Finite Group Databasebeta

← #22599120

Subgroups — #22599120

Order 9088.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986509C211yesmaximal, characteristic
2C2×#14556108C211yesmaximal, characteristic, complemented
3#17986342C212yesmaximal, complemented
4#17986329C211yesmaximal, characteristic, complemented
5#17986241C212yesmaximal, complemented
6#17986227C212yesmaximal, complemented
7#17986162C212yesmaximal, complemented
8#17986157C211yesmaximal, characteristic, complemented
9#17986137C211yesmaximal, characteristic, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#480×C71C211yesmaximal, characteristic, complemented
12C2×#11765864C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C2×Dic284C2212yescomplemented
16C2×#11765853C2212yescomplemented
17C2×C8×D71C2212yes—
18#14556110—21no—
19#14556108—21no—
20#14556108—21no—
21#14556108C2212yes—
22#14556108C2212yes—
23#14556108C2211yescharacteristic
24#14556107C2211yescharacteristic
25#14556099C2211yescharacteristic
26#14556096—21no—
27D4×Dic71—21no—
28(C22⋊D71)⋊C4—21no—
29#14556086C2211yescharacteristic
30C2×C4⋊Dic71—21no—
31C2×(C2×D71)⋊C4—21no—
32#14556059C2212yescomplemented
33#14556053C2212yescomplemented
34#14556051C2212yescomplemented
35#14556051C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yes—
38#121×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#107×C71C2212yescomplemented
41#102×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44#11765864—42no—
45C2×Dic142—21no—
46C2×Dic142—42no—
47C2×Dic142C2312yes—
48C2×Dic142C2311yescharacteristic
49C2×C22⋊D71—21no—
50C2×C22⋊D71—21no—
51C22×Dic71—21no—
52C22×Dic71—21no—
53C22×Dic71D411yescharacteristic
54C2×C4×D71—42no—
55C2×C4×D71D411yescharacteristic
56C2×C4×D71D411yescharacteristic
57C2×C4×D71C2311yescharacteristic
58C2×C4×D71C2311yescharacteristic
59Dic284—42no—
60#11765853—42no—
61C2×C71⋊C8C2312yes—
62C8×D71—42no—
63C4⋊Dic71—22no—
64C4⋊Dic71—22no—
65C4⋊Dic71—21no—
66C4⋊Dic71C2311yescharacteristic
67Dic71⋊C4—21no—
68C22⋊Dic71—21no—
69C22⋊Dic71—21no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4—22no—
72(C2×D71)⋊C4—21no—
73(C2×D71)⋊C4—21no—
74C4×Dic71—21no—
75C2×D4×C71—21no—
76C2×D4×C71C2311yescharacteristic
77C22×C4×C71—21no—
78C22×C4×C71C2311yescharacteristic
79C2×C8×C71C2312yes—
80C4⋊C4×C71—21no—
81C4⋊C4×C71C2312yes—
82C4⋊C4×C71C2311yescharacteristic
83C22⋊C4×C71—21no—
84C23×C71—21no—
85C23×C71C2×D411yescharacteristic
86C22×D71—22no—
87C22×D71—22no—
88C22×D71—41no—
89C22×D71C2×D411yescharacteristic
90C22×D71C2×D411yescharacteristic
91D4×C71—22no—
92D4×C71—42no—
93C2×C4×C71—22no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—42no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—42no—
100C22⋊D71—42no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—21no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71C2×D411yescharacteristic
108C2×Dic71C2×D411yescharacteristic
109C4×D71—22no—
110C4×D71—22no—
111C4×D71—41no—
112C4×D71—41no—
113Dic142—22no—
114Dic142—22no—
115Dic142—22no—
116Dic142—42no—
117C8×C71—22no—
118C71⋊C8—22no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71#13512yes—
125C22×C71C22×D411yescharacteristic
126C2×D71—21no—
127C2×D71—21no—
128C2×D71—21no—
129C2×D71—21no—
130C2×D71—21no—
131C2×D71—44no—
132C2×D71#13712yes—
133C4×C71—21no—
134C4×C71—42no—
135C4×C71—41no—
136C4×C71H(5,𝔽2)12yes—
137Dic71—21no—
138Dic71—21no—
139Dic71—42no—
140Dic71—41no—
141Dic71—41no—
142C2×C71—22no—
143C2×C71—81no—
144C2×C71#53411yescharacteristic
145C2×C71C2×#13511yescharacteristic
