Finite Group Databasebeta

← #22599338

Subgroups — #22599338

Order 9088.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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
2#124×D71C211yesmaximal, characteristic
3#17986391C211yesmaximal, characteristic, complemented
4#17986359C211yesmaximal, characteristic, complemented
5#17986358C211yesmaximal, characteristic, complemented
6#17986223C212yesmaximal, complemented
7#17986219C212yesmaximal, complemented
8#17986153C212yesmaximal, complemented
9#17986148C212yesmaximal
10#17986054C211yesmaximal, characteristic, complemented
11#497×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765855C2212yescomplemented
15C2×#11765852C2212yes—
16#14556128C2211yescharacteristic, complemented
17#14556119—21no—
18#14556117C2211yescharacteristic
19#14556116C2211yescharacteristic
20#14556115C2211yescharacteristic
21#14556114—21no—
22#14556114—21no—
23#14556108—21no—
24#14556104C2212yes—
25Q8×Dic71C2212yescomplemented
26#14556093—21no—
27#14556086—21no—
28C22⋊C4×D71—21no—
29C22⋊C4×D71—21no—
30C42×D71C2211yescharacteristic
31Dic71⋊C8C2212yescomplemented
32#14556053C2212yescomplemented
33#14556052C2212yescomplemented
34#14556050C2212yes—
35#14556049C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556039C2211yescharacteristic, complemented
38C2×Q16×C71C2212yescomplemented
39C2×SD16×C71C2212yes—
40C4⋊Q8×C71C2211yescharacteristic, complemented
41#124×C71C2211yescharacteristic
42#97×C71C2211yescharacteristic, complemented
43C23×D71—21no—
44C2×Dic142—21no—
45C2×Dic142—21no—
46C2×Dic142C2311yescharacteristic
47Q8×D71—42no—
48C2×D284C2311yescharacteristic
49C2×C22⋊D71—21no—
50D4×D71—42no—
51C2×C4×D71—21no—
52C2×C4×D71D411yescharacteristic
53C2×C4×D71D411yescharacteristic
54C2×C4×D71C2311yescharacteristic
55#11765855—42no—
56#11765852—42no—
57C2×C71⋊C8C2312yes—
58C4⋊Dic71—21no—
59C4⋊Dic71—22no—
60C4⋊Dic71C2312yes—
61Dic71⋊C4—22no—
62C22⋊Dic71—21no—
63C22⋊Dic71—21no—
64(C2×D71)⋊C4—22no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67(C2×D71)⋊C4—21no—
68(C2×D71)⋊C4—21no—
69C4×Dic71—22no—
70C4×Dic71D411yescharacteristic
71C4×Dic71D411yescharacteristic
72C4×Dic71C2311yescharacteristic
73C2×Q8×C71C2311yescharacteristic
74C2×Q8×C71C2312yes—
75C2×D4×C71C2311yescharacteristic
76Q16×C71—42no—
77SD16×C71—42no—
78C2×C8×C71C2312yes—
79C4⋊C4×C71—21no—
80C4⋊C4×C71—21no—
81C22⋊C4×C71—21no—
82C22⋊C4×C71—21no—
83C42×C71C2311yescharacteristic
84C23×C71—21no—
85C22×D71—21no—
86C22×D71—42no—
87C22×D71—42no—
88C22×D71—42no—
89C22×D71C2×D411yescharacteristic
90Q8×C71—22no—
91Q8×C71—22no—
92Q8×C71—22no—
93D4×C71—22no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—42no—
100C2×Dic71—22no—
101C2×Dic71—21no—
102C2×Dic71C2×D411yescharacteristic
103C2×Dic71C2×D411yescharacteristic
104C2×Dic71C2×D411yescharacteristic
105D284—22no—
106C4×D71—22no—
107C4×D71—22no—
108C4×D71—41no—
109C4×D71—42no—
110Dic142—22no—
111Dic142—42no—
112Dic142—42no—
113C8×C71—22no—
114C71⋊C8—22no—
115C22×C71—41no—
116C22×C71—41no—
117C22×C71—41no—
118C22×C71C22×D411yescharacteristic
119C2×D71—21no—
120C2×D71—21no—
121C2×D71—41no—
122C2×D71—41no—
123C2×D71—41no—
124C2×D71—84no—
125C2×D71#13712yes—
126C4×C71—21no—
127C4×C71—21no—
128C4×C71—41no—
129C4×C71—42no—
130C4×C71H(5,𝔽2)12yes—
131Dic71—21no—
132Dic71—21no—
133Dic71—21no—
134Dic71—41no—
135Dic71—42no—
136C2×C71—81no—
137C2×C71#53511yescharacteristic
138C2×C71C2×#13711yescharacteristic
139C2×C71#57511yescharacteristic
