Finite Group Databasebeta

← #22599943

Subgroups — #22599943

Order 9088.

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

normal subgroups only

This group has 546 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×#14556155C212yesmaximal, complemented
2#17986652C211yesmaximal, characteristic, complemented
3D4×D284C211yesmaximal, characteristic, complemented
4#17986373C211yesmaximal, characteristic, complemented
5#17986248C212yesmaximal, complemented
6#17986195C211yesmaximal, characteristic, complemented
7#17986148C212yesmaximal, complemented
8#17986147C212yesmaximal, complemented
9C4×#11765850C211yesmaximal, characteristic, complemented
10#17986076C211yesmaximal, characteristic, complemented
11#461×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yescomplemented
13C22×D284—21no—
14C2×D4×D71C2212yes—
15#14556155—28no—
16C2×#11765852C2212yescomplemented
17C2×#11765852C2212yescomplemented
18C2×D568C2212yescomplemented
19C2×#11765850C2211yescharacteristic
20C2×#11765848C2212yescomplemented
21#14556125C2211yescharacteristic
22#14556113—22no—
23#14556104C2212yescomplemented
24#14556103—21no—
25#14556103C2212yes—
26#14556102—21no—
27#14556098—22no—
28C4×D284C2211yescharacteristic
29C4×D284C2211yescharacteristic, complemented
30C4⋊C4×D71—21no—
31#14556061C2211yescharacteristic, complemented
32#14556050C2212yescomplemented
33#14556049C2211yescharacteristic
34#14556048C2211yescharacteristic, complemented
35(C2×D71)⋊C8C2212yescomplemented
36C4×C71⋊C8C2211yescharacteristic, complemented
37C2×SD16×C71C2212yes—
38C4⋊Q8×C71C2211yescharacteristic, complemented
39C4×D4×C71C2211yescharacteristic
40C4⋊C8×C71C2211yescharacteristic, complemented
41#103×C71C2212yescomplemented
42C23×D71—22no—
43#11765866—24no—
44#11765866—42no—
45C2×D284—22no—
46C2×D284—22no—
47C2×D284—21no—
48C2×D284—21no—
49C2×D284—44no—
50C2×D284D411yescharacteristic
51C2×D284C2312yes—
52C2×D284C2311yescharacteristic
53C2×C22⋊D71—22no—
54D4×D71—24no—
55D4×D71—42no—
56C2×C4×D71—21no—
57C2×C4×D71—22no—
58C2×C4×D71C2312yes—
59#11765852—24no—
60#11765852—42no—
61D568—24no—
62#11765850D414yescomplemented
63#11765848—24no—
64C2×C71⋊C8C2311yescharacteristic
65C2×C71⋊C8C2311yescharacteristic
66C4⋊Dic71D411yescharacteristic
67C4⋊Dic71C2311yescharacteristic
68Dic71⋊C4—22no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—22no—
72(C2×D71)⋊C4D412yes—
73C2×Q8×C71C2312yes—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76SD16×C71—24no—
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71—21no—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C42×C71C2311yescharacteristic
83C23×C71—21no—
84C22×D71—21no—
85C22×D71—22no—
86C22×D71—44no—
87C22×D71—44no—
88C22×D71—82no—
89C22×D71C2×D412yes—
90Q8×C71—22no—
91Q8×C71—22no—
92D4×C71—21no—
93D4×C71C2×D412yes—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—42no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2411yescharacteristic
100C22⋊D71—44no—
101C2×Dic71—21no—
102C2×Dic71C2×D412yes—
103D284—22no—
104D284—22no—
105D284—21no—
106D284—22no—
107D284—44no—
108D284—44no—
109D284—42no—
110D284—81no—
111D284C2×D412yescomplemented
112C4×D71—22no—
113C4×D71—22no—
114C4×D71—44no—
115C4×D71—42no—
116C8×C71—22no—
117C71⋊C8—21no—
118C71⋊C8C2×D412yescomplemented
119C22×C71—22no—
120C22×C71—41no—
121C22×C71—41no—
122C22×C71C22×D411yescharacteristic
123C2×D71—22no—
124C2×D71—22no—
125C2×D71—22no—
126C2×D71—22no—
127C2×D71—41no—
128C2×D71—41no—
129C2×D71—42no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—88no—
133C4×C71—21no—
134C4×C71—42no—
135C4×C71—41no—
136C4×C71AGL(1,ℤ/8)12yescomplemented
137C4×C71C22×D411yescharacteristic
138C4×C71H(5,𝔽2)11yescharacteristic
139Dic71—22no—
140Dic71—41no—
141C2×C71—42no—
142C2×C71D4211yescharacteristic
143C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C71#57711yescharacteristic
145D71—42no—
146D71—42no—
147D71—82no—
