Finite Group Databasebeta

← #22599962

Subgroups — #22599962

Order 9088.

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

normal subgroups only

This group has 556 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
1C2×#14556156C212yesmaximal, complemented
2#17986670C211yesmaximal, characteristic, complemented
3D4×D284C211yesmaximal, characteristic, complemented
4#17986376C211yesmaximal, characteristic, complemented
5#17986252C212yesmaximal, complemented
6#17986200C211yesmaximal, characteristic, complemented
7#17986195C211yesmaximal, characteristic, complemented
8#17986146C212yesmaximal, complemented
9#17986135C212yesmaximal, complemented
10C4×#11765854C211yesmaximal, characteristic, complemented
11#436×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2212yescomplemented
13C22×D284C2212yes—
14C2×D4×D71—21no—
15#14556156—28no—
16C2×#11765854C2211yescharacteristic
17C2×#11765854C2212yescomplemented
18C2×D568C2212yes—
19C2×D568C2212yescomplemented
20C2×D568C2212yescomplemented
21#14556125C2211yescharacteristic
22#14556125C2211yescharacteristic, complemented
23#14556124—21no—
24#14556103—21no—
25#14556102C2212yes—
26#14556102C2212yescomplemented
27#14556101—22no—
28#14556098—22no—
29C4×Dic142C2211yescharacteristic, complemented
30C4×D284C2211yescharacteristic
31#14556090—21no—
32#14556060C2211yescharacteristic, complemented
33#14556053C2211yescharacteristic, complemented
34#14556050C2211yescharacteristic
35#14556050C2212yescomplemented
36#14556050C2212yescomplemented
37C2×M16×C71C2212yescomplemented
38C4×D4×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic, complemented
40C22⋊C8×C71C2212yescomplemented
41C4×C8×C71C2211yescharacteristic, complemented
42C23×D71—22no—
43#11765865—24no—
44#11765865—42no—
45C2×Dic142C2311yescharacteristic
46C2×D284—24no—
47C2×D284—21no—
48C2×D284—22no—
49C2×D284—21no—
50C2×D284—21no—
51C2×D284—21no—
52C2×D284—42no—
53C2×D284C2312yes—
54C2×D284C2312yes—
55C2×D284C2311yescharacteristic
56C2×C22⋊D71—22no—
57C2×C22⋊D71—22no—
58C2×C22⋊D71—22no—
59D4×D71—44no—
60C2×C4×D71—21no—
61C2×C4×D71—22no—
62#11765854—24no—
63#11765854D414yescomplemented
64D568—24no—
65D568—24no—
66D568—42no—
67C4⋊Dic71C2311yescharacteristic
68C4⋊Dic71C2311yescharacteristic
69Dic71⋊C4—21no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4—21no—
72(C2×D71)⋊C4—22no—
73C4×Dic71—21no—
74C2×D4×C71D411yescharacteristic
75C22×C4×C71C2312yes—
76M16×C71—24no—
77C2×C8×C71C2311yescharacteristic
78C2×C8×C71C2312yes—
79C2×C8×C71C2311yescharacteristic
80C4⋊C4×C71D411yescharacteristic
81C22⋊C4×C71D412yes—
82C42×C71C2311yescharacteristic
83C23×C71C2×D412yes—
84C22×D71—21no—
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—44no—
88C22×D71—44no—
89C22×D71—82no—
90D4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2411yescharacteristic
97C22⋊D71—44no—
98C22⋊D71—44no—
99C22⋊D71—82no—
100C2×Dic71—21no—
101C2×Dic71—21no—
102C2×Dic71—21no—
103D284—22no—
104D284—22no—
105D284—21no—
106D284—22no—
107D284—22no—
108D284—44no—
109D284—42no—
110D284—42no—
111D284—81no—
112D284—81no—
113D284C2×D412yes—
114C4×D71—44no—
115C4×D71—42no—
116Dic142—21no—
117Dic142C2×D412yescomplemented
118C8×C71—22no—
119C8×C71—21no—
120C8×C71C2×D412yescomplemented
121C22×C71—22no—
122C22×C71—22no—
123C22×C71—22no—
124C22×C71C22×D411yescharacteristic
125C2×D71—22no—
126C2×D71—41no—
127C2×D71—41no—
128C2×D71—42no—
129C2×D71—42no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—42no—
134C2×D71—88no—
135C4×C71—22no—
136C4×C71—21no—
137C4×C71AGL(1,ℤ/8)12yescomplemented
138C4×C71C22×D411yescharacteristic
139C4×C71H(5,𝔽2)11yescharacteristic
140Dic71—22no—
141Dic71—41no—
142Dic71—41no—
143C2×C71—42no—
144C2×C71D4211yescharacteristic
145C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C71#57611yescharacteristic
147D71—42no—
148D71—82no—
149D71—82no—
150#3274—711nomaximal, Sylow
151C71#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—712no—
