Finite Group Databasebeta

← #22594146

Subgroups — #22594146

Order 9088.

668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14556108C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D71C211yesmaximal, characteristic
3#17985934C211yesmaximal, characteristic, complemented
4#17985922C212yesmaximal, complemented
5#17985921C212yesmaximal, complemented
6#17985912C211yesmaximal, characteristic, complemented
7#17985911C211yesmaximal, characteristic, complemented
8#17985870C212yesmaximal, complemented
9#17985858C211yesmaximal, characteristic, complemented
10#17985854C212yesmaximal
11#399×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C22×Dic142C2211yescharacteristic, complemented
14C22×C4×D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16#14556108—24no—
17C2×C4⋊Dic71C2212yescomplemented
18C2×Dic71⋊C4C2212yescomplemented
19C2×Dic71⋊C4C2212yescomplemented
20C2×C22⋊Dic71C2211yescharacteristic
21C2×C22⋊Dic71C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C22⋊C4×D71—24no—
29C71⋊#95C2211yescharacteristic, complemented
30C71⋊#95C2212yescomplemented
31C71⋊#95C2212yescomplemented
32C2×C4⋊C4×C71C2211yescharacteristic, complemented
33C2×C4⋊C4×C71C2212yescomplemented
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2212yes—
36#95×C71C2211yescharacteristic, complemented
37C23×D71—22no—
38C23×D71—22no—
39C23×D71—42no—
40C23×D71—42no—
41C23×D71Q814yescomplemented
42C23×D71C2311yescharacteristic
43C23×D71C2311yescharacteristic
44C2×Dic142—22no—
45C2×Dic142—42no—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2312yes—
48C22×Dic71C2312yes—
49C22×Dic71C2311yescharacteristic
50C2×C4×D71—22no—
51C2×C4×D71—42no—
52C2×C4×D71—42no—
53C2×C4×D71D414yes—
54C4⋊Dic71—24no—
55Dic71⋊C4—42no—
56Dic71⋊C4—42no—
57C22⋊Dic71—22no—
58C22⋊Dic71—42no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—42no—
64(C2×D71)⋊C4—42no—
65C24×C71C2311yescharacteristic
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2312yes—
68C22×C4×C71C2312yes—
69C22×C4×C71C2311yescharacteristic
70C4⋊C4×C71—22no—
71C4⋊C4×C71—42no—
72C22⋊C4×C71—22no—
73C22⋊C4×C71—42no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71—22no—
77C23×C71—22no—
78C23×C71C2×Q812yes—
79C23×C71C2411yescharacteristic
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—21no—
83C22×D71—21no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—28no—
87C22×D71—48no—
88C22×D71—44no—
89C22×D71—44no—
90C22×D71—48no—
91C22×D71C2×D412yes—
92C22×D71C2×Q812yes—
93C22×D71C2×Q812yes—
94C22×D71#4112yes—
95C2×C4×C71—22no—
96C2×C4×C71—21no—
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—22no—
104C2×C4×C71C2×D412yes—
105C2×Dic71—22no—
106C2×Dic71—21no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—22no—
114C2×Dic71C2×D412yes—
115C4×D71—44no—
116C4×D71—44no—
117Dic142—44no—
118C22×C71—22no—
119C22×C71—24no—
120C22×C71—41no—
121C22×C71—41no—
122C22×C71—42no—
123C22×C71C22×D411yescharacteristic
124C22×C71C22×Q811yescharacteristic
125C22×C71C2×#4111yescharacteristic
126C22×C71H(5,𝔽2)12yes—
127C22×C71H(5,𝔽2)12yes—
128C2×D71—22no—
129C2×D71—22no—
130C2×D71—24no—
131C2×D71—24no—
132C2×D71—22no—
133C2×D71—24no—
134C2×D71—416no—
135C2×D71—41no—
136C2×D71—41no—
137C2×D71—42no—
138C2×D71C22⋊Q814yes—
139C4×C71—22no—
140C4×C71—41no—
141C4×C71—42no—
142C4×C71—42no—
143Dic71—22no—
144Dic71—42no—
145Dic71—41no—
146Dic71—42no—
147C2×C71—42no—
148C2×C71C2×C22⋊Q811yescharacteristic
149C2×C71#53411yescharacteristic
150C2×C71#53511yescharacteristic
151C2×C71#53812yes—
152C2×C71#54312yes—
153D71—24no—
154D71—42no—
155#2626—711nomaximal, Sylow
