Finite Group Databasebeta

← #22600022

Subgroups — #22600022

Order 9088.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C22×D568C211yesmaximal, characteristic, complemented
3#17986640C212yesmaximal, complemented
4#17986384C211yesmaximal, characteristic, complemented
5#17986236C211yesmaximal, characteristic, complemented
6#17986217C212yesmaximal, complemented
7#17986149C212yesmaximal, complemented
8#17986147C212yesmaximal, complemented
9#17986081C211yesmaximal, characteristic, complemented
10C8×C22⋊D71C211yesmaximal, characteristic, complemented
11#467×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yescomplemented
13C22×D284C2212yes—
14#14556151—24no—
15C2×#11765852C2212yescomplemented
16C2×#11765852C2212yescomplemented
17C2×D568—24no—
18C2×D568C2211yescharacteristic
19C2×D568C2212yes—
20C2×D568C2211yescharacteristic
21C2×C8×D71C2211yescharacteristic, complemented
22#14556117C2212yescomplemented
23#14556113—22no—
24#14556103—22no—
25#14556103C2212yescomplemented
26#14556098—22no—
27#14556090C2212yescomplemented
28C4×C22⋊D71C2211yescharacteristic
29C22⋊C4×D71—22no—
30Dic71⋊C8C2211yescharacteristic, complemented
31#14556048C2212yescomplemented
32#14556048C2212yescomplemented
33C22⋊(C71⋊C8)C2211yescharacteristic
34(C2×D71)⋊C8C2211yescharacteristic, complemented
35C8×Dic71C2211yescharacteristic, complemented
36C2×Q16×C71C2211yescharacteristic, complemented
37C22×C8×C71C2211yescharacteristic
38C22⋊Q8×C71C2212yes—
39#107×C71C2211yescharacteristic, complemented
40#103×C71C2212yescomplemented
41C23×D71—22no—
42#11765866—24no—
43#11765866—42no—
44C2×D284—24no—
45C2×D284—22no—
46C2×D284—22no—
47C2×D284—22no—
48C2×D284—42no—
49C2×D284C2312yes—
50C2×D284C2312yes—
51C2×C22⋊D71D411yescharacteristic
52C2×C4×D71—22no—
53C2×C4×D71—22no—
54C2×C4×D71C2311yescharacteristic
55#11765852—24no—
56#11765852—42no—
57D568—24no—
58D568—22no—
59D568D414yescomplemented
60C2×C71⋊C8C2311yescharacteristic
61C2×C71⋊C8C2311yescharacteristic
62C8×D71—22no—
63Dic71⋊C4D411yescharacteristic
64C22⋊Dic71D411yescharacteristic
65(C2×D71)⋊C4—22no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4D411yescharacteristic
70C4×Dic71C2311yescharacteristic
71C2×Q8×C71C2312yes—
72C22×C4×C71C2311yescharacteristic
73Q16×C71—22no—
74C2×C8×C71—22no—
75C2×C8×C71C2311yescharacteristic
76C2×C8×C71C2311yescharacteristic
77C4⋊C4×C71—22no—
78C4⋊C4×C71C2312yes—
79C22⋊C4×C71—22no—
80C23×C71C2×D411yescharacteristic
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—44no—
84C22×D71—44no—
85C22×D71—82no—
86C22×D71C2×D411yescharacteristic
87Q8×C71—22no—
88Q8×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2411yescharacteristic
95C22⋊D71—22no—
96C2×Dic71C2×D411yescharacteristic
97C2×Dic71C2×D411yescharacteristic
98C2×Dic71C2×D411yescharacteristic
99D284—22no—
100D284—22no—
101D284—22no—
102D284—44no—
103D284—44no—
104D284—82no—
105D284C2×D414yes—
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—21no—
115C22×C71—21no—
116C22×C71#13512yescomplemented
117C22×C71C22×D411yescharacteristic
118C2×D71—21no—
119C2×D71—24no—
120C2×D71—21no—
121C2×D71—21no—
122C2×D71—42no—
123C2×D71—42no—
124C2×D71—42no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—88no—
128C4×C71—21no—
129C4×C71—42no—
130C4×C71—42no—
131C4×C71C22×D411yescharacteristic
132C4×C71H(5,𝔽2)11yescharacteristic
133Dic71—21no—
134Dic71—21no—
135Dic71—21no—
136C2×C71—22no—
137C2×C71D4211yescharacteristic
138C2×C71C2×#13511yescharacteristic
139C2×C71#57611yescharacteristic
140D71—41no—
141D71—44no—
142D71—82no—
