Finite Group Databasebeta

← #22598274

Subgroups — #22598274

Order 9088.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986511C211yesmaximal, characteristic
2C2×#14556086C211yesmaximal, characteristic, complemented
3#17986343C212yesmaximal, complemented
4#17986342C212yesmaximal, complemented
5#17986326C212yesmaximal, complemented
6#17986325C212yesmaximal, complemented
7#17986042C211yesmaximal, characteristic, complemented
8#17986001C211yesmaximal, characteristic, complemented
9#17985998C211yesmaximal, characteristic, complemented
10#17985962C211yesmaximal, characteristic, complemented
11#428×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C2×#11765849C2211yescharacteristic
15#14556112—21no—
16#14556108C2211yescharacteristic
17#14556108C2211yescharacteristic
18#14556103C2211yescharacteristic
19#14556103C2211yescharacteristic
20#14556098—21no—
21C4×D284—21no—
22C4×D284—21no—
23#14556086—21no—
24#14556086—21no—
25#14556086C2212yes—
26#14556086C2212yes—
27C2×C4⋊Dic71C2211yescharacteristic
28C2×(C2×D71)⋊C4—21no—
29C2×C4×Dic71—21no—
30#14556061C2212yescomplemented
31#14556059C2212yescomplemented
32#14556053C2212yescomplemented
33#14556051C2212yescomplemented
34#14556050C2212yescomplemented
35#14556048C2212yescomplemented
36(C2×D71)⋊C8C2212yes—
37(C2×D71)⋊C8C2212yes—
38C2×M16×C71C2211yescharacteristic
39#117×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#107×C71C2212yescomplemented
42#106×C71C2212yescomplemented
43C23×D71D411yescharacteristic
44#11765865—22no—
45#11765865—22no—
46C2×Dic142C2311yescharacteristic
47C2×D284—21no—
48C2×D284—21no—
49C2×D284C2311yescharacteristic
50C2×C22⋊D71—21no—
51C22×Dic71—21no—
52C22×Dic71D411yescharacteristic
53C2×C4×D71—22no—
54C2×C4×D71—22no—
55C2×C4×D71—21no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59C2×C4×D71C2311yescharacteristic
60#11765849—22no—
61C2×C71⋊C8C2312yes—
62C4⋊Dic71—21no—
63C4⋊Dic71—21no—
64C4⋊Dic71—21no—
65C4⋊Dic71C2312yes—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—21no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—22no—
73C4×Dic71—22no—
74C22×C4×C71—21no—
75C22×C4×C71C2311yescharacteristic
76M16×C71—22no—
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71C2311yescharacteristic
80C4⋊C4×C71C2312yes—
81C4⋊C4×C71C2311yescharacteristic
82C22⋊C4×C71—21no—
83C42×C71—21no—
84C23×C71C2×D411yescharacteristic
85C22×D71—22no—
86C22×D71—21no—
87C22×D71—41no—
88C22×D71—41no—
89C22×D71C2×D411yescharacteristic
90C22×D71C2×D411yescharacteristic
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—22no—
94C2×C4×C71—42no—
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71#4112yes—
97C2×C4×C71#4112yes—
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—42no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—42no—
104C2×Dic71C2×D411yescharacteristic
105C2×Dic71C2×D411yescharacteristic
106D284—21no—
107D284—21no—
108D284—42no—
109D284—42no—
110C4×D71—22no—
111C4×D71—22no—
112C4×D71—21no—
113C4×D71—21no—
114C4×D71—21no—
115C4×D71—21no—
116C4×D71—42no—
117Dic142—21no—
118Dic142—21no—
119C8×C71—22no—
120C71⋊C8—22no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71#12312yes—
124C22×C71C22×D411yescharacteristic
125C2×D71—21no—
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—
135C4×C71—41no—
136C4×C71—41no—
137C4×C71—42no—
138C4×C71#12312yes—
139C4×C71C2×#4111yescharacteristic
140C4×C71C2×#4111yescharacteristic
141Dic71—21no—
142Dic71—21no—
143Dic71—41no—
144Dic71—42no—
145C2×C71—22no—
146C2×C71C2×#12311yescharacteristic
147C2×C71#57511yescharacteristic
