Finite Group Databasebeta

← #22599094

Subgroups — #22599094

Order 9088.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D71C211yesmaximal, characteristic, complemented
2C2×#14556099C211yesmaximal, characteristic
3#17986330C211yesmaximal, characteristic, complemented
4#17986327C211yesmaximal, characteristic, complemented
5#17986316C212yesmaximal, complemented
6#17986227C212yesmaximal, complemented
7#17986225C212yesmaximal, complemented
8#17986150C211yesmaximal, characteristic, complemented
9#17986136C212yesmaximal, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#450×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic, complemented
13C22×C22⋊D71—21no—
14C22×C4×D71C2211yescharacteristic
15C2×#11765853C2212yescomplemented
16C2×C8×D71C2212yescomplemented
17#14556108C2211yescharacteristic, complemented
18#14556107C2211yescharacteristic
19#14556106—21no—
20#14556105—21no—
21#14556104C2211yescharacteristic, complemented
22#14556099—21no—
23#14556099—21no—
24#14556099C2212yes—
25#14556099C2212yes—
26C2×C4⋊Dic71C2211yescharacteristic
27C4⋊C4×D71—21no—
28C4⋊C4×D71C2211yescharacteristic, complemented
29C2×C22⋊Dic71—21no—
30C22⋊C4×D71—21no—
31#14556060C2212yescomplemented
32#14556059C2212yescomplemented
33#14556053C2212yescomplemented
34#14556052C2212yescomplemented
35#14556049C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yescomplemented
38C22×D4×C71C2211yescharacteristic
39C2×SD16×C71C2212yescomplemented
40C22⋊Q8×C71C2211yescharacteristic
41#102×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44C2×Dic142—21no—
45C2×Dic142C2311yescharacteristic
46Q8×D71—42no—
47C2×C22⋊D71—22no—
48C2×C22⋊D71—22no—
49C2×C22⋊D71—44no—
50C22×Dic71—21no—
51C22×Dic71D411yescharacteristic
52C2×C4×D71—21no—
53C2×C4×D71—21no—
54C2×C4×D71—42no—
55C2×C4×D71D411yescharacteristic
56C2×C4×D71D411yescharacteristic
57C2×C4×D71C2311yescharacteristic
58C2×C4×D71C2311yescharacteristic
59#11765853—42no—
60C2×C71⋊C8C2312yes—
61C8×D71—42no—
62C4⋊Dic71—21no—
63C4⋊Dic71—21no—
64C4⋊Dic71C2311yescharacteristic
65C4⋊Dic71C2312yes—
66Dic71⋊C4—21no—
67Dic71⋊C4—21no—
68Dic71⋊C4—21no—
69C22⋊Dic71—21no—
70C22⋊Dic71—22no—
71C22⋊Dic71—22no—
72(C2×D71)⋊C4—21no—
73(C2×D71)⋊C4—21no—
74C24×C71—21no—
75C2×Q8×C71C2311yescharacteristic
76C2×D4×C71—21no—
77C2×D4×C71—42no—
78C2×D4×C71C2312yes—
79C22×C4×C71C2311yescharacteristic
80SD16×C71—42no—
81C2×C8×C71C2312yes—
82C4⋊C4×C71—21no—
83C4⋊C4×C71C2311yescharacteristic
84C22⋊C4×C71—21no—
85C23×C71—22no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71—42no—
89C23×C71C2×D411yescharacteristic
90C22×D71—41no—
91C22×D71—41no—
92C22×D71SD1614yescomplemented
93C22×D71C2×D411yescharacteristic
94C22×D71C2×D411yescharacteristic
95Q8×C71—22no—
96D4×C71—22no—
97D4×C71—22no—
98D4×C71—42no—
99C2×C4×C71—22no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2411yescharacteristic
104C22⋊D71—44no—
105C22⋊D71—44no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71—22no—
109C2×Dic71—41no—
110C2×Dic71—42no—
111C2×Dic71C2×D411yescharacteristic
112C2×Dic71C2×D411yescharacteristic
113C4×D71—22no—
114C4×D71—22no—
115C4×D71—41no—
116C4×D71—41no—
117C4×D71—42no—
118C4×D71—42no—
119Dic142—22no—
120Dic142—42no—
121C8×C71—22no—
122C71⋊C8—22no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—42no—
126C22×C71—42no—
127C22×C71—42no—
128C22×C71—84no—
129C22×C71C2×SD1612yes—
130C22×C71C22×D411yescharacteristic
131C2×D71—21no—
132C2×D71—21no—
133C2×D71—21no—
134C2×D71—21no—
135C2×D71—44no—
136C2×D71C2×SD1612yes—
137C2×D71C2×SD1612yes—
138C4×C71—21no—
139C4×C71—41no—
140C4×C71—41no—
141C4×C71H(5,𝔽2)12yes—
142Dic71—21no—
143Dic71—21no—
144Dic71—41no—
145Dic71—42no—
146Dic71—41no—
147C2×C71—22no—
148C2×C71—82no—
149C2×C71#53411yescharacteristic
