Finite Group Databasebeta

← #22599936

Subgroups — #22599936

Order 9088.

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

normal subgroups only

This group has 546 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×#14556154C212yesmaximal, complemented
2#17986654C211yesmaximal, characteristic, complemented
3#17986371C211yesmaximal, characteristic, complemented
4#17986243C212yesmaximal, complemented
5#17986204C211yesmaximal, characteristic, complemented
6#17986191C211yesmaximal, characteristic, complemented
7#17986150C212yesmaximal, complemented
8#17986142C212yesmaximal, complemented
9C4×#11765850C211yesmaximal, characteristic, complemented
10#17986077C211yesmaximal, characteristic, complemented
11D42×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2212yescomplemented
13#14556154—28no—
14C2×#11765853C2212yes—
15C2×#11765853C2212yescomplemented
16C2×#11765853C2212yescomplemented
17C2×#11765850C2211yescharacteristic
18C2×#11765850C2212yescomplemented
19C2×#11765849C2212yescomplemented
20#14556128C2211yescharacteristic, complemented
21#14556110—22no—
22#14556107C2212yescomplemented
23#14556099—21no—
24C4×D284C2211yescharacteristic, complemented
25C4⋊C4×D71—21no—
26#14556061C2211yescharacteristic, complemented
27C4⋊(C71⋊C8)C2211yescharacteristic, complemented
28#14556051C2212yescomplemented
29#14556049C2211yescharacteristic
30#14556049C2212yescomplemented
31#14556048C2211yescharacteristic, complemented
32C22⋊(C71⋊C8)C2212yescomplemented
33C4×C71⋊C8C2211yescharacteristic, complemented
34C22×D4×C71—21no—
35C22×D4×C71C2212yes—
36#127×C71C2211yescharacteristic
37#121×C71—21no—
38#121×C71C2212yes—
39#120×C71—22no—
40C4×D4×C71C2211yescharacteristic
41C4×D4×C71C2211yescharacteristic
42#11765865—24no—
43#11765865—42no—
44C2×Dic142C2312yes—
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—22no—
47C2×C4×D71—22no—
48C2×C4×D71—21no—
49#11765853—24no—
50#11765853—24no—
51#11765853—42no—
52#11765850—24no—
53#11765850D414yescomplemented
54#11765849—24no—
55C2×C71⋊C8C2311yescharacteristic
56C2×C71⋊C8C2312yes—
57C2×C71⋊C8C2311yescharacteristic
58C4⋊Dic71—21no—
59C4⋊Dic71—21no—
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—22no—
62C22⋊Dic71—22no—
63(C2×D71)⋊C4—21no—
64C24×C71—22no—
65C2×D4×C71—24no—
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—21no—
69C2×D4×C71—21no—
70C2×D4×C71—42no—
71C2×D4×C71—44no—
72C2×D4×C71D411yescharacteristic
73C2×D4×C71C2312yes—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76C22×C4×C71C2312yes—
77C4⋊C4×C71D411yescharacteristic
78C4⋊C4×C71C2311yescharacteristic
79C22⋊C4×C71—21no—
80C22⋊C4×C71—22no—
81C22⋊C4×C71D412yes—
82C42×C71C2311yescharacteristic
83C23×C71—21no—
84C23×C71—22no—
85C23×C71—44no—
86C23×C71—44no—
87C23×C71—82no—
88C23×C71C2×D412yes—
89C22×D71—21no—
90D4×C71—22no—
91D4×C71—22no—
92D4×C71—21no—
93D4×C71—22no—
94D4×C71—44no—
95D4×C71—44no—
96D4×C71—42no—
97D4×C71—81no—
98D4×C71C2×D412yes—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—21no—
102C2×C4×C71—42no—
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2×D412yes—
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2411yescharacteristic
107C22⋊D71—44no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110D284—21no—
111D284C2×D412yescomplemented
112C4×D71—44no—
113C4×D71—42no—
114Dic142—22no—
115Dic142—22no—
116C71⋊C8—22no—
117C71⋊C8—21no—
118C71⋊C8C2×D412yescomplemented
119C22×C71—22no—
120C22×C71—22no—
121C22×C71—22no—
122C22×C71—22no—
123C22×C71—41no—
124C22×C71—41no—
125C22×C71—42no—
126C22×C71—42no—
127C22×C71—42no—
128C22×C71—88no—
129C22×C71C22×D411yescharacteristic
130C2×D71—22no—
131C2×D71—41no—
132C2×D71—41no—
133C4×C71—22no—
134C4×C71—21no—
135C4×C71—41no—
136C4×C71AGL(1,ℤ/8)12yescomplemented
137C4×C71C22×D411yescharacteristic
