Finite Group Databasebeta

← #22594151

Subgroups — #22594151

Order 9088.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556105C211yesmaximal, characteristic, complemented
2C22×(C2×D71)⋊C4C211yesmaximal, characteristic
3#17985934C212yesmaximal, complemented
4#17985923C212yesmaximal, complemented
5#17985921C212yesmaximal, complemented
6#17985912C212yesmaximal, complemented
7#17985872C211yesmaximal, characteristic, complemented
8#17985857C211yesmaximal, characteristic, complemented
9#17985854C211yesmaximal, characteristic
10#17985854C211yesmaximal, characteristic
11#385×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C22×Dic142C2211yescharacteristic, complemented
14C23×Dic71C2211yescharacteristic
15#14556105—24no—
16C2×C4⋊Dic71C2211yescharacteristic, complemented
17C2×Dic71⋊C4C2211yescharacteristic, complemented
18C2×Dic71⋊C4C2211yescharacteristic, complemented
19C2×Dic71⋊C4C2212yescomplemented
20C2×Dic71⋊C4C2212yescomplemented
21C2×C22⋊Dic71C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×(C2×D71)⋊C4—24no—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×(C2×D71)⋊C4C2212yes—
30C71⋊#95C2212yescomplemented
31C71⋊#95C2212yescomplemented
32C23×C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2212yescomplemented
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2211yescharacteristic
36#95×C71C2212yescomplemented
37C23×D71—22no—
38C23×D71—22no—
39C23×D71—42no—
40C23×D71—42no—
41C23×D71Q814yescomplemented
42C23×D71C2312yes—
43C2×Dic142—22no—
44C2×Dic142—42no—
45C22×Dic71—42no—
46C22×Dic71D414yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2312yes—
49C22×Dic71C2311yescharacteristic
50C22×Dic71C2311yescharacteristic
51C22×Dic71C2311yescharacteristic
52C4⋊Dic71—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4—42no—
56Dic71⋊C4—42no—
57C22⋊Dic71—22no—
58C22⋊Dic71—22no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—42no—
62(C2×D71)⋊C4—42no—
63(C2×D71)⋊C4—42no—
64(C2×D71)⋊C4—42no—
65C24×C71C2311yescharacteristic
66C22×C4×C71—22no—
67C22×C4×C71—42no—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2312yes—
70C22×C4×C71C2312yes—
71C22×C4×C71C2311yescharacteristic
72C4⋊C4×C71—42no—
73C22⋊C4×C71—22no—
74C22⋊C4×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71C2×D412yes—
78C23×C71C2×Q812yes—
79C23×C71#4112yes—
80C23×C71C2411yescharacteristic
81C22×D71—28no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—22no—
88C22×D71—48no—
89C22×D71—44no—
90C22×D71—44no—
91C22×D71—48no—
92C22×D71C2×Q814yes—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—44no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—44no—
114C2×Dic71C2×D412yes—
115C2×Dic71C2×D412yes—
116Dic142—44no—
117C22×C71—22no—
118C22×C71—22no—
119C22×C71—24no—
120C22×C71—24no—
121C22×C71C22⋊Q814yes—
122C22×C71C22×D411yescharacteristic
123C22×C71C22×Q811yescharacteristic
124C22×C71C2×#4111yescharacteristic
125C22×C71H(5,𝔽2)12yes—
126C22×C71H(5,𝔽2)12yes—
127C2×D71—24no—
128C2×D71—24no—
129C2×D71—24no—
130C2×D71—416no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—42no—
134C2×D71—42no—
135C4×C71—41no—
136C4×C71—41no—
137C4×C71—42no—
138C4×C71—42no—
139Dic71—22no—
140Dic71—22no—
141Dic71—41no—
142Dic71—41no—
143Dic71—42no—
144C2×C71—24no—
145C2×C71C2×C22⋊Q811yescharacteristic
146C2×C71#53411yescharacteristic
147C2×C71#53511yescharacteristic
148C2×C71#53812yes—
149C2×C71#54312yes—
150D71—44no—
151#2626—711nomaximal, Sylow
152C71#262611yescharacteristic, complemented, Sylow