148#3272—711nomaximal, Sylow
149C71#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—712no—
151#547—711no—
152D42—711no—
153#492—711no—
154#468—712no—
155#461D7111yescharacteristic, complemented
156#459—711no—
157#450—712no—
158#449—712no—
159C4×D8—711no—
160#436—711no—
161C2×#41—712no—
162C22×D4—712no—
163C22×D4—1421no—
164AGL(1,ℤ/8)—1428no—
165C2×SD16—712no—
166C2×SD16—712no—
167C2×SD16C2×D7112yes—
168C2×D8—711no—
169C2×D8—712no—
170C2×M16—712no—
171C4⋊Q8C2×D7111yescharacteristic, complemented
172#127—711no—
173#123—1422no—
174C22⋊Q8—712no—
175#121—712no—
176#121—1421no—
177#121—1421no—
178#120—1422no—
179C4×D4—711no—
180C4×D4—711no—
181C4×D4C2×D7111yescharacteristic
182C2×C4⋊C4—1421no—
183#107—711no—
184C4⋊C8C2×D7111yescharacteristic, complemented
185#103C2×D7112yescomplemented
186#102—711no—
187#102—712no—
188#102—711no—
189C22⋊C8—712no—
190C4×C8—711no—
191C24—1422no—
192#41—2842no—
193#41—1424no—
194C2×Q8C22×D7112yes—
195C2×D4—2844no—
196C2×D4—2842no—
197C2×D4—711no—
198C2×D4—711no—
199C2×D4—712no—
200C2×D4—1422no—
201C2×D4—1422no—
202C2×D4—1422no—
203C2×D4—1424no—
204C2×D4—1421no—
205C2×D4—1421no—
206C2×D4C22×D7111yescharacteristic
207C22×C4—21no—
208C22×C4—712no—
209C22×C4—1422no—
210C22×C4—1421no—
211SD16—24no—
212SD16—2842no—
213SD16—1424no—
214D8—714no—
215D8—1424no—
216M16—1424no—
217C2×C8—711no—
218C2×C8—711no—
219C2×C8C22×D7112yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—711no—
223C4⋊C4—711no—
224C4⋊C4—1422no—
225C4⋊C4C22×D7111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—712no—
228C22⋊C4—1422no—
229C22⋊C4—1422no—
230C22⋊C4—1421no—
231C42C22×D7111yescharacteristic
232C23—21no—
233C23—2844no—
234C23—2844no—
235C23—5682no—
236C23—712no—
237C23—1422no—
238C23—1421no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—2844no—
243D4—2844no—
244D4—2844no—
245D4—2842no—
246D4—5681no—
247D4—712no—
248D4—1422no—
249D4—1422no—
250D4—1421no—
251D4—1422no—
252D4D4×D7112yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—2844no—
257C2×C4—2842no—
258C2×C4—712no—
259C2×C4—1421no—
260C2×C4—1422no—
261C2×C4—1422no—
262C2×C4D4×D7111yescharacteristic
263C2×C4D4×D7111yescharacteristic
264C2×C4C23×D7111yescharacteristic
265C8—22no—
266C8—712no—
267C8—1421no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—2841no—
272C22—2841no—
273C22—2842no—
274C22—2842no—
275C22—2842no—
276C22—5688no—
277C22—1422no—
278C22—1422no—
279C22—1422no—
280C22—1422no—
281C22C2×D4×D7111yescharacteristic
282C4—21no—
283C4—42no—
284C4—41no—
285C4—2841no—
286C4—1422no—
287C4#1455615512yescomplemented
288C4C2×D4×D7111yescharacteristic
289C4#1455617411yescharacteristic
290C2—42no—
291C2—2842no—
292C2—2842no—
293C2—5682no—
294C2#1798663311yescharacteristic
295C2C2×#1455615511yescharacteristic
296C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#461×C71C2
socleC22×C71C22×D4
hypercenter#461D71
hypocenterC71#3272
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#14556155C2semidirect
2. #17986652C2semidirect
3. D4×D284C2semidirect
4. #17986373C2semidirect
5. #17986248C2semidirect
6. #17986195C2semidirect
7. #17986148C2semidirect
8. #17986147C2semidirect
9. C4×#11765850C2semidirect
10. #17986076C2semidirect
11. #461×C71C2semidirect
12. C2×#11765866C22semidirect
16. C2×#11765852C22semidirect
17. C2×#11765852C22semidirect
18. C2×D568C22semidirect
20. C2×#11765848C22semidirect
23. #14556104C22semidirect
29. C4×D284C22semidirect
31. #14556061C22semidirect
32. #14556050C22semidirect
34. #14556048C22semidirect
35. (C2×D71)⋊C8C22semidirect
36. C4×C71⋊C8C22semidirect
38. C4⋊Q8×C71C22semidirect
40. C4⋊C8×C71C22semidirect
41. #103×C71C22semidirect
62. #11765850D4semidirect
111. D284C2×D4semidirect
118. C71⋊C8C2×D4semidirect
136. C4×C71AGL(1,ℤ/8)semidirect
149. C71#3272semidirect
155. #461D71semidirect
171. C4⋊Q8C2×D71semidirect
184. C4⋊C8C2×D71semidirect
185. #103C2×D71semidirect
287. C4#14556155semidirect