Finite Group Databasebeta

← #22600014

Subgroups — #22600014

Order 9088.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×D568C211yesmaximal, characteristic, complemented
2C2×D8×D71C211yesmaximal, characteristic, complemented
3#17986633C212yesmaximal, complemented
4#17986374C211yesmaximal, characteristic, complemented
5#17986236C211yesmaximal, characteristic, complemented
6#17986193C212yesmaximal, complemented
7#17986134C212yesmaximal, complemented
8#17986132C212yesmaximal, complemented
9#17986081C211yesmaximal, characteristic, complemented
10C8×C22⋊D71C211yesmaximal, characteristic, complemented
11#466×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yescomplemented
13C2×D4×D71—22no—
14C2×D4×D71C2212yescomplemented
15C2×D568—24no—
16C2×D568C2211yescharacteristic
17C2×D568C2211yescharacteristic, complemented
18C2×D568C2212yescomplemented
19C2×#11765850C2212yescomplemented
20C2×#11765850C2212yescomplemented
21D8×D71—24no—
22C2×C8×D71C2211yescharacteristic, complemented
23#14556124C2212yescomplemented
24#14556103C2212yescomplemented
25#14556101—22no—
26#14556098—22no—
27#14556096—22no—
28#14556090C2212yescomplemented
29C4×C22⋊D71C2211yescharacteristic
30Dic71⋊C8C2211yescharacteristic, complemented
31#14556048C2212yescomplemented
32#14556048C2212yescomplemented
33C22⋊(C71⋊C8)C2211yescharacteristic, complemented
34(C2×D71)⋊C8C2211yescharacteristic, complemented
35C8×Dic71C2211yescharacteristic, complemented
36C2×D8×C71C2211yescharacteristic, complemented
37C22×C8×C71C2211yescharacteristic, complemented
38#121×C71C2212yescomplemented
39#107×C71C2211yescharacteristic, complemented
40#102×C71C2212yescomplemented
41C23×D71—22no—
42C23×D71—22no—
43C2×D284—24no—
44C2×D284—22no—
45C2×D284—22no—
46C2×D284—42no—
47C2×D284C2312yes—
48C2×D284C2312yes—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C2×C22⋊D71—22no—
52C2×C22⋊D71D411yescharacteristic
53D4×D71—24no—
54D4×D71—42no—
55D4×D71—48no—
56C2×C4×D71—22no—
57C2×C4×D71C2311yescharacteristic
58D568—24no—
59D568—22no—
60D568D414yescomplemented
61#11765850—24no—
62#11765850—42no—
63C2×C71⋊C8C2311yescharacteristic
64C2×C71⋊C8C2311yescharacteristic
65C8×D71—22no—
66Dic71⋊C4D411yescharacteristic
67C22⋊Dic71D411yescharacteristic
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C2×D4×C71—22no—
73C2×D4×C71C2312yes—
74C22×C4×C71C2311yescharacteristic
75D8×C71—22no—
76C2×C8×C71—22no—
77C2×C8×C71C2311yescharacteristic
78C2×C8×C71C2311yescharacteristic
79C4⋊C4×C71C2312yes—
80C22⋊C4×C71—22no—
81C23×C71—22no—
82C23×C71C2×D411yescharacteristic
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—44no—
86C22×D71—44no—
87C22×D71—44no—
88C22×D71—44no—
89C22×D71—82no—
90C22×D71—82no—
91C22×D71C2×D411yescharacteristic
92D4×C71—22no—
93D4×C71—22no—
94D4×C71—44no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2411yescharacteristic
100C22⋊D71—44no—
101C22⋊D71—44no—
102C22⋊D71—82no—
103C22⋊D71D814yescomplemented
104C2×Dic71C2×D411yescharacteristic
105C2×Dic71C2×D411yescharacteristic
106C2×Dic71C2×D411yescharacteristic
107D284—22no—
108D284—22no—
109D284—22no—
110D284—44no—
111D284—44no—
112D284C2×D414yescomplemented
113C4×D71—21no—
114C4×D71—21no—
115C4×D71—44no—
116C8×C71—21no—
117C8×C71C2×D412yescomplemented
118C71⋊C8—21no—
119C71⋊C8—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—42no—
123C22×C71—42no—
124C22×C71—42no—
125C22×C71C2×D812yescomplemented
126C22×C71C22×D411yescharacteristic
127C2×D71—21no—
128C2×D71—21no—
129C2×D71—24no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—42no—
134C2×D71—42no—
135C2×D71—88no—
136C2×D71—88no—
137C2×D71C2×D812yescomplemented
138C4×C71—21no—
139C4×C71—42no—
140C4×C71C22×D411yescharacteristic
141C4×C71H(5,𝔽2)11yescharacteristic
142Dic71—21no—
143Dic71—21no—
144Dic71C2×D812yescomplemented
145C2×C71—22no—
146C2×C71—82no—
147C2×C71D4211yescharacteristic
148C2×C71C22×D811yescharacteristic
149C2×C71#57611yescharacteristic
150D71—22no—
151D71—44no—
152D71—82no—
153D4×D8—711nomaximal, Sylow
154C71D4×D811yescharacteristic, complemented, Sylow
155C22×D8—711no—
156C22×D8—711no—
157D42—712no—