153C2×C22⋊Q8—711no—
154C22×C22⋊C4—711no—
155#399—712no—
156#396—712no—
157#396—712no—
158#393—712no—
159#385—711no—
160#385D7111yescharacteristic, complemented
161#380—711no—
162#379—711no—
163#379—711no—
164C25—711no—
165C22×Q8—711no—
166C23×C4—711no—
167C23×C4C2×D7111yescharacteristic
168C22⋊Q8—1424no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—712no—
171C2×C4⋊C4—711no—
172C2×C4⋊C4—711no—
173C2×C4⋊C4—711no—
174C2×C4⋊C4C2×D7112yescomplemented
175C2×C22⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C22⋊C4—712no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4—711no—
180C2×C22⋊C4—711no—
181C2×C22⋊C4—712no—
182C2×C22⋊C4—712no—
183C2×C22⋊C4—1424no—
184C2×C22⋊C4C2×D7111yescharacteristic
185C2×C22⋊C4C2×D7111yescharacteristic
186#95—712no—
187#95—712no—
188#95C2×D7112yescomplemented
189C24—2842no—
190C24—2842no—
191C24—714no—
192C24—712no—
193C24—1422no—
194C24—1422no—
195C24C22×D7111yescharacteristic
196C2×Q8—2842no—
197C2×Q8—1422no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—2842no—
201C22×C4—714no—
202C22×C4—711no—
203C22×C4—711no—
204C22×C4—712no—
205C22×C4—711no—
206C22×C4—711no—
207C22×C4C22×D7112yes—
208C22×C4C22×D7112yes—
209C22×C4C22×D7111yescharacteristic
210C22×C4C22×D7111yescharacteristic
211C4⋊C4—42no—
212C4⋊C4—2842no—
213C4⋊C4—2842no—
214C4⋊C4—1422no—
215C4⋊C4—1422no—
216C4⋊C4—1422no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—2842no—
220C22⋊C4—2842no—
221C22⋊C4—2842no—
222C22⋊C4—2842no—
223C22⋊C4—1424no—
224C22⋊C4—1424no—
225C22⋊C4—1422no—
226C22⋊C4—1422no—
227C23—22no—
228C23—22no—
229C23—2844no—
230C23—2844no—
231C23—2848no—
232C23—2848no—
233C23—714no—
234C23—1422no—
235C23—1422no—
236C23—1422no—
237C23—1422no—
238C23—1422no—
239C23—1422no—
240C23—1428no—
241C23D4×D7112yes—
242C23Q8×D7112yes—
243C23#1176586512yes—
244C23C23×D7111yescharacteristic
245Q8—2844no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—2844no—
258C2×C4—712no—
259C2×C4—712no—
260C2×C4—1421no—
261C2×C4—1421no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4—1422no—
268C2×C4—1422no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—28416no—
274C22—2842no—
275C22—2842no—
276C22—2842no—
277C22—2842no—
278C22—1424no—
279C22—1424no—
280C22—1424no—
281C22#1455610614yes—
282C22C2×D4×D7111yescharacteristic
283C22C2×Q8×D7111yescharacteristic
284C22C2×#1176586511yescharacteristic
285C22#1455617312yes—
286C22#1455617412yes—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—42no—
291C4—2841no—
292C4—2841no—
293C4—2842no—
294C4—1422no—
295C4—1422no—
296C2—24no—
297C2—2844no—
298C2C2×#1455610611yescharacteristic
299C2#1798656011yescharacteristic
300C2#1798656411yescharacteristic
301C2#1798658212yes—
302C2#1798662412yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#385×C71C2
hypercenter#385D71
hypocenterC71#2626
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#14556105C2semidirect
3. #17985934C2semidirect
4. #17985923C2semidirect
5. #17985921C2semidirect
6. #17985912C2semidirect
7. #17985872C2semidirect
8. #17985857C2semidirect
11. #385×C71C2semidirect
13. C22×Dic142C22semidirect
16. C2×C4⋊Dic71C22semidirect
17. C2×Dic71⋊C4C22semidirect
18. C2×Dic71⋊C4C22semidirect
19. C2×Dic71⋊C4C22semidirect
20. C2×Dic71⋊C4C22semidirect
30. C71⋊#95C22semidirect
31. C71⋊#95C22semidirect
33. C2×C4⋊C4×C71C22semidirect
36. #95×C71C22semidirect
41. C23×D71Q8semidirect
152. C71#2626semidirect
160. #385D71semidirect
174. C2×C4⋊C4C2×D71semidirect
188. #95C2×D71semidirect