Finite Group Databasebeta

← #22600024

Subgroups — #22600024

Order 9088.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14556151C211yesmaximal, characteristic, complemented
2C2×D8×D71C211yesmaximal, characteristic, complemented
3#17986642C212yesmaximal, complemented
4#17986237C211yesmaximal, characteristic, complemented
5#17986232C211yesmaximal, characteristic, complemented
6#17986211C212yesmaximal, complemented
7#17986148C212yesmaximal, complemented
8#17986143C212yesmaximal, complemented
9C4×D568C211yesmaximal, characteristic, complemented
10C8×D284C211yesmaximal, characteristic, complemented
11#495×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yescomplemented
13C2×D4×D71C2212yes—
14#14556151—24no—
15C2×#11765852C2212yescomplemented
16C2×D568C2211yescharacteristic
17C2×#11765850C2212yes—
18D8×D71—24no—
19C2×C8×D71C2211yescharacteristic
20C2×C8×D71C2211yescharacteristic, complemented
21#14556110—22no—
22#14556104C2212yes—
23#14556101—22no—
24#14556099C2212yescomplemented
25#14556098—22no—
26C4×D284C2211yescharacteristic
27C4×D284C2212yescomplemented
28C22⋊C4×D71—22no—
29#14556062C2211yescharacteristic, complemented
30C4⋊(C71⋊C8)C2211yescharacteristic, complemented
31#14556052C2212yescomplemented
32#14556050C2212yescomplemented
33#14556049C2212yescomplemented
34(C2×D71)⋊C8C2211yescharacteristic
35(C2×D71)⋊C8C2211yescharacteristic, complemented
36C2×Q16×C71C2211yescharacteristic, complemented
37C2×SD16×C71C2212yescomplemented
38C2×D8×C71C2211yescharacteristic
39#124×C71C2212yescomplemented
40C4×C8×C71C2211yescharacteristic, complemented
41C23×D71—22no—
42#11765866—24no—
43#11765866—42no—
44C2×D284—22no—
45C2×D284D411yescharacteristic
46C2×D284C2312yes—
47C2×C22⋊D71—22no—
48C2×C22⋊D71—22no—
49D4×D71—24no—
50D4×D71—42no—
51C2×C4×D71—22no—
52C2×C4×D71—22no—
53C2×C4×D71C2311yescharacteristic
54C2×C4×D71C2311yescharacteristic
55#11765852—24no—
56D568D414yescomplemented
57#11765850—24no—
58C2×C71⋊C8C2311yescharacteristic
59C2×C71⋊C8C2311yescharacteristic
60C8×D71—22no—
61C8×D71—22no—
62C4⋊Dic71D411yescharacteristic
63C4⋊Dic71C2312yes—
64Dic71⋊C4—22no—
65C22⋊Dic71—22no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4D411yescharacteristic
69(C2×D71)⋊C4D411yescharacteristic
70C2×Q8×C71C2312yes—
71C2×D4×C71C2312yes—
72Q16×C71—22no—
73SD16×C71—42no—
74D8×C71—22no—
75C2×C8×C71C2311yescharacteristic
76C2×C8×C71C2311yescharacteristic
77C22⋊C4×C71—22no—
78C22⋊C4×C71—22no—
79C42×C71C2311yescharacteristic
80C23×C71—22no—
81C22×D71—22no—
82C22×D71—44no—
83C22×D71—44no—
84C22×D71—82no—
85C22×D71C2×D411yescharacteristic
86C22×D71C2×D411yescharacteristic
87Q8×C71—22no—
88Q8×C71—22no—
89D4×C71—22no—
90D4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2411yescharacteristic
95C22⋊D71—44no—
96C22⋊D71—82no—
97C2×Dic71—22no—
98C2×Dic71C2×D411yescharacteristic
99C2×Dic71C2×D411yescharacteristic
100D284—22no—
101D284—22no—
102D284—44no—
103D284C2×D414yes—
104C4×D71—21no—
105C4×D71—21no—
106C4×D71—21no—
107C4×D71—21no—
108C4×D71—44no—
109C4×D71—82no—
110C8×C71—21no—
111C8×C71C2×D412yes—
112C71⋊C8—21no—
113C71⋊C8—21no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71—42no—
117C22×C71C22×D411yescharacteristic
118C2×D71—21no—
119C2×D71—21no—
120C2×D71—21no—
121C2×D71—24no—
122C2×D71—21no—
123C2×D71—21no—
124C2×D71—42no—
125C2×D71—42no—
126C2×D71—88no—
127C2×D71#13512yescomplemented
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—42no—
131C4×C71C22×D411yescharacteristic
132C4×C71H(5,𝔽2)11yescharacteristic
133Dic71—21no—
134Dic71—21no—
135Dic71—42no—
136C2×C71—82no—
137C2×C71D4211yescharacteristic
138C2×C71C2×#13511yescharacteristic
139C2×C71#57611yescharacteristic
140D71—22no—
141D71—41no—
142D71—44no—
143#3280—711nomaximal, Sylow
