Finite Group Databasebeta

← #22598282

Subgroups — #22598282

Order 9088.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#14556103C211yesmaximal, characteristic
2#117×D71C211yesmaximal, characteristic, complemented
3#17986326C212yesmaximal, complemented
4#17986325C212yesmaximal, complemented
5#17986319C212yesmaximal, complemented
6#17986314C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic, complemented
8#17986005C211yesmaximal, characteristic, complemented
9#17986003C211yesmaximal, characteristic, complemented
10#17985962C211yesmaximal, characteristic, complemented
11#428×C71C211yesmaximal, characteristic, complemented
12C22×D284—21no—
13C22×D284C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C2×#11765849C2211yescharacteristic
16#14556103—21no—
17#14556103—21no—
18#14556103C2212yes—
19#14556103C2212yes—
20#14556087C2211yescharacteristic
21#14556086C2211yescharacteristic, complemented
22#14556086C2211yescharacteristic, complemented
23#14556085—21no—
24#14556084—21no—
25C4⋊C4×D71C2211yescharacteristic, complemented
26C4⋊C4×D71C2211yescharacteristic, complemented
27C2×(C2×D71)⋊C4—21no—
28C22⋊C4×D71—21no—
29C42×D71—21no—
30#14556061C2212yescomplemented
31#14556059C2212yescomplemented
32#14556050C2212yescomplemented
33#14556050C2212yescomplemented
34#14556048C2212yescomplemented
35#14556048C2212yescomplemented
36(C2×D71)⋊C8C2212yescomplemented
37(C2×D71)⋊C8C2212yescomplemented
38C2×M16×C71C2211yescharacteristic
39#117×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#107×C71C2212yescomplemented
42#106×C71C2212yescomplemented
43C23×D71—21no—
44C23×D71D411yescharacteristic
45C2×D284—22no—
46C2×D284—22no—
47C2×D284—22no—
48C2×D284—22no—
49C2×D284—21no—
50C2×D284—44no—
51C2×D284C2312yes—
52C22×Dic71D411yescharacteristic
53C2×C4×D71—22no—
54C2×C4×D71—22no—
55C2×C4×D71—21no—
56C2×C4×D71—21no—
57C2×C4×D71D411yescharacteristic
58C2×C4×D71D411yescharacteristic
59C2×C4×D71C2311yescharacteristic
60C2×C4×D71C2311yescharacteristic
61#11765849—22no—
62C2×C71⋊C8C2312yes—
63C4⋊Dic71C2311yescharacteristic
64C4⋊Dic71C2311yescharacteristic
65Dic71⋊C4—21no—
66Dic71⋊C4—21no—
67C22⋊Dic71—21no—
68(C2×D71)⋊C4—21no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4—22no—
72C4×Dic71—21no—
73C4×Dic71—21no—
74C4×Dic71—21no—
75C22×C4×C71—21no—
76C22×C4×C71C2311yescharacteristic
77M16×C71—22no—
78C2×C8×C71C2312yes—
79C4⋊C4×C71—21no—
80C4⋊C4×C71C2311yescharacteristic
81C4⋊C4×C71C2312yes—
82C4⋊C4×C71C2311yescharacteristic
83C22⋊C4×C71—21no—
84C42×C71—21no—
85C23×C71C2×D411yescharacteristic
86C22×D71—22no—
87C22×D71—22no—
88C22×D71—22no—
89C22×D71—41no—
90C22×D71—42no—
91C22×D71—42no—
92C22×D71—42no—
93C22×D71C2×D411yescharacteristic
94C22×D71C2×D411yescharacteristic
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—42no—
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71#4112yes—
101C2×C4×C71#4112yes—
102C2×C4×C71C2411yescharacteristic
103C2×Dic71—22no—
104C2×Dic71—21no—
105C2×Dic71—21no—
106C2×Dic71C2×D411yescharacteristic
107C2×Dic71C2×D411yescharacteristic
108D284—22no—
109D284—22no—
110D284—44no—
111D284—44no—
112D284—42no—
113C4×D71—22no—
114C4×D71—22no—
115C4×D71—21no—
116C4×D71—21no—
117C4×D71—21no—
118C4×D71—21no—
119C4×D71—42no—
120C4×D71—42no—
121C8×C71—22no—
122C71⋊C8—22no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71#12312yes—
126C22×C71C22×D411yescharacteristic
127C2×D71—21no—
128C2×D71—21no—
129C2×D71—21no—
130C2×D71—21no—
131C2×D71—21no—
132C2×D71—44no—
133C2×D71—42no—
134C2×D71—42no—
135C2×D71—42no—
136C2×D71—84no—
137C2×D71AGL(1,ℤ/8)12yes—
138C4×C71—41no—
139C4×C71—42no—
140C4×C71—41no—
141C4×C71#12312yes—
142C4×C71C2×#4111yescharacteristic
143C4×C71C2×#4111yescharacteristic
144Dic71—21no—
145Dic71—21no—
146Dic71—41no—
147Dic71—41no—
148C2×C71—22no—
149C2×C71C2×#12311yescharacteristic
