Finite Group Databasebeta

← #22598138

Subgroups — #22598138

Order 9088.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#17986554C211yesmaximal, characteristic, complemented
2C4×D4×D71C211yesmaximal, characteristic
3#17986298C212yesmaximal, complemented
4#17986297C212yesmaximal, complemented
5#17986286C211yesmaximal, characteristic, complemented
6#17986280C211yesmaximal, characteristic, complemented
7#17986279C211yesmaximal, characteristic, complemented
8#17986038C212yesmaximal, complemented
9#17986021C211yesmaximal, characteristic, complemented
10#17986002C212yesmaximal
11#476×C71C211yesmaximal, characteristic, complemented
12C2×D4×D71C2211yescharacteristic
13C22×C4×D71—21no—
14C2×#11765848C2212yes—
15#14556128C2211yescharacteristic, complemented
16#14556119C2212yescomplemented
17#14556108—21no—
18#14556104—21no—
19Q8×Dic71—21no—
20#14556093—21no—
21C4×D284C2211yescharacteristic
22D4×Dic71C2211yescharacteristic
23#14556090C2211yescharacteristic
24C4×C22⋊D71—21no—
25(C22⋊D71)⋊C4—21no—
26#14556086C2212yes—
27C4⋊C4×D71C2211yescharacteristic
28C22⋊C4×D71—21no—
29C42×D71C2211yescharacteristic
30#14556062C2212yescomplemented
31#14556061C2212yescomplemented
32#14556059C2212yescomplemented
33C4⋊(C71⋊C8)C2211yescharacteristic, complemented
34Dic71⋊C8C2212yescomplemented
35#14556050C2212yes—
36#14556049C2212yes—
37#14556048C2212yes—
38C4⋊Q8×C71C2211yescharacteristic, complemented
39C4×D4×C71C2211yescharacteristic
40#106×C71C2212yescomplemented
41C4⋊C8×C71C2211yescharacteristic, complemented
42#102×C71C2212yes—
43C23×D71—21no—
44C2×Dic142—21no—
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—21no—
47D4×D71—22no—
48D4×D71Q814yescomplemented
49C22×Dic71—21no—
50C2×C4×D71—21no—
51C2×C4×D71—21no—
52C2×C4×D71—21no—
53C2×C4×D71—44no—
54C2×C4×D71D411yescharacteristic
55C2×C4×D71D411yescharacteristic
56C2×C4×D71C2311yescharacteristic
57#11765848—24no—
58C2×C71⋊C8C2312yes—
59C4⋊Dic71—21no—
60C4⋊Dic71—22no—
61C4⋊Dic71C2312yes—
62C4⋊Dic71C2311yescharacteristic
63Dic71⋊C4—21no—
64Dic71⋊C4—22no—
65C22⋊Dic71—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—21no—
68(C2×D71)⋊C4—21no—
69C4×Dic71—22no—
70C4×Dic71D411yescharacteristic
71C4×Dic71D411yescharacteristic
72C4×Dic71C2311yescharacteristic
73C2×Q8×C71—21no—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76C2×C8×C71C2312yes—
77C4⋊C4×C71—21no—
78C4⋊C4×C71C2312yes—
79C4⋊C4×C71C2311yescharacteristic
80C22⋊C4×C71—21no—
81C42×C71C2311yescharacteristic
82C23×C71—21no—
83C22×D71—21no—
84C22×D71—24no—
85C22×D71—42no—
86C22×D71—42no—
87C22×D71C2×D411yescharacteristic
88Q8×C71—42no—
89D4×C71—21no—
90D4×C71C2×Q812yes—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—42no—
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2411yescharacteristic
97C22⋊D71—24no—
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—42no—
101C2×Dic71C2×D411yescharacteristic
102C2×Dic71C2×D411yescharacteristic
103C2×Dic71C2×D411yescharacteristic
104D284—21no—
105D284C2×Q812yes—
106C4×D71—22no—
107C4×D71—41no—
108C4×D71—42no—
109C4×D71—42no—
110C4×D71—42no—
111C4×D71C2×Q812yes—
112C4×D71#4112yes—
113Dic142—42no—
114C8×C71—22no—
115C71⋊C8—22no—
116C22×C71—22no—
117C22×C71—41no—
118C22×C71—41no—
119C22×C71C22×D411yescharacteristic
120C2×D71—21no—
121C2×D71—21no—
122C2×D71—22no—
123C2×D71—48no—
124C2×D71—41no—
125C2×D71—41no—
126C2×D71C22⋊Q812yes—
127C4×C71—21no—
128C4×C71—41no—
129C4×C71—42no—
130C4×C71AGL(1,ℤ/8)12yes—
131C4×C71C22×Q811yescharacteristic
132C4×C71C2×#4111yescharacteristic
133Dic71—21no—
134Dic71—21no—
135Dic71—42no—
136Dic71—41no—
137Dic71C22⋊Q812yes—
138C2×C71—42no—