148C2×C71#57511yescharacteristic
149D71—41no—
150D71—41no—
151D71—81no—
152#3093—711nomaximal, Sylow
153C71#309311yescharacteristic, complemented, Sylow
154#525—711no—
155C2×#117—711no—
156#485—712no—
157#485—712no—
158#482—712no—
159#482—712no—
160#428—711no—
161#428D7111yescharacteristic, complemented
162#417—711no—
163#417—711no—
164#407—711no—
165C2×#41—711no—
166C23×C4—711no—
167C2×M16—711no—
168C2×M16C2×D7111yescharacteristic
169#123—1421no—
170C22⋊Q8—711no—
171C22⋊Q8—711no—
172#121—711no—
173#121—711no—
174#120—1421no—
175C4×D4—1421no—
176C4×D4—1421no—
177#117—712no—
178#117—712no—
179#117—1421no—
180#117—1421no—
181#117C2×D7111yescharacteristic
182C2×C4⋊C4—711no—
183C2×C4⋊C4C2×D7111yescharacteristic
184C2×C22⋊C4—1421no—
185C2×C42—1421no—
186#107—712no—
187#107C2×D7112yescomplemented
188#106—712no—
189#106C2×D7112yescomplemented
190#103—712no—
191#103—712no—
192#102—712no—
193#102—712no—
194C22⋊C8—712no—
195C22⋊C8—712no—
196C24—711no—
197#41—1422no—
198#41—1422no—
199C2×Q8—711no—
200C2×D4—711no—
201C2×D4—1421no—
202C2×D4—1421no—
203C2×D4—1421no—
204C22×C4—21no—
205C22×C4—711no—
206C22×C4—711no—
207C22×C4—711no—
208C22×C4—711no—
209C22×C4—711no—
210C22×C4—1421no—
211C22×C4—1421no—
212C22×C4—1422no—
213C22×C4—1422no—
214C22×C4C22×D7111yescharacteristic
215M16—22no—
216M16—1422no—
217C2×C8—712no—
218C2×C8C22×D7112yes—
219C4⋊C4—21no—
220C4⋊C4—712no—
221C4⋊C4—1421no—
222C4⋊C4—1421no—
223C4⋊C4—1421no—
224C4⋊C4C22×D7112yes—
225C4⋊C4C22×D7111yescharacteristic
226C4⋊C4C22×D7111yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—1421no—
229C22⋊C4—1421no—
230C22⋊C4—1422no—
231C22⋊C4—1422no—
232C22⋊C4—1421no—
233C22⋊C4—1421no—
234C42—21no—
235C42—1422no—
236C42—1422no—
237C23—2841no—
238C23—2841no—
239C23—711no—
240C23—711no—
241C23—1421no—
242C23—1422no—
243C23D4×D7111yescharacteristic
244Q8—1421no—
245Q8—1421no—
246D4—2842no—
247D4—2842no—
248D4—2842no—
249D4—1421no—
250D4—1421no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—711no—
258C2×C4—711no—
259C2×C4—1422no—
260C2×C4—1421no—
261C2×C4—1422no—
262C2×C4—1421no—
263C2×C4—1421no—
264C2×C4—1421no—
265C2×C4—1421no—
266C2×C4—1422no—
267C2×C4—1422no—
268C2×C4D4×D7111yescharacteristic
269C2×C4#1176586512yes—
270C2×C4#1176586612yes—
271C2×C4C23×D7111yescharacteristic
272C8—22no—
273C8—1422no—
274C22—21no—
275C22—21no—
276C22—2844no—
277C22—2841no—
278C22—2841no—
279C22—2841no—
280C22—1421no—
281C22—1421no—
282C22—1421no—
283C22—1421no—
284C22—1421no—
285C22—1421no—
286C22#1455611312yes—
287C22C2×D4×D7111yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—2841no—
292C4—2842no—
293C4—1421no—
294C4—1421no—
295C4#1455611312yes—
296C4C2×#1176586511yescharacteristic
297C4C2×#1176586611yescharacteristic
298C2—22no—
299C2—2841no—
300C2—2841no—
301C2—5681no—
302C2C2×#1455611311yescharacteristic
303C2#1798679511yescharacteristic
304C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3093
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. C2×#14556086C2semidirect
3. #17986343C2semidirect
4. #17986342C2semidirect
5. #17986326C2semidirect
6. #17986325C2semidirect
7. #17986042C2semidirect
8. #17986001C2semidirect
9. #17985998C2semidirect
10. #17985962C2semidirect
11. #428×C71C2semidirect
30. #14556061C22semidirect
31. #14556059C22semidirect
32. #14556053C22semidirect
33. #14556051C22semidirect
34. #14556050C22semidirect
35. #14556048C22semidirect
41. #107×C71C22semidirect
42. #106×C71C22semidirect
153. C71#3093semidirect
161. #428D71semidirect
187. #107C2×D71semidirect
189. #106C2×D71semidirect