150C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C71#57511yescharacteristic
152D71—22no—
153D71—41no—
154D71—82no—
155#3094—711nomaximal, Sylow
156C71#309411yescharacteristic, complemented, Sylow
157C2×#121—711no—
158C2×#117—711no—
159#482—712no—
160#482—712no—
161#481—712no—
162#480—712no—
163#428—711no—
164#428D7111yescharacteristic, complemented
165#418—711no—
166#418—711no—
167#407—711no—
168C22×D4—711no—
169C22×D4—1421no—
170C23×C4—711no—
171C2×M16—711no—
172C2×M16C2×D7111yescharacteristic
173#121—712no—
174#121—712no—
175#121—1421no—
176#121—1421no—
177#117—711no—
178#117—711no—
179#117—711no—
180#117—1421no—
181#117—1421no—
182#117C2×D7111yescharacteristic
183C2×C4⋊C4—711no—
184C2×C4⋊C4—711no—
185C2×C4⋊C4C2×D7111yescharacteristic
186C2×C22⋊C4—1421no—
187C2×C22⋊C4—1421no—
188C2×C42—1421no—
189#107—712no—
190#107C2×D7112yescomplemented
191#106—712no—
192#106C2×D7112yescomplemented
193#102—712no—
194#102—712no—
195#102—712no—
196#102—712no—
197C22⋊C8—712no—
198C22⋊C8—712no—
199C24—711no—
200C24—1421no—
201C2×D4—2844no—
202C2×D4—712no—
203C2×D4—1422no—
204C2×D4—1422no—
205C2×D4—1421no—
206C2×D4—1422no—
207C2×D4—1422no—
208C22×C4—21no—
209C22×C4—711no—
210C22×C4—711no—
211C22×C4—711no—
212C22×C4—711no—
213C22×C4—711no—
214C22×C4—1421no—
215C22×C4—1421no—
216C22×C4—1422no—
217C22×C4—1422no—
218C22×C4C22×D7111yescharacteristic
219M16—22no—
220M16—1422no—
221C2×C8—712no—
222C2×C8C22×D7112yes—
223C4⋊C4—21no—
224C4⋊C4—711no—
225C4⋊C4—711no—
226C4⋊C4—1421no—
227C4⋊C4—1421no—
228C4⋊C4C22×D7111yescharacteristic
229C4⋊C4C22×D7111yescharacteristic
230C4⋊C4C22×D7112yes—
231C22⋊C4—21no—
232C22⋊C4—1422no—
233C22⋊C4—1422no—
234C22⋊C4—1421no—
235C22⋊C4—1421no—
236C22⋊C4—1421no—
237C42—21no—
238C42—1421no—
239C42—1421no—
240C42—1421no—
241C23—2841no—
242C23—2842no—
243C23—2842no—
244C23—2842no—
245C23—711no—
246C23—711no—
247C23—1422no—
248C23—1422no—
249C23—1422no—
250C23D4×D7111yescharacteristic
251D4—2844no—
252D4—2844no—
253D4—2842no—
254D4—1422no—
255D4—1422no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—2842no—
261C2×C4—2842no—
262C2×C4—711no—
263C2×C4—711no—
264C2×C4—1421no—
265C2×C4—1421no—
266C2×C4—1422no—
267C2×C4—1421no—
268C2×C4—1421no—
269C2×C4—1421no—
270C2×C4—1421no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4D4×D7111yescharacteristic
274C2×C4#1176586512yes—
275C2×C4#1176586612yes—
276C2×C4C23×D7111yescharacteristic
277C8—22no—
278C8—1422no—
279C22—21no—
280C22—21no—
281C22—2844no—
282C22—2842no—
283C22—2842no—
284C22—2842no—
285C22—5684no—
286C22—712no—
287C22—1421no—
288C22—1421no—
289C22—1421no—
290C22—1421no—
291C22—1421no—
292C22#1455611312yes—
293C22C2×D4×D7111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—2841no—
298C4—2841no—
299C4—1421no—
300C4—1421no—
301C4#1455611312yes—
302C4C2×#1176586511yescharacteristic
303C4C2×#1176586611yescharacteristic
304C2—22no—
305C2—2841no—
306C2—5682no—
307C2—1422no—
308C2C2×#1455611311yescharacteristic
309C2AGL(1,ℤ/8)×D7111yescharacteristic
310C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3094
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D71C2semidirect
3. #17986326C2semidirect
4. #17986325C2semidirect
5. #17986319C2semidirect
6. #17986314C2semidirect
7. #17986039C2semidirect
8. #17986005C2semidirect
9. #17986003C2semidirect
10. #17985962C2semidirect
11. #428×C71C2semidirect
21. #14556086C22semidirect
22. #14556086C22semidirect
25. C4⋊C4×D71C22semidirect
26. C4⋊C4×D71C22semidirect
30. #14556061C22semidirect
31. #14556059C22semidirect
32. #14556050C22semidirect
33. #14556050C22semidirect
34. #14556048C22semidirect
35. #14556048C22semidirect
36. (C2×D71)⋊C8C22semidirect
37. (C2×D71)⋊C8C22semidirect
41. #107×C71C22semidirect
42. #106×C71C22semidirect
156. C71#3094semidirect
164. #428D71semidirect
190. #107C2×D71semidirect
192. #106C2×D71semidirect