Finite Group Databasebeta

← #22598277

Subgroups — #22598277

Order 9088.

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

normal subgroups only

This group has 462 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
1#121×D71C211yesmaximal, characteristic
2C2×#14556086C211yesmaximal, characteristic, complemented
3#17986325C212yesmaximal, complemented
4#17986321C212yesmaximal, complemented
5#17986317C211yesmaximal, characteristic, complemented
6#17986313C211yesmaximal, characteristic, complemented
7#17986310C211yesmaximal, characteristic, complemented
8#17986038C212yesmaximal, complemented
9#17986002C212yesmaximal, complemented
10#17985961C211yesmaximal, characteristic, complemented
11#481×C71C211yesmaximal, characteristic, complemented
12C2×D4×D71—21no—
13C2×D4×D71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C2×#11765848C2212yes—
16#14556103C2211yescharacteristic
17#14556102C2211yescharacteristic
18#14556101—21no—
19#14556100—21no—
20#14556099C2211yescharacteristic
21#14556086—21no—
22#14556086—21no—
23#14556086C2212yes—
24#14556086C2212yescomplemented
25C2×C4⋊Dic71C2211yescharacteristic, complemented
26C4⋊C4×D71C2211yescharacteristic
27C2×(C2×D71)⋊C4—21no—
28C22⋊C4×D71—21no—
29C2×C4×Dic71—21no—
30#14556062C2212yescomplemented
31#14556061C2212yescomplemented
32#14556059C2212yescomplemented
33#14556050C2212yescomplemented
34#14556049C2212yescomplemented
35#14556048C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic, complemented
37(C2×D71)⋊C8C2212yescomplemented
38#121×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic, complemented
40#106×C71C2212yescomplemented
41#102×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic, complemented
43C23×D71—21no—
44C23×D71D411yescharacteristic
45C2×D284—21no—
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C2×C22⋊D71—21no—
49C2×C22⋊D71—21no—
50D4×D71—24no—
51D4×D71—44no—
52C22×Dic71—21no—
53C22×Dic71D411yescharacteristic
54C2×C4×D71—24no—
55C2×C4×D71—21no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59C2×C4×D71C2311yescharacteristic
60#11765848—24no—
61C2×C71⋊C8C2312yes—
62C4⋊Dic71—21no—
63C4⋊Dic71C2312yes—
64C4⋊Dic71C2311yescharacteristic
65Dic71⋊C4—21no—
66C22⋊Dic71—21no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—21no—
71C4×Dic71—22no—
72C4×Dic71—22no—
73C2×D4×C71—21no—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76C22×C4×C71C2311yescharacteristic
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71C2312yes—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C23×C71—21no—
83C23×C71C2×D411yescharacteristic
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—21no—
87C22×D71—41no—
88C22×D71—42no—
89C22×D71—42no—
90C22×D71—42no—
91C22×D71C2×D411yescharacteristic
92C22×D71C2×D411yescharacteristic
93D4×C71—22no—
94D4×C71—42no—
95C2×C4×C71—22no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—42no—
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2411yescharacteristic
101C22⋊D71—42no—
102C22⋊D71—42no—
103C22⋊D71—42no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—21no—
107C2×Dic71—42no—
108C2×Dic71C2×D411yescharacteristic
109C2×Dic71C2×D411yescharacteristic
110D284—22no—
111D284—42no—
112C4×D71—22no—
113C4×D71—22no—
114C4×D71—22no—
115C4×D71—42no—
116C4×D71#4114yes—
117C8×C71—22no—
118C71⋊C8—22no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71AGL(1,ℤ/8)12yes—
125C22×C71C22×D411yescharacteristic
126C2×D71—21no—
127C2×D71—21no—
128C2×D71—21no—
129C2×D71—21no—
130C2×D71—21no—
131C2×D71—44no—
132C2×D71—41no—
133C2×D71—41no—
134C2×D71—41no—
135C2×D71—84no—
136C2×D71#12312yes—
137C4×C71—21no—
138C4×C71—41no—
139C4×C71—42no—
140C4×C71C2×#4112yes—
141Dic71—21no—
142Dic71—41no—
143Dic71—42no—
144Dic71#12312yes—
145C2×C71—22no—
146C2×C71—81no—
147C2×C71C2×#12311yescharacteristic