146C2×C71C2×#13711yescharacteristic
147D71—22no—
148D71—41no—
149#3190—711nomaximal, Sylow
150C71#319011yescharacteristic, complemented, Sylow
151#525—711no—
152C2×C22⋊Q8—711no—
153#485—712no—
154#483—711no—
155#480D7111yescharacteristic, complemented
156#467—712no—
157#465—712no—
158#452—712no—
159#451—711no—
160#448—711no—
161C2×C22⋊C8—711no—
162C2×#41—711no—
163C22×Q8—711no—
164C23×C4—711no—
165C2×Q16—712no—
166C2×SD16—712no—
167C22×C8—712no—
168#123—1421no—
169C22⋊Q8—711no—
170C22⋊Q8—712no—
171C22⋊Q8—712no—
172C22⋊Q8—711no—
173C22⋊Q8—1421no—
174C22⋊Q8—1421no—
175#121—711no—
176#121C2×D7111yescharacteristic
177#120—1421no—
178C4×D4—1421no—
179C4×D4—1421no—
180#117—711no—
181C2×C4⋊C4—1421no—
182C2×C4⋊C4C2×D7111yescharacteristic
183C2×C22⋊C4—1421no—
184#107C2×D7112yescomplemented
185#106—712no—
186#103—712no—
187#103—712no—
188#103—712no—
189#102C2×D7112yescomplemented
190C22⋊C8—711no—
191C22⋊C8—712no—
192C22⋊C8C2×D7111yescharacteristic
193C24—711no—
194#41—2842no—
195C2×Q8—2842no—
196C2×Q8—712no—
197C2×Q8—711no—
198C2×Q8—1421no—
199C2×D4—21no—
200C2×D4—1421no—
201C2×D4—1421no—
202C2×D4C22×D7111yescharacteristic
203C22×C4—21no—
204C22×C4—2842no—
205C22×C4—711no—
206C22×C4—711no—
207C22×C4—711no—
208C22×C4—711no—
209C22×C4—711no—
210C22×C4—1421no—
211C22×C4—1421no—
212C22×C4C22×D7111yescharacteristic
213Q16—2842no—
214SD16—2842no—
215C2×C8—2842no—
216C2×C8—712no—
217C2×C8C22×D7112yes—
218C4⋊C4—21no—
219C4⋊C4—711no—
220C4⋊C4—1422no—
221C4⋊C4—1422no—
222C4⋊C4—1421no—
223C4⋊C4—1421no—
224C4⋊C4C22×D7112yes—
225C4⋊C4C22×D7111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1422no—
228C22⋊C4—1422no—
229C22⋊C4—1421no—
230C22⋊C4—1421no—
231C22⋊C4—1421no—
232C22⋊C4—1421no—
233C42—1421no—
234C23—21no—
235C23—2841no—
236C23—711no—
237C23—711no—
238C23—1422no—
239C23—1422no—
240C23D4×D7111yescharacteristic
241Q8—2842no—
242Q8—1422no—
243Q8—1422no—
244Q8—1422no—
245D4—22no—
246D4—42no—
247D4—2842no—
248D4—2842no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—2841no—
254C2×C4—2841no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—2841no—
258C2×C4—711no—
259C2×C4—711no—
260C2×C4—1421no—
261C2×C4—1421no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4D4×D7111yescharacteristic
266C2×C4C23×D7111yescharacteristic
267C8—22no—
268C8—1422no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—2844no—
275C22—712no—
276C22—1421no—
277C22—1421no—
278C22—1421no—
279C22—1421no—
280C22—1421no—
281C22#1455614712yes—
282C22C2×D4×D7111yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—2841no—
287C4—2842no—
288C4—2841no—
289C4—1421no—
290C4—1421no—
291C4#1455617412yes—
292C2—22no—
293C2—81no—
294C2—2841no—
295C2—1422no—
296C2#1798656011yescharacteristic
297C2C2×#1455614711yescharacteristic
298C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#480×C71C2
socleC22×C71C22×D4
hypercenter#480D71
hypocenterC71#3190
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#14556108C2semidirect
3. #17986342C2semidirect
4. #17986329C2semidirect
5. #17986241C2semidirect
6. #17986227C2semidirect
7. #17986162C2semidirect
8. #17986157C2semidirect
9. #17986137C2semidirect
10. C22⋊C8×D71C2semidirect
11. #480×C71C2semidirect
15. C2×Dic284C22semidirect
16. C2×#11765853C22semidirect
32. #14556059C22semidirect
33. #14556053C22semidirect
34. #14556051C22semidirect
35. #14556051C22semidirect
40. #107×C71C22semidirect
41. #102×C71C22semidirect
150. C71#3190semidirect
155. #480D71semidirect
184. #107C2×D71semidirect
189. #102C2×D71semidirect