158#493—711no—
159#466—711no—
160#466D7111yescharacteristic, complemented
161#459—712no—
162#447—712no—
163#447—712no—
164C4×D8—711no—
165C8×D4—711no—
166C22×D4—712no—
167C22×D4—712no—
168C22×D4—1422no—
169C2×D8—712no—
170C2×D8—712no—
171C2×D8—711no—
172C2×D8—712no—
173C2×D8—711no—
174C2×D8—1424no—
175C2×D8—1424no—
176C2×D8C2×D7111yescharacteristic, complemented
177C22×C8—711no—
178C22×C8C2×D7111yescharacteristic, complemented
179#127—712no—
180#121—712no—
181#121—1422no—
182#121C2×D7112yescomplemented
183#120—1422no—
184#120—1422no—
185C4×D4—711no—
186C4×D4—712no—
187#107C2×D7111yescharacteristic, complemented
188C4⋊C8—711no—
189#102—712no—
190#102—712no—
191#102C2×D7112yescomplemented
192C22⋊C8—711no—
193C22⋊C8—711no—
194C4×C8—711no—
195C24—1422no—
196C24—1422no—
197C2×D4—22no—
198C2×D4—2842no—
199C2×D4—2848no—
200C2×D4—2842no—
201C2×D4—711no—
202C2×D4—712no—
203C2×D4—712no—
204C2×D4—1422no—
205C2×D4—1424no—
206C2×D4—1422no—
207C2×D4—1422no—
208C2×D4—1422no—
209C2×D4—1424no—
210C2×D4—1422no—
211C2×D4C22×D7112yes—
212C22×C4—711no—
213C22×C4—1422no—
214C22×C4C22×D7111yescharacteristic
215D8—22no—
216D8—2842no—
217D8—714no—
218D8—1424no—
219D8—1424no—
220D8—1422no—
221C2×C8—22no—
222C2×C8—711no—
223C2×C8—711no—
224C2×C8—1422no—
225C2×C8C22×D7111yescharacteristic
226C2×C8C22×D7111yescharacteristic
227C4⋊C4—711no—
228C4⋊C4C22×D7112yes—
229C22⋊C4—22no—
230C22⋊C4—711no—
231C22⋊C4—711no—
232C22⋊C4—1422no—
233C22⋊C4—1422no—
234C42—711no—
235C23—22no—
236C23—2844no—
237C23—2844no—
238C23—2844no—
239C23—2844no—
240C23—5682no—
241C23—5682no—
242C23—711no—
243C23—1422no—
244C23—1422no—
245C23D4×D7111yescharacteristic
246D4—22no—
247D4—22no—
248D4—44no—
249D4—2844no—
250D4—2844no—
251D4—2844no—
252D4—2844no—
253D4—5682no—
254D4—714no—
255D4—714no—
256D4—1422no—
257D4—1422no—
258D4—1422no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—2844no—
263C2×C4—711no—
264C2×C4—711no—
265C2×C4—711no—
266C2×C4—1421no—
267C2×C4—1421no—
268C2×C4D4×D7111yescharacteristic
269C2×C4C23×D7111yescharacteristic
270C8—21no—
271C8—1421no—
272C8—1421no—
273C8D4×D7112yescomplemented
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—2842no—
280C22—2842no—
281C22—2842no—
282C22—2842no—
283C22—2842no—
284C22—5688no—
285C22—5688no—
286C22—712no—
287C22—1424no—
288C22—1421no—
289C22—1421no—
290C22D8×D7112yescomplemented
291C22C2×D4×D7111yescharacteristic
292C4—21no—
293C4—42no—
294C4—712no—
295C4—1421no—
296C4—1421no—
297C4C2×D4×D7111yescharacteristic
298C4#1455617411yescharacteristic
299C2—22no—
300C2—82no—
301C2—2844no—
302C2—5682no—
303C2—1422no—
304C2#1798663311yescharacteristic
305C2C2×D8×D7111yescharacteristic
306C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#466×C71C2
socleC22×C71C22×D4
hypercenter#466D71
hypocenterC71D4×D8
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C22×D568C2semidirect
2. C2×D8×D71C2semidirect
3. #17986633C2semidirect
4. #17986374C2semidirect
5. #17986236C2semidirect
6. #17986193C2semidirect
7. #17986134C2semidirect
8. #17986132C2semidirect
9. #17986081C2semidirect
10. C8×C22⋊D71C2semidirect
11. #466×C71C2semidirect
12. C22×D284C22semidirect
14. C2×D4×D71C22semidirect
17. C2×D568C22semidirect
18. C2×D568C22semidirect
19. C2×#11765850C22semidirect
20. C2×#11765850C22semidirect
22. C2×C8×D71C22semidirect
23. #14556124C22semidirect
24. #14556103C22semidirect
28. #14556090C22semidirect
30. Dic71⋊C8C22semidirect
31. #14556048C22semidirect
32. #14556048C22semidirect
33. C22⋊(C71⋊C8)C22semidirect
34. (C2×D71)⋊C8C22semidirect
35. C8×Dic71C22semidirect
36. C2×D8×C71C22semidirect
37. C22×C8×C71C22semidirect
38. #121×C71C22semidirect
39. #107×C71C22semidirect
40. #102×C71C22semidirect
60. D568D4semidirect
103. C22⋊D71D8semidirect
112. D284C2×D4semidirect
117. C8×C71C2×D4semidirect
125. C22×C71C2×D8semidirect
137. C2×D71C2×D8semidirect
144. Dic71C2×D8semidirect
154. C71D4×D8semidirect
160. #466D71semidirect
176. C2×D8C2×D71semidirect
178. C22×C8C2×D71semidirect
182. #121C2×D71semidirect
187. #107C2×D71semidirect
191. #102C2×D71semidirect
273. C8D4×D71semidirect
290. C22D8×D71semidirect