153#550—711no—
154D42—711no—
155#493—711no—
156#469—712no—
157#460—711no—
158#459—711no—
159#449—712no—
160#447—712no—
161C4×SD16—711no—
162#436D7111yescharacteristic, complemented
163C2×#41—712no—
164C22×D4—712no—
165C22×D4—1421no—
166AGL(1,ℤ/8)—1428no—
167C2×SD16—711no—
168C2×SD16—712no—
169C2×D8—712no—
170C2×D8—712no—
171C2×D8—712no—
172C2×M16C2×D7112yescomplemented
173#127—711no—
174#127—711no—
175#127—1421no—
176#121—712no—
177#121—712no—
178#121—1421no—
179#121—1422no—
180#120—1422no—
181C4×Q8—711no—
182C4×D4—711no—
183C4×D4—1421no—
184C4×D4C2×D7111yescharacteristic
185#106—711no—
186C4⋊C8C2×D7111yescharacteristic, complemented
187#103—711no—
188#102—711no—
189#102—712no—
190#102—712no—
191C22⋊C8C2×D7112yescomplemented
192C4×C8C2×D7111yescharacteristic, complemented
193C24—1422no—
194#41—2842no—
195#41—1424no—
196C2×Q8—711no—
197C2×D4—2842no—
198C2×D4—2844no—
199C2×D4—711no—
200C2×D4—712no—
201C2×D4—712no—
202C2×D4—1424no—
203C2×D4—1422no—
204C2×D4—1422no—
205C2×D4—1422no—
206C2×D4—1421no—
207C2×D4—1422no—
208C2×D4—1421no—
209C2×D4—1421no—
210C2×D4—1421no—
211C2×D4D28411yescharacteristic
212C22×C4—1421no—
213C22×C4—1422no—
214C22×C4C22×D7112yes—
215SD16—714no—
216SD16—1424no—
217D8—2842no—
218D8—1424no—
219D8—1424no—
220M16—24no—
221C2×C8C22×D7111yescharacteristic
222C2×C8C22×D7112yes—
223C2×C8C22×D7111yescharacteristic
224C4⋊C4—711no—
225C4⋊C4—711no—
226C4⋊C4—1421no—
227C4⋊C4D28411yescharacteristic
228C22⋊C4—1421no—
229C22⋊C4—1422no—
230C22⋊C4—1422no—
231C22⋊C4D28412yes—
232C42—1421no—
233C42C22×D7111yescharacteristic
234C23—2844no—
235C23—2844no—
236C23—5682no—
237C23—1422no—
238C23—1422no—
239C23—1421no—
240C23C2×D28412yes—
241Q8—712no—
242Q8—1421no—
243D4—22no—
244D4—2844no—
245D4—2844no—
246D4—2844no—
247D4—2842no—
248D4—2842no—
249D4—5681no—
250D4—5681no—
251D4—5682no—
252D4—712no—
253D4—1422no—
254D4—1422no—
255D4—1422no—
256D4—1422no—
257D4—1421no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—2844no—
261C2×C4—2842no—
262C2×C4—1421no—
263C2×C4—1421no—
264C2×C4—1421no—
265C2×C4C2×D28411yescharacteristic
266C2×C4C2×D28412yes—
267C2×C4C2×D28411yescharacteristic
268C2×C4C23×D7111yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D7112yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—2841no—
276C22—2841no—
277C22—2842no—
278C22—2842no—
279C22—2842no—
280C22—2842no—
281C22—2842no—
282C22—2842no—
283C22—5688no—
284C22—1422no—
285C22C22×D28411yescharacteristic
286C4—22no—
287C4—21no—
288C4—2841no—
289C4—2841no—
290C4—1422no—
291C4#1455615612yescomplemented
292C4C2×D4×D7111yescharacteristic
293C4#1455617411yescharacteristic
294C2—42no—
295C2—2842no—
296C2—5682no—
297C2—5682no—
298C2D4×D28411yescharacteristic
299C2C2×#1455615611yescharacteristic
300C2#1798680411yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#436×C71C2
socleC22×C71C22×D4
hypercenter#436D71
hypocenterC71#3274
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#14556156C2semidirect
2. #17986670C2semidirect
3. D4×D284C2semidirect
4. #17986376C2semidirect
5. #17986252C2semidirect
6. #17986200C2semidirect
7. #17986195C2semidirect
8. #17986146C2semidirect
9. #17986135C2semidirect
10. C4×#11765854C2semidirect
11. #436×C71C2semidirect
12. C2×#11765865C22semidirect
17. C2×#11765854C22semidirect
19. C2×D568C22semidirect
20. C2×D568C22semidirect
22. #14556125C22semidirect
26. #14556102C22semidirect
29. C4×Dic142C22semidirect
32. #14556060C22semidirect
33. #14556053C22semidirect
35. #14556050C22semidirect
36. #14556050C22semidirect
37. C2×M16×C71C22semidirect
39. C4⋊C8×C71C22semidirect
40. C22⋊C8×C71C22semidirect
41. C4×C8×C71C22semidirect
63. #11765854D4semidirect
117. Dic142C2×D4semidirect
120. C8×C71C2×D4semidirect
137. C4×C71AGL(1,ℤ/8)semidirect
151. C71#3274semidirect
162. #436D71semidirect
172. C2×M16C2×D71semidirect
186. C4⋊C8C2×D71semidirect
191. C22⋊C8C2×D71semidirect
192. C4×C8C2×D71semidirect
271. C8D4×D71semidirect
291. C4#14556156semidirect