71—22no—
82C23×C71C2×D414yes—
83C23×C71#4112yes—
84C23×C71C2411yescharacteristic
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—22no—
88C22×D71—22no—
89C22×D71—22no—
90C22×D71—22no—
91C22×D71—44no—
92C22×D71—48no—
93C22×D71—48no—
94C22×D71—44no—
95C22×D71—44no—
96C22×D71C2×D414yes—
97D4×C71—44no—
98D4×C71—48no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C22⋊D71—48no—
107C22⋊D71—48no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—22no—
114C2×Dic71—22no—
115C2×Dic71—22no—
116C2×Dic71—48no—
117C2×Dic71C2×D414yes—
118C22×C71—24no—
119C22×C71—22no—
120C22×C71—24no—
121C22×C71—24no—
122C22×C71—24no—
123C22×C71—42no—
124C22×C71—42no—
125C22×C71—42no—
126C22×C71—41no—
127C22×C71—42no—
128C22×C71—41no—
129C22×C71—42no—
130C22×C71C22×D412yes—
131C22×C71C2×#4111yescharacteristic
132C22×C71H(5,𝔽2)12yes—
133C22×C71H(5,𝔽2)11yescharacteristic
134C22×C71H(5,𝔽2)11yescharacteristic
135C2×D71—24no—
136C2×D71—24no—
137C2×D71—24no—
138C2×D71—416no—
139C2×D71—42no—
140C2×D71—42no—
141C2×D71—42no—
142C2×D71—42no—
143C4×C71—41no—
144C4×C71—42no—
145C4×C71—41no—
146Dic71—24no—
147Dic71—42no—
148Dic71—42no—
149C2×C71—42no—
150C2×C71—44no—
151C2×C71#53412yes—
152C2×C71#53811yescharacteristic
153C2×C71D4211yescharacteristic
154C2×C71#54612yes—
155C2×C71#56011yescharacteristic
156D71—44no—
157#2668—711nomaximal, Sylow
158C71#266811yescharacteristic, complemented, Sylow
159C2×#121—712no—
160C2×#120—712no—
161#399—711no—
162#397—712no—
163#394—712no—
164#392D7111yescharacteristic, complemented
165#386—712no—
166#386—712no—
167#379—711no—
168C25—711no—
169C22×D4—712no—
170C22×D4—712no—
171C22×D4C2×D7112yes—
172C22×D4C2×D7111yescharacteristic, complemented
173C23×C4—712no—
174#121—1428no—
175#120—1428no—
176C2×C4⋊C4—711no—
177C2×C4⋊C4—712no—
178C2×C22⋊C4—711no—
179C2×C22⋊C4—712no—
180C2×C22⋊C4—712no—
181C2×C22⋊C4—712no—
182C2×C22⋊C4—712no—
183C2×C22⋊C4—712no—
184C2×C22⋊C4—712no—
185C2×C22⋊C4C2×D7111yescharacteristic
186C2×C22⋊C4C2×D7112yescomplemented
187#95—712no—
188#95—711no—
189#95—712no—
190#95C2×D7111yescharacteristic, complemented
191C24—2841no—
192C24—2842no—
193C24—2841no—
194C24—712no—
195C24—1424no—
196C24—1422no—
197C24C22×D7112yes—
198C24C22×D7111yescharacteristic
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—2842no—
205C2×D4—2842no—
206C2×D4—2842no—
207C2×D4—718no—
208C2×D4—1424no—
209C2×D4—1424no—
210C22×C4—2842no—
211C22×C4—2842no—
212C22×C4—712no—
213C22×C4—712no—
214C22×C4—712no—
215C22×C4—1424no—
216C22×C4C22×D7112yes—
217C22×C4C22×D7111yescharacteristic
218C22×C4C22×D7111yescharacteristic
219C4⋊C4—2841no—
220C4⋊C4—1424no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—2842no—
224C22⋊C4—2841no—
225C22⋊C4—1424no—
226C22⋊C4—1424no—
227C22⋊C4—1424no—
228C22⋊C4—1424no—
229C22⋊C4—1424no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—2844no—
241C23—2848no—
242C23—2844no—
243C23—2848no—
244C23—2844no—
245C23—714no—
246C23—1422no—
247C23—1422no—
248C23—1422no—
249C23—1422no—
250C23—1422no—
251C23—1422no—
252C23D4×D7114yes—
253C23#1176586412yes—
254C23C23×D7111yescharacteristic
255D4—44no—
256D4—48no—
257D4—2848no—
258D4—2848no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—2848no—
267C2×C4—714no—
268C2×C4—1422no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4—1422no—
274C2×C4—1422no—
275C2×C4—1422no—
276C22—22no—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—28416no—
289C22—2842no—
290C22—2842no—
291C22—2842no—
292C22—2842no—
293C22—1424no—
294C22—1424no—
295C22—1424no—
296C22C2×D4×D7112yes—
297C22C2×#1176586411yescharacteristic
298C22#1455617311yescharacteristic
299C22#1455617312yes—
300C22#1455617411yescharacteristic
301C4—41no—
302C4—42no—
303C4—41no—
304C4—2842no—
305C4—2842no—
306C4—1424no—
307C2—42no—
308C2—44no—
309C2—2844no—
310C2#1798655712yes—
311C2#1798658611yescharacteristic
312C2#1798663111yescharacteristic
313C2#1798663612yes—
314C2#1798673611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#392×C71C2
hypercenter#392D71
hypocenterC71#2668
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14556100C2semidirect
2. C2×#14556096C2semidirect
3. #17985935C2semidirect
4. #17985928C2semidirect
5. #17985915C2semidirect
6. #17985879C2semidirect
7. #17985876C2semidirect
8. #17985854C2semidirect
9. #392×C71C2semidirect
13. C23×Dic71C22semidirect
16. C2×C4⋊Dic71C22semidirect
17. C2×Dic71⋊C4C22semidirect
20. C2×C22⋊Dic71C22semidirect
25. C71⋊#95C22semidirect
26. C71⋊#95C22semidirect
27. C71⋊#95C22semidirect
29. C22×D4×C71C22semidirect
31. C2×C22⋊C4×C71C22semidirect
32. #95×C71C22semidirect
44. C2×C22⋊D71D4semidirect
158. C71#2668semidirect
164. #392D71semidirect
172. C22×D4C2×D71semidirect
186. C2×C22⋊C4C2×D71semidirect
190. #95C2×D71semidirect