140D71—22no—
141D71—81no—
142#3215—711nomaximal, Sylow
143C71#321511yescharacteristic, complemented, Sylow
144#533—711no—
145C2×#124—711no—
146#497—711no—
147#497D7111yescharacteristic, complemented
148#489—711no—
149#489—711no—
150#464—712no—
151#464—712no—
152#451—712no—
153#450—712no—
154#432—711no—
155C22×Q8—711no—
156C22×D4—711no—
157C2×Q16—712no—
158C2×Q16C2×D7112yescomplemented
159C2×SD16—712no—
160C2×SD16C2×D7112yes—
161C4⋊Q8—711no—
162C4⋊Q8C2×D7111yescharacteristic, complemented
163#125—1421no—
164#124—711no—
165#124—711no—
166#124—711no—
167#124—1421no—
168#124—1421no—
169#124C2×D7111yescharacteristic
170C22⋊Q8—712no—
171C22⋊Q8—1421no—
172C4×Q8—712no—
173C4×Q8—1421no—
174#117—1421no—
175C2×C22⋊C4—1421no—
176C2×C22⋊C4—1421no—
177C2×C42—711no—
178C4⋊C8—712no—
179#103—712no—
180#103—712no—
181#102—712no—
182#102—712no—
183C22⋊C8—712no—
184#97—711no—
185#97C2×D7111yescharacteristic, complemented
186C24—1421no—
187C2×Q8—2842no—
188C2×Q8—711no—
189C2×Q8—1421no—
190C2×Q8—1421no—
191C2×Q8C22×D7111yescharacteristic
192C2×Q8C22×D7112yes—
193C2×D4—2842no—
194C2×D4—711no—
195C2×D4—1421no—
196C2×D4C22×D7111yescharacteristic
197C22×C4—711no—
198C22×C4—711no—
199C22×C4—711no—
200C22×C4—1421no—
201Q16—42no—
202Q16—2842no—
203SD16—42no—
204SD16—2842no—
205C2×C8—712no—
206C2×C8C22×D7112yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—712no—
210C4⋊C4—1422no—
211C4⋊C4—1422no—
212C4⋊C4—1421no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1421no—
216C22⋊C4—1421no—
217C22⋊C4—1422no—
218C22⋊C4—1421no—
219C22⋊C4—1421no—
220C22⋊C4—1421no—
221C22⋊C4—1421no—
222C42—711no—
223C42—711no—
224C42—711no—
225C42—1422no—
226C42C22×D7111yescharacteristic
227C23—21no—
228C23—2842no—
229C23—2842no—
230C23—2842no—
231C23—711no—
232C23—1421no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—2842no—
237Q8—2842no—
238Q8—1422no—
239D4—22no—
240D4—2842no—
241D4—1422no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—2841no—
245C2×C4—2842no—
246C2×C4—711no—
247C2×C4—711no—
248C2×C4—711no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4—1421no—
252C2×C4—1422no—
253C2×C4D4×D7111yescharacteristic
254C2×C4D4×D7111yescharacteristic
255C2×C4C23×D7111yescharacteristic
256C8—22no—
257C8—1422no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—2841no—
262C22—2841no—
263C22—2841no—
264C22—5684no—
265C22—712no—
266C22—1421no—
267C22—1421no—
268C22C2×D4×D7111yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—2841no—
274C4—2842no—
275C4—1421no—
276C4—1421no—
277C4—1421no—
278C4#1455617312yes—
279C2—81no—
280C2—5681no—
281C2—1422no—
282C2#1798656211yescharacteristic
283C2#137×D7111yescharacteristic
284C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#497×C71C2
socleC22×C71C22×D4
hypercenter#497D71
hypocenterC71#3215
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986391C2semidirect
4. #17986359C2semidirect
5. #17986358C2semidirect
6. #17986223C2semidirect
7. #17986219C2semidirect
8. #17986153C2semidirect
10. #17986054C2semidirect
11. #497×C71C2semidirect
14. C2×#11765855C22semidirect
16. #14556128C22semidirect
25. Q8×Dic71C22semidirect
31. Dic71⋊C8C22semidirect
32. #14556053C22semidirect
33. #14556052C22semidirect
37. #14556039C22semidirect
38. C2×Q16×C71C22semidirect
40. C4⋊Q8×C71C22semidirect
42. #97×C71C22semidirect
143. C71#3215semidirect
147. #497D71semidirect
158. C2×Q16C2×D71semidirect
162. C4⋊Q8C2×D71semidirect
185. #97C2×D71semidirect