Finite Group Databasebeta

← #22599280

Subgroups — #22599280

Order 9088.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986553C211yesmaximal, characteristic, complemented
2#127×D71C211yesmaximal, characteristic
3#17986370C212yesmaximal, complemented
4#17986344C212yesmaximal, complemented
5#17986200C211yesmaximal, characteristic, complemented
6#17986199C211yesmaximal, characteristic, complemented
7#17986196C211yesmaximal, characteristic, complemented
8#17986151C212yesmaximal, complemented
9#17986148C212yesmaximal, complemented
10#17986051C211yesmaximal, characteristic, complemented
11#460×C71C211yesmaximal, characteristic, complemented
12C2×D4×D71—21no—
13C2×D4×D71—21no—
14C2×D4×D71C2212yes—
15C2×#11765854C2212yescomplemented
16C2×#11765853C2212yescomplemented
17C2×#11765852C2212yescomplemented
18#14556127C2211yescharacteristic, complemented
19#14556126C2211yescharacteristic, complemented
20#14556125C2211yescharacteristic
21#14556124—21no—
22#14556124—21no—
23#14556124C2212yes—
24#14556118—21no—
25#14556108C2211yescharacteristic
26#14556106—21no—
27#14556104C2211yescharacteristic
28C4×Dic142—21no—
29C4×Dic142C2211yescharacteristic, complemented
30#14556085—21no—
31C42×D71C2211yescharacteristic
32C4⋊(C71⋊C8)C2211yescharacteristic, complemented
33#14556050C2212yescomplemented
34#14556049C2212yescomplemented
35#14556048C2212yescomplemented
36(C2×D71)⋊C8C2212yes—
37C8×Dic71C2212yescomplemented
38C2×SD16×C71C2212yescomplemented
39#127×C71C2211yescharacteristic
40C4×Q8×C71C2211yescharacteristic, complemented
41C4⋊C8×C71C2211yescharacteristic, complemented
42#102×C71C2212yescomplemented
43C23×D71—22no—
44C2×Dic142—21no—
45C2×Dic142C2311yescharacteristic
46C2×D284—21no—
47C2×D284—21no—
48C2×D284C2312yes—
49C2×C22⋊D71—22no—
50C2×C22⋊D71—22no—
51C2×C22⋊D71—22no—
52D4×D71—42no—
53D4×D71—42no—
54D4×D71—44no—
55D4×D71—44no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59#11765854—42no—
60#11765853—42no—
61#11765852—42no—
62C2×C71⋊C8C2312yes—
63C4⋊Dic71—21no—
64C4⋊Dic71C2311yescharacteristic
65Dic71⋊C4—21no—
66Dic71⋊C4—21no—
67Dic71⋊C4—21no—
68Dic71⋊C4—21no—
69Dic71⋊C4—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—21no—
73C4×Dic71D411yescharacteristic
74C4×Dic71D411yescharacteristic
75C4×Dic71C2311yescharacteristic
76C2×Q8×C71C2311yescharacteristic
77C2×D4×C71—21no—
78C2×D4×C71—21no—
79C2×D4×C71C2312yes—
80SD16×C71—42no—
81C2×C8×C71C2312yes—
82C4⋊C4×C71—21no—
83C4⋊C4×C71C2311yescharacteristic
84C42×C71—21no—
85C42×C71C2311yescharacteristic
86C23×C71—22no—
87C22×D71—22no—
88C22×D71—44no—
89C22×D71—44no—
90C22×D71—44no—
91C22×D71C2×D411yescharacteristic
92Q8×C71—21no—
93Q8×C71—21no—
94D4×C71—22no—
95D4×C71—22no—
96D4×C71—42no—
97D4×C71—42no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71C2×D411yescharacteristic
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71C2411yescharacteristic
103C22⋊D71—44no—
104C22⋊D71—44no—
105C22⋊D71—44no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71C2×D411yescharacteristic
109C2×Dic71C2×D411yescharacteristic
110C2×Dic71C2×D411yescharacteristic
111D284—22no—
112D284—22no—
113D284—42no—
114D284—42no—
115C4×D71—21no—
116C4×D71—21no—
117C4×D71—22no—
118C4×D71SD1614yescomplemented
119Dic142—21no—
120Dic142—21no—
121Dic142—42no—
122C8×C71—22no—
123C71⋊C8—22no—
124C22×C71—42no—
125C22×C71—42no—
126C22×C71—42no—
127C22×C71C22×D411yescharacteristic
128C2×D71—21no—
129C2×D71—21no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—88no—
134C2×D71C2×SD1612yes—
135C4×C71—21no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71C2×SD1612yes—
139C4×C71H(5,𝔽2)11yescharacteristic
140C4×C71H(5,𝔽2)11yescharacteristic
141Dic71—21no—
142Dic71—41no—
143Dic71—41no—
144Dic71C2×SD1612yes—
145Dic71AGL(1,ℤ/8)12yes—
146C2×C71—82no—
147C2×C71#53511yescharacteristic
148C2×C71C22×SD1611yescharacteristic
149C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