150C2×C71C22×SD1611yescharacteristic
151C2×C71#57511yescharacteristic
152D71—22no—
153D71—22no—
154#3187—711nomaximal, Sylow
155C71#318711yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—711no—
157C2×#121—711no—
158#483—711no—
159#483—711no—
160#481—712no—
161#465—712no—
162#465—712no—
163#450—711no—
164#450D7111yescharacteristic, complemented
165#448—712no—
166C2×C22⋊C8—711no—
167C22×Q8—711no—
168C22×D4—1421no—
169C22×D4C2×D7111yescharacteristic
170C23×C4—711no—
171C2×SD16—712no—
172C2×SD16C2×D7112yescomplemented
173C22×C8—712no—
174C22⋊Q8—711no—
175C22⋊Q8—711no—
176C22⋊Q8—711no—
177C22⋊Q8—1421no—
178C22⋊Q8—1421no—
179C22⋊Q8C2×D7111yescharacteristic
180#121—712no—
181#121—712no—
182#121—1421no—
183#121—1421no—
184C2×C4⋊C4—711no—
185C2×C4⋊C4—711no—
186C2×C4⋊C4—1421no—
187C2×C22⋊C4—1421no—
188C2×C22⋊C4—1421no—
189#106—712no—
190#106—712no—
191#103—712no—
192#103—712no—
193#102—712no—
194#102C2×D7112yescomplemented
195C22⋊C8—711no—
196C22⋊C8—712no—
197C22⋊C8C2×D7111yescharacteristic
198C24—21no—
199C24—711no—
200C2×Q8—2842no—
201C2×Q8—711no—
202C2×Q8—1421no—
203C2×Q8C22×D7111yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—2844no—
207C2×D4—1422no—
208C2×D4—1422no—
209C2×D4C22×D7112yes—
210C22×C4—2842no—
211C22×C4—711no—
212C22×C4—711no—
213C22×C4—711no—
214C22×C4—711no—
215C22×C4—711no—
216C22×C4—1421no—
217C22×C4—1421no—
218C22×C4—1421no—
219C22×C4C22×D7111yescharacteristic
220SD16—42no—
221SD16—2842no—
222C2×C8—2842no—
223C2×C8—712no—
224C2×C8C22×D7112yes—
225C4⋊C4—21no—
226C4⋊C4—711no—
227C4⋊C4—712no—
228C4⋊C4—1421no—
229C4⋊C4—1421no—
230C4⋊C4—1421no—
231C4⋊C4—1421no—
232C4⋊C4—1421no—
233C4⋊C4C22×D7111yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—1421no—
236C22⋊C4—1421no—
237C22⋊C4—1422no—
238C22⋊C4—1422no—
239C22⋊C4—1421no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—2841no—
245C23—2841no—
246C23—714no—
247C23—711no—
248C23—711no—
249C23D4×D7111yescharacteristic
250Q8—22no—
251Q8—2842no—
252Q8—1422no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2844no—
257D4—2844no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—2841no—
262C2×C4—2841no—
263C2×C4—2842no—
264C2×C4—2842no—
265C2×C4—2842no—
266C2×C4—2841no—
267C2×C4—711no—
268C2×C4—711no—
269C2×C4—1421no—
270C2×C4—1422no—
271C2×C4—1421no—
272C2×C4—1422no—
273C2×C4—1422no—
274C2×C4D4×D7111yescharacteristic
275C2×C4C23×D7111yescharacteristic
276C8—22no—
277C8—1422no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—2844no—
285C22—712no—
286C22—712no—
287C22—1421no—
288C22—1421no—
289C22—1421no—
290C22—1421no—
291C22SD16×D7112yes—
292C22C2×D4×D7111yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—2842no—
297C4—2841no—
298C4—2841no—
299C4—1421no—
300C4—1421no—
301C4#1455617312yes—
302C2—22no—
303C2—82no—
304C2—1422no—
305C2—1422no—
306C2#1798655911yescharacteristic
307C2C2×SD16×D7111yescharacteristic
308C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3187
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. C22⋊Q8×D71C2semidirect
3. #17986330C2semidirect
4. #17986327C2semidirect
5. #17986316C2semidirect
6. #17986227C2semidirect
7. #17986225C2semidirect
8. #17986150C2semidirect
9. #17986136C2semidirect
10. C22⋊C8×D71C2semidirect
11. #450×C71C2semidirect
12. C2×Q8×D71C22semidirect
15. C2×#11765853C22semidirect
16. C2×C8×D71C22semidirect
17. #14556108C22semidirect
21. #14556104C22semidirect
28. C4⋊C4×D71C22semidirect
31. #14556060C22semidirect
32. #14556059C22semidirect
33. #14556053C22semidirect
34. #14556052C22semidirect
35. #14556049C22semidirect
37. (C2×D71)⋊C8C22semidirect
39. C2×SD16×C71C22semidirect
41. #102×C71C22semidirect
92. C22×D71SD16semidirect
155. C71#3187semidirect
164. #450D71semidirect
172. C2×SD16C2×D71semidirect
194. #102C2×D71semidirect