148C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C71#57511yescharacteristic
150D71—22no—
151D71—41no—
152D71—81no—
153#3094—711nomaximal, Sylow
154C71#309411yescharacteristic, complemented, Sylow
155C2×#121—711no—
156C2×#117—711no—
157#482—712no—
158#482—712no—
159#481—711no—
160#481D7111yescharacteristic, complemented
161#480—711no—
162#480—711no—
163#428—712no—
164#418—712no—
165#407—711no—
166C22×D4—711no—
167C22×D4—1421no—
168C23×C4—711no—
169C2×M16—712no—
170#121—711no—
171#121—711no—
172#121—711no—
173#121—1421no—
174#121—1421no—
175#121C2×D7111yescharacteristic
176#117—712no—
177#117—712no—
178#117—1421no—
179#117—1421no—
180C2×C4⋊C4—711no—
181C2×C4⋊C4—711no—
182C2×C4⋊C4C2×D7111yescharacteristic, complemented
183C2×C22⋊C4—1421no—
184C2×C22⋊C4—1421no—
185C2×C42—1421no—
186#107—712no—
187#107—712no—
188#106—712no—
189#106C2×D7112yescomplemented
190#102—712no—
191#102—712no—
192#102—712no—
193#102C2×D7112yescomplemented
194C22⋊C8—711no—
195C22⋊C8—712no—
196C22⋊C8C2×D7111yescharacteristic, complemented
197C24—711no—
198C24—1421no—
199C2×D4—21no—
200C2×D4—2844no—
201C2×D4—711no—
202C2×D4—1421no—
203C2×D4—1424no—
204C2×D4—1421no—
205C2×D4—1421no—
206C2×D4—1421no—
207C2×D4C22×D7111yescharacteristic
208C22×C4—21no—
209C22×C4—711no—
210C22×C4—711no—
211C22×C4—711no—
212C22×C4—711no—
213C22×C4—711no—
214C22×C4—1421no—
215C22×C4—1424no—
216C22×C4—1421no—
217C22×C4C22×D7111yescharacteristic
218M16—1424no—
219C2×C8—712no—
220C2×C8C22×D7112yes—
221C4⋊C4—21no—
222C4⋊C4—712no—
223C4⋊C4—711no—
224C4⋊C4—1421no—
225C4⋊C4—1421no—
226C4⋊C4C22×D7112yes—
227C4⋊C4C22×D7111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1421no—
230C22⋊C4—1422no—
231C22⋊C4—1422no—
232C22⋊C4—1421no—
233C22⋊C4—1421no—
234C42—1422no—
235C42—1422no—
236C23—21no—
237C23—2841no—
238C23—2842no—
239C23—2842no—
240C23—2842no—
241C23—711no—
242C23—711no—
243C23—1422no—
244C23—1421no—
245C23—1422no—
246C23D4×D7111yescharacteristic
247D4—22no—
248D4—42no—
249D4—2842no—
250D4—2842no—
251D4—2842no—
252D4—2842no—
253D4—1422no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—2842no—
259C2×C4—2842no—
260C2×C4—711no—
261C2×C4—714no—
262C2×C4—711no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1421no—
267C2×C4—1422no—
268C2×C4—1422no—
269C2×C4D4×D7111yescharacteristic
270C2×C4C23×D7111yescharacteristic
271C8—22no—
272C8—1422no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—2844no—
279C22—2841no—
280C22—2841no—
281C22—2841no—
282C22—5684no—
283C22—712no—
284C22—1421no—
285C22—1421no—
286C22—1421no—
287C22—1421no—
288C22—1421no—
289C22#1455615512yes—
290C22C2×D4×D7111yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—2841no—
295C4—2842no—
296C4—712no—
297C4—1421no—
298C4#41×D7112yes—
299C2—22no—
300C2—81no—
301C2—2841no—
302C2—5681no—
303C2—1422no—
304C2#123×D7111yescharacteristic
305C2C2×#1455615511yescharacteristic
306C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#481×C71C2
socleC22×C71C22×D4
hypercenter#481D71
hypocenterC71#3094
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#14556086C2semidirect
3. #17986325C2semidirect
4. #17986321C2semidirect
5. #17986317C2semidirect
6. #17986313C2semidirect
7. #17986310C2semidirect
8. #17986038C2semidirect
9. #17986002C2semidirect
10. #17985961C2semidirect
11. #481×C71C2semidirect
24. #14556086C22semidirect
25. C2×C4⋊Dic71C22semidirect
30. #14556062C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. #14556050C22semidirect
34. #14556049C22semidirect
35. #14556048C22semidirect
36. C22⋊(C71⋊C8)C22semidirect
37. (C2×D71)⋊C8C22semidirect
39. C2×C4⋊C4×C71C22semidirect
40. #106×C71C22semidirect
41. #102×C71C22semidirect
42. C22⋊C8×C71C22semidirect
154. C71#3094semidirect
160. #481D71semidirect
182. C2×C4⋊C4C2×D71semidirect
189. #106C2×D71semidirect
193. #102C2×D71semidirect
196. C22⋊C8C2×D71semidirect