150D71—22no—
151D71—82no—
152#3208—711nomaximal, Sylow
153C71#320811yescharacteristic, complemented, Sylow
154#533—711no—
155C2×#127—711no—
156#492—712no—
157#486—712no—
158#460—711no—
159#460—711no—
160#460—711no—
161#460D7111yescharacteristic, complemented
162#450—712no—
163#450—712no—
164#431—711no—
165C22×D4—712no—
166C22×D4—1421no—
167C22×D4—1421no—
168C2×SD16—712no—
169C2×SD16—712no—
170C2×SD16—712no—
171C2×SD16C2×D7112yescomplemented
172C4⋊Q8—711no—
173C4⋊Q8—711no—
174#127—712no—
175#127—711no—
176#127—1421no—
177#127—1421no—
178#127C2×D7111yescharacteristic
179#125—1421no—
180C22⋊Q8—711no—
181C22⋊Q8—711no—
182C22⋊Q8—1421no—
183C4×Q8—711no—
184C4×Q8—1421no—
185C4×Q8C2×D7111yescharacteristic, complemented
186#117—1421no—
187C2×C42—711no—
188C4⋊C8—711no—
189C4⋊C8C2×D7111yescharacteristic, complemented
190#102—712no—
191#102—712no—
192#102—712no—
193#102C2×D7112yescomplemented
194C22⋊C8—712no—
195C4×C8—712no—
196C24—1422no—
197C2×Q8—711no—
198C2×Q8—1421no—
199C2×Q8C22×D7111yescharacteristic
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—2844no—
203C2×D4—2842no—
204C2×D4—2842no—
205C2×D4—2844no—
206C2×D4—712no—
207C2×D4—1422no—
208C2×D4—1422no—
209C2×D4—1422no—
210C2×D4—1421no—
211C2×D4—1421no—
212C2×D4C22×D7112yes—
213C22×C4—711no—
214C22×C4—711no—
215C22×C4—711no—
216SD16—42no—
217SD16—2842no—
218SD16—2842no—
219SD16—2842no—
220C2×C8—712no—
221C2×C8C22×D7112yes—
222C4⋊C4—21no—
223C4⋊C4—711no—
224C4⋊C4—1421no—
225C4⋊C4—1421no—
226C4⋊C4—1421no—
227C4⋊C4—1421no—
228C4⋊C4—1421no—
229C4⋊C4—1421no—
230C4⋊C4C22×D7111yescharacteristic
231C22⋊C4—1421no—
232C22⋊C4—1421no—
233C42—21no—
234C42—711no—
235C42—711no—
236C42—711no—
237C42—1421no—
238C42C22×D7111yescharacteristic
239C23—22no—
240C23—2844no—
241C23—2844no—
242C23—2844no—
243C23—711no—
244C23—1422no—
245Q8—21no—
246Q8—21no—
247Q8—2842no—
248Q8—1421no—
249Q8—1421no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—42no—
254D4—2842no—
255D4—2844no—
256D4—2844no—
257D4—2844no—
258D4—2842no—
259D4—1422no—
260D4—1422no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—711no—
264C2×C4—711no—
265C2×C4—714no—
266C2×C4—711no—
267C2×C4—1422no—
268C2×C4—1421no—
269C2×C4—1421no—
270C2×C4—1421no—
271C2×C4—1421no—
272C2×C4D4×D7111yescharacteristic
273C2×C4D4×D7111yescharacteristic
274C2×C4C23×D7111yescharacteristic
275C8—22no—
276C8—1422no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—2842no—
281C22—2842no—
282C22—2842no—
283C22—5688no—
284C22—712no—
285C22—1421no—
286C22—1421no—
287C22C2×D4×D7111yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—2841no—
292C4—2841no—
293C4—712no—
294C4—712no—
295C4—1421no—
296C4SD16×D7112yes—
297C4#1455617311yescharacteristic
298C4#1455617411yescharacteristic
299C2—82no—
300C2—5682no—
301C2—1422no—
302C2#1798656511yescharacteristic
303C2C2×SD16×D7111yescharacteristic
304C2AGL(1,ℤ/8)×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#460×C71C2
socleC22×C71C22×D4
hypercenter#460D71
hypocenterC71#3208
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #17986553C2semidirect
3. #17986370C2semidirect
4. #17986344C2semidirect
5. #17986200C2semidirect
6. #17986199C2semidirect
7. #17986196C2semidirect
8. #17986151C2semidirect
9. #17986148C2semidirect
10. #17986051C2semidirect
11. #460×C71C2semidirect
15. C2×#11765854C22semidirect
16. C2×#11765853C22semidirect
17. C2×#11765852C22semidirect
18. #14556127C22semidirect
19. #14556126C22semidirect
29. C4×Dic142C22semidirect
32. C4⋊(C71⋊C8)C22semidirect
33. #14556050C22semidirect
34. #14556049C22semidirect
35. #14556048C22semidirect
37. C8×Dic71C22semidirect
38. C2×SD16×C71C22semidirect
40. C4×Q8×C71C22semidirect
41. C4⋊C8×C71C22semidirect
42. #102×C71C22semidirect
118. C4×D71SD16semidirect
153. C71#3208semidirect
161. #460D71semidirect
171. C2×SD16C2×D71semidirect
185. C4×Q8C2×D71semidirect
189. C4⋊C8C2×D71semidirect
193. #102C2×D71semidirect