139C2×C71C2×C22⋊Q811yescharacteristic
140C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C71#57511yescharacteristic
142D71—22no—
143D71—42no—
144#3077—711nomaximal, Sylow
145C71#307711yescharacteristic, complemented, Sylow
146#533—711no—
147C2×C4×D4—711no—
148#478—712no—
149#478—712no—
150#476—711no—
151#476D7111yescharacteristic, complemented
152#474—711no—
153#474—711no—
154#428—712no—
155#423—711no—
156#418—712no—
157C22×D4—711no—
158C23×C4—1421no—
159C2×M16—712no—
160C4⋊Q8—711no—
161C4⋊Q8C2×D7111yescharacteristic, complemented
162#125—712no—
163C22⋊Q8—1421no—
164C22⋊Q8—1421no—
165C4×Q8—1421no—
166C4×Q8—1421no—
167C4×D4—711no—
168C4×D4—711no—
169C4×D4—711no—
170C4×D4—1421no—
171C4×D4—1421no—
172C4×D4C2×D7111yescharacteristic
173#117—712no—
174C2×C4⋊C4—711no—
175C2×C22⋊C4—1421no—
176C2×C42—711no—
177#107—712no—
178#107—712no—
179#106—712no—
180#106C2×D7112yescomplemented
181C4⋊C8—712no—
182C4⋊C8—711no—
183C4⋊C8C2×D7111yescharacteristic, complemented
184#102—712no—
185#102—712no—
186#102—712no—
187#102C2×D7112yes—
188C24—1421no—
189C2×Q8—21no—
190C2×Q8—1421no—
191C2×D4—714no—
192C2×D4—711no—
193C2×D4—1422no—
194C2×D4—1421no—
195C2×D4C22×D7111yescharacteristic
196C22×C4—21no—
197C22×C4—2844no—
198C22×C4—711no—
199C22×C4—711no—
200C22×C4—711no—
201C22×C4—1421no—
202C22×C4—1421no—
203C22×C4—1421no—
204C22×C4—1421no—
205M16—1424no—
206C2×C8—712no—
207C2×C8C22×D7112yes—
208C4⋊C4—21no—
209C4⋊C4—712no—
210C4⋊C4—711no—
211C4⋊C4—1421no—
212C4⋊C4—1422no—
213C4⋊C4—1422no—
214C4⋊C4—1421no—
215C4⋊C4C22×D7112yes—
216C4⋊C4C22×D7111yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1421no—
219C22⋊C4—1422no—
220C22⋊C4—1421no—
221C22⋊C4—1421no—
222C42—711no—
223C42—711no—
224C42—711no—
225C42—1422no—
226C42C22×D7111yescharacteristic
227C23—21no—
228C23—2842no—
229C23—2842no—
230C23—711no—
231C23—1424no—
232C23—1421no—
233Q8—42no—
234Q8—2842no—
235D4—21no—
236D4—712no—
237D4—1424no—
238D4—1421no—
239D4Q8×D7112yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—2841no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—711no—
249C2×C4—712no—
250C2×C4—712no—
251C2×C4—711no—
252C2×C4—711no—
253C2×C4—1422no—
254C2×C4—1421no—
255C2×C4—1422no—
256C2×C4D4×D7111yescharacteristic
257C2×C4D4×D7111yescharacteristic
258C2×C4C23×D7111yescharacteristic
259C8—22no—
260C8—1422no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—2848no—
265C22—2841no—
266C22—2841no—
267C22—712no—
268C22—1422no—
269C22—1421no—
270C22—1421no—
271C22C2×D4×D7111yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—2841no—
276C4—2842no—
277C4—712no—
278C4—1421no—
279C4—1421no—
280C4#1455615512yes—
281C4C2×Q8×D7111yescharacteristic
282C4#41×D7111yescharacteristic
283C2—42no—
284C2—2842no—
285C2—1422no—
286C2C22⋊Q8×D7111yescharacteristic
287C2C2×#1455615511yescharacteristic
288C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#476×C71C2
socleC22×C71C22×D4
hypercenter#476D71
hypocenterC71#3077
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986298C2semidirect
4. #17986297C2semidirect
5. #17986286C2semidirect
6. #17986280C2semidirect
7. #17986279C2semidirect
8. #17986038C2semidirect
9. #17986021C2semidirect
11. #476×C71C2semidirect
15. #14556128C22semidirect
16. #14556119C22semidirect
30. #14556062C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. C4⋊(C71⋊C8)C22semidirect
34. Dic71⋊C8C22semidirect
38. C4⋊Q8×C71C22semidirect
40. #106×C71C22semidirect
41. C4⋊C8×C71C22semidirect
48. D4×D71Q8semidirect
145. C71#3077semidirect
151. #476D71semidirect
161. C4⋊Q8C2×D71semidirect
180. #106C2×D71semidirect
183. C4⋊C8C2×D71semidirect