138C4×C71H(5,𝔽2)11yescharacteristic
139Dic71—42no—
140Dic71—41no—
141C2×C71—42no—
142C2×C71—42no—
143C2×C71—82no—
144C2×C71D4211yescharacteristic
145C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C71#57711yescharacteristic
147D71—42no—
148#3272—711nomaximal, Sylow
149C71#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—712no—
151#547—711no—
152D42D7111yescharacteristic, complemented
153#492—711no—
154#468—712no—
155#461—711no—
156#459—711no—
157#450—712no—
158#449—712no—
159C4×D8—711no—
160#436—711no—
161C2×#41—712no—
162C22×D4—21no—
163C22×D4C2×D7112yes—
164AGL(1,ℤ/8)—1428no—
165C2×SD16—712no—
166C2×SD16—712no—
167C2×SD16—712no—
168C2×D8—711no—
169C2×D8—712no—
170C2×M16—712no—
171C4⋊Q8—711no—
172#127C2×D7111yescharacteristic
173#123—1422no—
174C22⋊Q8—712no—
175#121—21no—
176#121—1421no—
177#121C2×D7112yes—
178#120—22no—
179C4×D4—711no—
180C4×D4C2×D7111yescharacteristic
181C4×D4C2×D7111yescharacteristic
182C2×C4⋊C4—1421no—
183#107—711no—
184C4⋊C8—711no—
185#103—712no—
186#102—711no—
187#102—712no—
188#102—711no—
189C22⋊C8—712no—
190C4×C8—711no—
191C24—22no—
192#41—2842no—
193#41—1424no—
194C2×Q8—712no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—711no—
203C2×D4—1422no—
204C2×D4C22⋊D7111yescharacteristic
205C2×D4C22×D7112yes—
206C2×D4C22×D7111yescharacteristic
207C22×C4—21no—
208C22×C4—1421no—
209C22×C4—1422no—
210C22×C4C22×D7112yes—
211SD16—2842no—
212SD16—1424no—
213SD16—1424no—
214D8—714no—
215D8—1424no—
216M16—1424no—
217C2×C8—711no—
218C2×C8—711no—
219C2×C8—712no—
220C4⋊C4—711no—
221C4⋊C4—1422no—
222C4⋊C4—1421no—
223C4⋊C4—1421no—
224C4⋊C4C22⋊D7111yescharacteristic
225C4⋊C4C22×D7111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—1422no—
229C22⋊C4—1421no—
230C22⋊C4C22⋊D7112yes—
231C42C22×D7111yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—1421no—
238C23C2×C22⋊D7112yes—
239Q8—1422no—
240Q8—1422no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—2844no—
250D4—712no—
251D4—1421no—
252D4D4×D7112yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—2844no—
258C2×C4—2842no—
259C2×C4—1421no—
260C2×C4—1422no—
261C2×C4C2×C22⋊D7111yescharacteristic
262C2×C4C2×C22⋊D7112yes—
263C2×C4C2×C22⋊D7111yescharacteristic
264C2×C4C23×D7111yescharacteristic
265C8—712no—
266C8—1422no—
267C8—1421no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—2841no—
279C22—2841no—
280C22—1422no—
281C22C22×C22⋊D7111yescharacteristic
282C4—22no—
283C4—21no—
284C4—41no—
285C4—2842no—
286C4—2841no—
287C4#1455615412yescomplemented
288C4C2×D4×D7111yescharacteristic
289C4#1455617311yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—2842no—
294C2D4×C22⋊D7111yescharacteristic
295C2C2×#1455615411yescharacteristic
296C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingD42×C71C2
socleC22×C71C22×D4
hypercenterD42D71
hypocenterC71#3272
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#14556154C2semidirect
2. #17986654C2semidirect
3. #17986371C2semidirect
4. #17986243C2semidirect
5. #17986204C2semidirect
6. #17986191C2semidirect
7. #17986150C2semidirect
8. #17986142C2semidirect
9. C4×#11765850C2semidirect
10. #17986077C2semidirect
11. D42×C71C2semidirect
12. C2×#11765865C22semidirect
15. C2×#11765853C22semidirect
16. C2×#11765853C22semidirect
18. C2×#11765850C22semidirect
19. C2×#11765849C22semidirect
20. #14556128C22semidirect
22. #14556107C22semidirect
24. C4×D284C22semidirect
26. #14556061C22semidirect
27. C4⋊(C71⋊C8)C22semidirect
28. #14556051C22semidirect
30. #14556049C22semidirect
31. #14556048C22semidirect
32. C22⋊(C71⋊C8)C22semidirect
33. C4×C71⋊C8C22semidirect
53. #11765850D4semidirect
111. D284C2×D4semidirect
118. C71⋊C8C2×D4semidirect
136. C4×C71AGL(1,ℤ/8)semidirect
149. C71#3272semidirect
152. D42D71semidirect
287. C4#14556154semidirect