143#3280—711nomaximal, Sylow
144C71#328011yescharacteristic, complemented, Sylow
145C2×#135—711no—
146C22×D8—711no—
147#546—712no—
148#495—711no—
149#467D7111yescharacteristic, complemented
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×SD16—712no—
162C2×D8—711no—
163C2×D8—712no—
164C2×D8—711no—
165C2×D8—1424no—
166C22×C8—711no—
167C22×C8C2×D7111yescharacteristic
168#124—712no—
169#123—1422no—
170C22⋊Q8C2×D7112yes—
171#121—712no—
172#121—1422no—
173#120—1422no—
174C4×D4—711no—
175C4×D4—712no—
176C2×C22⋊C4—1422no—
177#107C2×D7111yescharacteristic, complemented
178C4⋊C8—711no—
179#103C2×D7112yescomplemented
180#102—712no—
181#102—712no—
182C22⋊C8—711no—
183C22⋊C8—711no—
184C4×C8—711no—
185C24—1422no—
186#41—2842no—
187#41—1424no—
188C2×Q8C22×D7112yes—
189C2×D4—2842no—
190C2×D4—711no—
191C2×D4—712no—
192C2×D4—712no—
193C2×D4—1422no—
194C2×D4—1424no—
195C2×D4—1422no—
196C2×D4—1422no—
197C22×C4—711no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4C22×D7111yescharacteristic
201Q16—22no—
202SD16—2842no—
203SD16—1424no—
204D8—714no—
205D8—1424no—
206D8—1422no—
207C2×C8—22no—
208C2×C8—711no—
209C2×C8—711no—
210C2×C8—1422no—
211C2×C8C22×D7111yescharacteristic
212C2×C8C22×D7111yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—711no—
215C4⋊C4C22×D7112yes—
216C22⋊C4—22no—
217C22⋊C4—711no—
218C22⋊C4—711no—
219C22⋊C4—1422no—
220C22⋊C4—1422no—
221C22⋊C4—1422no—
222C22⋊C4—1422no—
223C42—711no—
224C23—2844no—
225C23—2844no—
226C23—5682no—
227C23—711no—
228C23—1422no—
229C23—1422no—
230C23D4×D7111yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—2844no—
234D4—2844no—
235D4—5682no—
236D4—714no—
237D4—1422no—
238D4—1422no—
239D4—1422no—
240D4—1422no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—2844no—
246C2×C4—5682no—
247C2×C4—711no—
248C2×C4—711no—
249C2×C4—711no—
250C2×C4—1421no—
251C2×C4—1421no—
252C2×C4D4×D7111yescharacteristic
253C2×C4C23×D7111yescharacteristic
254C8—21no—
255C8—1421no—
256C8—1421no—
257C8D4×D7112yes—
258C22—21no—
259C22—21no—
260C22—2842no—
261C22—2842no—
262C22—2842no—
263C22—2842no—
264C22—2842no—
265C22—5688no—
266C22—1424no—
267C22—1421no—
268C22—1421no—
269C22—1421no—
270C22#1455615112yescomplemented
271C22C2×D4×D7111yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—1421no—
276C4—1421no—
277C4—1421no—
278C4C2×D4×D7111yescharacteristic
279C4#1455617411yescharacteristic
280C2—22no—
281C2—2844no—
282C2—2841no—
283C2—5682no—
284C2#1798663311yescharacteristic
285C2C2×#1455615111yescharacteristic
286C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3280
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#14556151C2semidirect
2. C22×D568C2semidirect
3. #17986640C2semidirect
4. #17986384C2semidirect
5. #17986236C2semidirect
6. #17986217C2semidirect
7. #17986149C2semidirect
8. #17986147C2semidirect
9. #17986081C2semidirect
10. C8×C22⋊D71C2semidirect
11. #467×C71C2semidirect
12. C2×#11765866C22semidirect
15. C2×#11765852C22semidirect
16. C2×#11765852C22semidirect
21. C2×C8×D71C22semidirect
22. #14556117C22semidirect
25. #14556103C22semidirect
27. #14556090C22semidirect
30. Dic71⋊C8C22semidirect
31. #14556048C22semidirect
32. #14556048C22semidirect
34. (C2×D71)⋊C8C22semidirect
35. C8×Dic71C22semidirect
36. C2×Q16×C71C22semidirect
39. #107×C71C22semidirect
40. #103×C71C22semidirect
59. D568D4semidirect
116. C22×C71#135semidirect
144. C71#3280semidirect
149. #467D71semidirect
159. C2×Q16C2×D71semidirect
177. #107C2×D71semidirect
179. #103C2×D71semidirect
270. C22#14556151semidirect