144C71#328011yescharacteristic, complemented, Sylow
145C2×#135—711no—
146C22×D8—711no—
147#546—712no—
148#495D7111yescharacteristic, complemented
149#467—711no—
150#466—711no—
151#463—712no—
152#450—712no—
153#449—712no—
154C4×D8—711no—
155C8×D4—711no—
156C2×#41—712no—
157C22×D4—712no—
158#135—1424no—
159C2×Q16C2×D7111yescharacteristic, complemented
160C2×SD16—712no—
161C2×SD16C2×D7112yescomplemented
162C2×D8—712no—
163C2×D8—711no—
164C2×D8—1424no—
165C2×D8C2×D7111yescharacteristic
166C22×C8—711no—
167C22×C8—711no—
168#124C2×D7112yescomplemented
169#123—1422no—
170C22⋊Q8—712no—
171#121—712no—
172#121—1422no—
173#120—1422no—
174C4×D4—711no—
175C4×D4—712no—
176C2×C22⋊C4—1422no—
177#107—711no—
178C4⋊C8—711no—
179#103—712no—
180#102—712no—
181#102—712no—
182C22⋊C8—711no—
183C22⋊C8—711no—
184C4×C8C2×D7111yescharacteristic, complemented
185C24—1422no—
186#41—2842no—
187#41—1424no—
188C2×Q8C22×D7112yes—
189C2×D4—2842no—
190C2×D4—711no—
191C2×D4—712no—
192C2×D4—1422no—
193C2×D4—1422no—
194C2×D4—1424no—
195C2×D4—1422no—
196C2×D4C22×D7112yes—
197C22×C4—711no—
198C22×C4—711no—
199C22×C4—1422no—
200C22×C4—1422no—
201Q16—22no—
202SD16—42no—
203SD16—1424no—
204D8—22no—
205D8—714no—
206D8—1424no—
207C2×C8—711no—
208C2×C8—711no—
209C2×C8—1422no—
210C2×C8—1422no—
211C2×C8C22×D7111yescharacteristic
212C2×C8C22×D7111yescharacteristic
213C4⋊C4—711no—
214C4⋊C4—712no—
215C4⋊C4—1422no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—711no—
219C22⋊C4—711no—
220C22⋊C4—1422no—
221C22⋊C4—1422no—
222C22⋊C4—1422no—
223C42C22×D7111yescharacteristic
224C23—22no—
225C23—2844no—
226C23—2844no—
227C23—5682no—
228C23—711no—
229C23—711no—
230C23—1422no—
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—22no—
235D4—2844no—
236D4—2844no—
237D4—5682no—
238D4—714no—
239D4—1422no—
240D4—1422no—
241C2×C4—22no—
242C2×C4—2844no—
243C2×C4—5682no—
244C2×C4—711no—
245C2×C4—711no—
246C2×C4—1422no—
247C2×C4—1421no—
248C2×C4—1421no—
249C2×C4—1421no—
250C2×C4—1421no—
251C2×C4D4×D7111yescharacteristic
252C2×C4D4×D7111yescharacteristic
253C2×C4C23×D7111yescharacteristic
254C8—21no—
255C8—1421no—
256C8—1421no—
257C8D4×D7112yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—2842no—
262C22—2842no—
263C22—5688no—
264C22—712no—
265C22—1424no—
266C22—1421no—
267C22—1421no—
268C22—1421no—
269C22—1421no—
270C22—1421no—
271C22C2×D4×D7111yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—2842no—
276C4—1421no—
277C4—1421no—
278C4C2×D4×D7111yescharacteristic
279C4#1455617311yescharacteristic
280C2—82no—
281C2—2844no—
282C2—2841no—
283C2—1422no—
284C2#1798663411yescharacteristic
285C2#135×D7111yescharacteristic
286C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#495×C71C2
socleC22×C71C22×D4
hypercenter#495D71
hypocenterC71#3280
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#14556151C2semidirect
2. C2×D8×D71C2semidirect
3. #17986642C2semidirect
4. #17986237C2semidirect
5. #17986232C2semidirect
6. #17986211C2semidirect
7. #17986148C2semidirect
8. #17986143C2semidirect
9. C4×D568C2semidirect
10. C8×D284C2semidirect
11. #495×C71C2semidirect
12. C2×#11765866C22semidirect
15. C2×#11765852C22semidirect
20. C2×C8×D71C22semidirect
24. #14556099C22semidirect
27. C4×D284C22semidirect
29. #14556062C22semidirect
30. C4⋊(C71⋊C8)C22semidirect
31. #14556052C22semidirect
32. #14556050C22semidirect
33. #14556049C22semidirect
35. (C2×D71)⋊C8C22semidirect
36. C2×Q16×C71C22semidirect
37. C2×SD16×C71C22semidirect
39. #124×C71C22semidirect
40. C4×C8×C71C22semidirect
56. D568D4semidirect
127. C2×D71#135semidirect
144. C71#3280semidirect
148. #495D71semidirect
159. C2×Q16C2×D71semidirect
161. C2×SD16C2×D71semidirect
168. #124C2×D71semidirect
184. C4×C8C2×D71semidirect