Finite Group Databasebeta

← #22599340

Subgroups — #22599340

Order 9088.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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#17986551C211yesmaximal, characteristic, complemented
2C2×#14556115C211yesmaximal, characteristic
3#17986390C212yesmaximal, complemented
4#17986357C212yesmaximal, complemented
5#17986220C212yesmaximal, complemented
6#17986219C212yesmaximal, complemented
7#17986157C211yesmaximal, characteristic, complemented
8#17986154C211yesmaximal, characteristic, complemented
9#17986150C211yesmaximal, characteristic
10#17986053C211yesmaximal, characteristic, complemented
11#450×C71C211yesmaximal, characteristic
12C22×Dic142C2211yescharacteristic
13C2×Dic284C2212yescomplemented
14C2×#11765855C2212yescomplemented
15C2×#11765853C2212yes—
16#14556127C2211yescharacteristic, complemented
17#14556126C2211yescharacteristic, complemented
18#14556118—21no—
19#14556115—21no—
20#14556115—21no—
21#14556115C2212yes—
22#14556115C2212yes—
23#14556107C2211yescharacteristic
24#14556105—21no—
25Q8×Dic71C2211yescharacteristic, complemented
26#14556093—21no—
27#14556093C2211yescharacteristic, complemented
28#14556084—21no—
29C2×C22⋊Dic71—21no—
30C2×C22⋊Dic71—21no—
31C2×C4×Dic71C2211yescharacteristic
32Dic71⋊C8C2212yescomplemented
33#14556053C2212yescomplemented
34#14556051C2212yescomplemented
35#14556049C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37#14556038C2212yescomplemented
38C22×D4×C71C2211yescharacteristic
39C2×SD16×C71C2212yes—
40C22⋊Q8×C71C2211yescharacteristic
41#102×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142—21no—
45C2×Dic142—42no—
46C2×Dic142C2312yes—
47C2×Dic142C2311yescharacteristic
48C22×Dic71—21no—
49C22×Dic71D411yescharacteristic
50C22×Dic71D411yescharacteristic
51Dic284—42no—
52#11765855—42no—
53#11765853—42no—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71—21no—
56C4⋊Dic71C2311yescharacteristic
57Dic71⋊C4—21no—
58Dic71⋊C4—21no—
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61Dic71⋊C4—21no—
62C22⋊Dic71—21no—
63C22⋊Dic71—22no—
64C22⋊Dic71—22no—
65C22⋊Dic71—22no—
66C22⋊Dic71—22no—
67C4×Dic71—21no—
68C4×Dic71—21no—
69C4×Dic71D411yescharacteristic
70C4×Dic71D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C4×Dic71C2311yescharacteristic
73C24×C71—21no—
74C2×Q8×C71C2311yescharacteristic
75C2×D4×C71—21no—
76C2×D4×C71—42no—
77C2×D4×C71C2312yes—
78C22×C4×C71C2311yescharacteristic
79SD16×C71—42no—
80C2×C8×C71C2312yes—
81C4⋊C4×C71—21no—
82C4⋊C4×C71C2311yescharacteristic
83C22⋊C4×C71—21no—
84C23×C71—22no—
85C23×C71—42no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71C2×D411yescharacteristic
89Q8×C71—22no—
90D4×C71—22no—
91D4×C71—22no—
92D4×C71—42no—
93C2×C4×C71—22no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2411yescharacteristic
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—41no—
103C2×Dic71—42no—
104C2×Dic71C2×D411yescharacteristic
105C2×Dic71C2×D411yescharacteristic
106C2×Dic71C2×D411yescharacteristic
107C2×Dic71C2×D411yescharacteristic
108Dic142—22no—
109Dic142—22no—
110Dic142—22no—
111Dic142—42no—
112Dic142—42no—
113C8×C71—22no—
114C71⋊C8—22no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71—42no—
120C22×C71—84no—
121C22×C71#13712yes—
122C22×C71C22×D411yescharacteristic
123C4×C71—21no—
124C4×C71—41no—
125C4×C71—41no—
