Finite Group Databasebeta

← #22598475

Subgroups — #22598475

Order 9088.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986530C211yesmaximal, characteristic, complemented
2C2×#14556090C211yesmaximal, characteristic, complemented
3#17986329C211yesmaximal, characteristic, complemented
4#17986302C212yesmaximal, complemented
5#17986213C212yesmaximal, complemented
6#17986152C211yesmaximal, characteristic, complemented
7#17986132C211yesmaximal, characteristic
8#17986108C212yesmaximal, complemented
9#17986098C212yesmaximal, complemented
10#17986053C211yesmaximal, characteristic, complemented
11#480×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C4×D71—21no—
14C2×#11765854C2212yes—
15C2×#11765850C2212yes—
16#14556121—21no—
17#14556119C2211yescharacteristic, complemented
18#14556115C2211yescharacteristic, complemented
19#14556109—21no—
20#14556107C2211yescharacteristic
21#14556093C2211yescharacteristic, complemented
22D4×Dic71—21no—
23D4×Dic71C2211yescharacteristic, complemented
24#14556090—21no—
25#14556090—21no—
26#14556090C2212yes—
27#14556090C2212yes—
28#14556084—21no—
29C2×(C2×D71)⋊C4—21no—
30C2×C4×Dic71C2211yescharacteristic
31#14556059C2212yescomplemented
32Dic71⋊C8C2212yescomplemented
33#14556051C2212yescomplemented
34#14556050C2212yes—
35#14556048C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37#14556038C2212yescomplemented
38#121×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#107×C71C2212yescomplemented
41#102×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71—21no—
44C2×Dic142C2311yescharacteristic
45C2×D284—21no—
46C2×D284—24no—
47C2×D284C2312yes—
48C22×Dic71—21no—
49C22×Dic71D411yescharacteristic
50C22×Dic71D411yescharacteristic
51C2×C4×D71—22no—
52C2×C4×D71—22no—
53C2×C4×D71—44no—
54#11765854—24no—
55#11765850—24no—
56C2×C71⋊C8C2312yes—
57C4⋊Dic71—21no—
58C4⋊Dic71C2311yescharacteristic
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61Dic71⋊C4—21no—
62C22⋊Dic71—21no—
63C22⋊Dic71—21no—
64C22⋊Dic71—21no—
65C22⋊Dic71—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68C4×Dic71—21no—
69C4×Dic71D411yescharacteristic
70C4×Dic71D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C4×Dic71C2311yescharacteristic
73C2×D4×C71—21no—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76C22×C4×C71C2311yescharacteristic
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71C2312yes—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C23×C71—21no—
83C23×C71C2×D411yescharacteristic
84C22×D71—24no—
85C22×D71—22no—
86C22×D71—42no—
87C22×D71—42no—
88D4×C71—22no—
89D4×C71—42no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—42no—
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2411yescharacteristic
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—41no—
100C2×Dic71—42no—
101C2×Dic71C2×D411yescharacteristic
102C2×Dic71C2×D411yescharacteristic
103C2×Dic71C2×D411yescharacteristic
104C2×Dic71C2×D411yescharacteristic
105D284—24no—
106D284—22no—
107D284#4114yes—
108C4×D71—44no—
109C4×D71—44no—
110Dic142—22no—
111C8×C71—22no—
112C71⋊C8—22no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71—41no—
116C22×C71—41no—
117C22×C71—41no—
118C22×C71AGL(1,ℤ/8)12yes—
119C22×C71C22×D411yescharacteristic
120C2×D71—24no—
121C2×D71—48no—
122C2×D71—42no—
123C2×D71—42no—
124C4×C71—21no—
125C4×C71—42no—
126C4×C71—41no—
127C4×C71C2×#4112yes—
128Dic71—21no—
129Dic71—21no—
130Dic71—21no—
131Dic71—21no—
132Dic71—41no—
133Dic71—41no—
134C2×C71—22no—