156C71#262611yescharacteristic, complemented, Sylow
157C2×C22⋊Q8—711no—
158C22×C22⋊C4—711no—
159#399—711no—
160#399D7111yescharacteristic, complemented
161#396—712no—
162#396—712no—
163#393—711no—
164#393—711no—
165#385—712no—
166#380—711no—
167#379—712no—
168C25—711no—
169C22×Q8—711no—
170C23×C4—711no—
171C23×C4—711no—
172C22⋊Q8—1424no—
173C2×C4⋊C4—712no—
174C2×C4⋊C4—712no—
175C2×C4⋊C4—712no—
176C2×C4⋊C4C2×D7111yescharacteristic, complemented
177C2×C4⋊C4C2×D7112yescomplemented
178C2×C22⋊C4—712no—
179C2×C22⋊C4—712no—
180C2×C22⋊C4—712no—
181C2×C22⋊C4—711no—
182C2×C22⋊C4—711no—
183C2×C22⋊C4—711no—
184C2×C22⋊C4—712no—
185C2×C22⋊C4—712no—
186C2×C22⋊C4—1424no—
187C2×C22⋊C4C2×D7111yescharacteristic
188C2×C22⋊C4C2×D7112yes—
189#95—711no—
190#95—712no—
191#95—712no—
192#95C2×D7111yescharacteristic, complemented
193C24—2842no—
194C24—2842no—
195C24—714no—
196C24—711no—
197C24—711no—
198C24—1422no—
199C24—1422no—
200C24C22×D7111yescharacteristic
201C2×Q8—2842no—
202C2×Q8—1422no—
203C22×C4—2842no—
204C22×C4—2842no—
205C22×C4—714no—
206C22×C4—711no—
207C22×C4—712no—
208C22×C4—712no—
209C22×C4—711no—
210C22×C4—1422no—
211C22×C4C22×D7111yescharacteristic
212C22×C4C22×D7112yes—
213C22×C4C22×D7112yes—
214C22×C4C22×D7111yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—2842no—
218C4⋊C4—2842no—
219C4⋊C4—1424no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—2842no—
223C22⋊C4—2842no—
224C22⋊C4—2842no—
225C22⋊C4—1422no—
226C22⋊C4—1424no—
227C22⋊C4—1424no—
228C22⋊C4—1422no—
229C22⋊C4—1422no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—2848no—
235C23—2844no—
236C23—2844no—
237C23—2848no—
238C23—712no—
239C23—712no—
240C23—712no—
241C23—712no—
242C23—1428no—
243C23—1421no—
244C23—1421no—
245C23—1422no—
246C23—1422no—
247C23—1422no—
248C23—1422no—
249C23Q8×D7112yes—
250C23C23×D7111yescharacteristic
251Q8—2844no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—2844no—
262C2×C4—2844no—
263C2×C4—712no—
264C2×C4—1421no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4—1422no—
268C2×C4—1422no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4D4×D7112yes—
274C22—24no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—28416no—
280C22—2841no—
281C22—2841no—
282C22—2842no—
283C22—714no—
284C22—1424no—
285C22—1422no—
286C22—1422no—
287C22—1422no—
288C22—1424no—
289C22—1424no—
290C22C2×D4×D7111yescharacteristic
291C22C2×Q8×D7111yescharacteristic
292C22#41×D7111yescharacteristic
293C22#1455617312yes—
294C22#1455617412yes—
295C4—22no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—2842no—
300C4—2841no—
301C4—2842no—
302C4—1422no—
303C2—42no—
304C2—2842no—
305C2—1424no—
306C2C22⋊Q8×D7111yescharacteristic
307C2#1798656011yescharacteristic
308C2#1798656211yescharacteristic
309C2#1798658512yes—
310C2#1798662412yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2626
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14556108C2semidirect
3. #17985934C2semidirect
4. #17985922C2semidirect
5. #17985921C2semidirect
6. #17985912C2semidirect
7. #17985911C2semidirect
8. #17985870C2semidirect
9. #17985858C2semidirect
11. #399×C71C2semidirect
13. C22×Dic142C22semidirect
17. C2×C4⋊Dic71C22semidirect
18. C2×Dic71⋊C4C22semidirect
19. C2×Dic71⋊C4C22semidirect
29. C71⋊#95C22semidirect
30. C71⋊#95C22semidirect
31. C71⋊#95C22semidirect
32. C2×C4⋊C4×C71C22semidirect
33. C2×C4⋊C4×C71C22semidirect
36. #95×C71C22semidirect
41. C23×D71Q8semidirect
156. C71#2626semidirect
160. #399D71semidirect
176. C2×C4⋊C4C2×D71semidirect
177. C2×C4⋊C4C2×D71semidirect
192. #95C2×D71semidirect