126C4×C71H(5,𝔽2)12yes—
127Dic71—21no—
128Dic71—21no—
129Dic71—21no—
130Dic71—21no—
131Dic71—42no—
132Dic71—41no—
133Dic71—41no—
134C2×C71—22no—
135C2×C71—82no—
136C2×C71#53511yescharacteristic
137C2×C71C2×#13711yescharacteristic
138C2×C71#57511yescharacteristic
139#3215—711nomaximal, Sylow
140C71#321511yescharacteristic, complemented, Sylow
141#533—711no—
142C2×#124—711no—
143#497—712no—
144#489—712no—
145#464—712no—
146#464—712no—
147#451—711no—
148#451—711no—
149#450—711no—
150#450D7111yescharacteristic
151#432—711no—
152C22×Q8—711no—
153C22×D4C2×D7111yescharacteristic
154C2×Q16—712no—
155C2×Q16—712no—
156C2×SD16—712no—
157C2×SD16C2×D7112yes—
158C4⋊Q8—711no—
159C4⋊Q8—711no—
160#125—1421no—
161#124—712no—
162#124—712no—
163#124—1421no—
164#124—1421no—
165C22⋊Q8—711no—
166C22⋊Q8—1421no—
167C22⋊Q8C2×D7111yescharacteristic
168C4×Q8—711no—
169C4×Q8—711no—
170C4×Q8—1421no—
171#117—1421no—
172C2×C22⋊C4—1421no—
173C2×C22⋊C4—1421no—
174C2×C42—711no—
175C4⋊C8—712no—
176#103—712no—
177#103—712no—
178#102—712no—
179#102C2×D7112yes—
180C22⋊C8—711no—
181C22⋊C8C2×D7111yescharacteristic
182#97—712no—
183C24—21no—
184C2×Q8—2842no—
185C2×Q8—712no—
186C2×Q8—711no—
187C2×Q8—1421no—
188C2×Q8—1421no—
189C2×Q8C22×D7111yescharacteristic
190C2×D4—21no—
191C2×D4—42no—
192C2×D4C22×D7112yes—
193C22×C4—711no—
194C22×C4—711no—
195C22×C4—1421no—
196C22×C4C22×D7111yescharacteristic
197Q16—2842no—
198Q16—2842no—
199SD16—42no—
200SD16—2842no—
201C2×C8—712no—
202C2×C8C22×D7112yes—
203C4⋊C4—21no—
204C4⋊C4—711no—
205C4⋊C4—1421no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C4⋊C4—1421no—
209C4⋊C4—1421no—
210C4⋊C4—1421no—
211C4⋊C4C22×D7111yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—1422no—
214C22⋊C4—1422no—
215C22⋊C4—1422no—
216C22⋊C4—1422no—
217C22⋊C4—1421no—
218C42—711no—
219C42—711no—
220C42—711no—
221C42—711no—
222C42—1421no—
223C42—1421no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23D4×D7111yescharacteristic
229Q8—22no—
230Q8—2842no—
231Q8—2842no—
232Q8—1422no—
233Q8—1422no—
234Q8—1422no—
235D4—22no—
236D4—22no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—2842no—
242C2×C4—2841no—
243C2×C4—711no—
244C2×C4—711no—
245C2×C4—711no—
246C2×C4—711no—
247C2×C4—1421no—
248C2×C4—1421no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4D4×D7111yescharacteristic
252C2×C4C23×D7111yescharacteristic
253C8—22no—
254C8—1422no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—84no—
261C22#1455615712yes—
262C22C2×D4×D7111yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—2842no—
267C4—2841no—
268C4—2841no—
269C4—1421no—
270C4—1421no—
271C4—1421no—
272C4—1421no—
273C4#1455617312yes—
274C2—22no—
275C2—82no—
276C2#1798656411yescharacteristic
277C2C2×#1455615711yescharacteristic
278C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3215
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #17986551C2semidirect
3. #17986390C2semidirect
4. #17986357C2semidirect
5. #17986220C2semidirect
6. #17986219C2semidirect
7. #17986157C2semidirect
8. #17986154C2semidirect
10. #17986053C2semidirect
13. C2×Dic284C22semidirect
14. C2×#11765855C22semidirect
16. #14556127C22semidirect
17. #14556126C22semidirect
25. Q8×Dic71C22semidirect
27. #14556093C22semidirect
32. Dic71⋊C8C22semidirect
33. #14556053C22semidirect
34. #14556051C22semidirect
37. #14556038C22semidirect
140. C71#3215semidirect