135C2×C71—81no—
136C2×C71#52511yescharacteristic
137C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C71#57511yescharacteristic
139D71—44no—
140#3114—711nomaximal, Sylow
141C71#311411yescharacteristic, complemented, Sylow
142#529—711no—
143C2×C4×D4—711no—
144#483—711no—
145#480D7111yescharacteristic, complemented
146#478—712no—
147#463—712no—
148#450—711no—
149#447—711no—
150#442—712no—
151#440—712no—
152#432—711no—
153C22×D4—711no—
154C23×C4—1421no—
155C2×SD16—712no—
156C2×D8—712no—
157#126—1421no—
158#125—711no—
159#124—711no—
160#123—1421no—
161C22⋊Q8—711no—
162#121C2×D7111yescharacteristic
163C4×Q8—711no—
164C4×D4—711no—
165C4×D4—712no—
166C4×D4—712no—
167C4×D4—1421no—
168C4×D4—1421no—
169C4×D4—1421no—
170#117—1421no—
171C2×C4⋊C4C2×D7111yescharacteristic
172C2×C22⋊C4—1421no—
173C2×C42—711no—
174#107C2×D7112yescomplemented
175#106—712no—
176C4⋊C8—712no—
177#103—712no—
178#102—712no—
179#102—712no—
180#102C2×D7112yescomplemented
181C22⋊C8—711no—
182C22⋊C8C2×D7111yescharacteristic
183#97—712no—
184C24—1421no—
185C2×Q8—711no—
186C2×D4—21no—
187C2×D4—712no—
188C2×D4—1424no—
189C2×D4—1421no—
190C2×D4C22×D7111yescharacteristic
191C22×C4—21no—
192C22×C4—2844no—
193C22×C4—711no—
194C22×C4—711no—
195C22×C4—1422no—
196C22×C4—1422no—
197C22×C4—1421no—
198C22×C4C22×D7111yescharacteristic
199SD16—1424no—
200D8—1424no—
201C2×C8—712no—
202C2×C8C22×D7112yes—
203C4⋊C4—21no—
204C4⋊C4—711no—
205C4⋊C4—1421no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C4⋊C4—1421no—
209C4⋊C4C22×D7111yescharacteristic
210C4⋊C4C22×D7112yes—
211C22⋊C4—21no—
212C22⋊C4—1422no—
213C22⋊C4—1422no—
214C22⋊C4—1421no—
215C22⋊C4—1421no—
216C22⋊C4—1421no—
217C22⋊C4—1421no—
218C42—711no—
219C42—711no—
220C42—711no—
221C42—711no—
222C42—1421no—
223C23—21no—
224C23—2842no—
225C23—2842no—
226C23—1424no—
227C23—1422no—
228C23D4×D7111yescharacteristic
229Q8—1422no—
230D4—22no—
231D4—42no—
232D4—714no—
233D4—1424no—
234D4—1422no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—2844no—
240C2×C4—2844no—
241C2×C4—2842no—
242C2×C4—2841no—
243C2×C4—711no—
244C2×C4—711no—
245C2×C4—711no—
246C2×C4—711no—
247C2×C4—1421no—
248C2×C4—1421no—
249C2×C4—1422no—
250C2×C4D4×D7111yescharacteristic
251C2×C4C23×D7111yescharacteristic
252C8—22no—
253C8—1422no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—2848no—
260C22—2842no—
261C22—2842no—
262C22—1424no—
263C22#1455615212yes—
264C22C2×D4×D7111yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—2841no—
269C4—2841no—
270C4—1421no—
271C4—1421no—
272C4—1421no—
273C4—1421no—
274C4#41×D7112yes—
275C2—22no—
276C2—81no—
277C2—2844no—
278C2#1798650711yescharacteristic
279C2C2×#1455615211yescharacteristic
280C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#480×C71C2
socleC22×C71C22×D4
hypercenter#480D71
hypocenterC71#3114
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #17986530C2semidirect
2. C2×#14556090C2semidirect
3. #17986329C2semidirect
4. #17986302C2semidirect
5. #17986213C2semidirect
6. #17986152C2semidirect
8. #17986108C2semidirect
9. #17986098C2semidirect
10. #17986053C2semidirect
11. #480×C71C2semidirect
17. #14556119C22semidirect
18. #14556115C22semidirect
21. #14556093C22semidirect
23. D4×Dic71C22semidirect
31. #14556059C22semidirect
32. Dic71⋊C8C22semidirect
33. #14556051C22semidirect
37. #14556038C22semidirect
40. #107×C71C22semidirect
41. #102×C71C22semidirect
141. C71#3114semidirect
145. #480D71semidirect
174. #107C2×D71semidirect
180. #102C2×D71semidirect