Finite Group Databasebeta

← #22591698

Subgroups — #22591698

Order 9088.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×#14556105C212yesmaximal, complemented
3C2×(C22⋊D71)⋊C4C211yesmaximal, characteristic
4#17985920C211yesmaximal, characteristic, complemented
5#17985918C211yesmaximal, characteristic, complemented
6#17985911C211yesmaximal, characteristic, complemented
7#17985903C212yesmaximal, complemented
8#17985875C212yesmaximal
9#17985873C212yesmaximal, complemented
10#17985858C211yesmaximal, characteristic, complemented
11#393×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×Dic142C2212yes—
14C2×Q8×D71—28no—
15C2×Q8×D71—24no—
16C22×C22⋊D71C2211yescharacteristic
17C23×Dic71C2211yescharacteristic
18C22×C4×D71C2212yes—
19C22×C4×D71C2211yescharacteristic
20#14556105—28no—
21(C22⋊D71)⋊C4—24no—
22C2×C4⋊Dic71C2212yes—
23C2×Dic71⋊C4C2212yes—
24C2×Dic71⋊C4C2212yes—
25C2×Dic71⋊C4C2211yescharacteristic
26C2×C22⋊Dic71C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2211yescharacteristic
29C2×C4×Dic71C2211yescharacteristic
30C71⋊#95C2211yescharacteristic
31C71⋊#95C2212yes—
32C71⋊#95C2212yes—
33C71⋊#95C2212yes—
34C22×Q8×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2212yes—
36C2×C22⋊C4×C71C2211yescharacteristic
37#95×C71C2211yescharacteristic
38#95×C71C2212yes—
39C23×D71C2311yescharacteristic
40C2×Dic142—22no—
41C2×Dic142—24no—
42C2×Dic142—24no—
43C2×Dic142—24no—
44C2×Dic142D418yescomplemented
45Q8×D71—232no—
46C2×C22⋊D71—22no—
47C2×C22⋊D71—22no—
48C2×C22⋊D71Q814yescomplemented
49C22×Dic71—22no—
50C22×Dic71—24no—
51C22×Dic71C2311yescharacteristic
52C22×Dic71C2312yes—
53C22×Dic71C2312yes—
54C22×Dic71C2311yescharacteristic
55C22×Dic71C2311yescharacteristic
56C2×C4×D71—24no—
57C2×C4×D71—24no—
58C2×C4×D71—24no—
59C2×C4×D71—24no—
60C2×C4×D71D414yes—
61C4⋊Dic71—24no—
62Dic71⋊C4—22no—
63Dic71⋊C4—24no—
64Dic71⋊C4—24no—
65C22⋊Dic71—24no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—24no—
68C4×Dic71—22no—
69C24×C71C2311yescharacteristic
70C2×Q8×C71—24no—
71C2×Q8×C71—22no—
72C22×C4×C71C2311yescharacteristic
73C22×C4×C71C2312yes—
74C22×C4×C71C2312yes—
75C22×C4×C71C2311yescharacteristic
76C22⋊C4×C71—22no—
77C22⋊C4×C71—24no—
78C23×C71—22no—
79C23×C71—21no—
80C23×C71—21no—
81C23×C71—22no—
82C23×C71C2×Q812yescomplemented
83C23×C71C2411yescharacteristic
84C22×D71—22no—
85C22×D71—22no—
86C22×D71C2×D412yes—
87C22×D71C2×Q812yes—
88C22×D71#4112yes—
89Q8×C71—28no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71C2×D414yes—
99C2×C4×C71C2×D412yes—
100C22⋊D71—28no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—21no—
105C2×Dic71—21no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—44no—
111C2×Dic71C2×D414yes—
112C2×Dic71C2×D412yes—
113C2×Dic71C2×D414yes—
114C2×Dic71C2×Q812yescomplemented
115C4×D71—216no—
116C4×D71—28no—
117Dic142—28no—
118Dic142—216no—
119C22×C71—22no—
120C22×C71—24no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—42no—
124C22×C71C22×D411yescharacteristic
125C22×C71C22×D412yes—
126C22×C71C22×Q811yescharacteristic
127C22×C71C2×#4111yescharacteristic
128C22×C712−1+412yes—
129C2×D71—22no—
130C2×D71—22no—
131C2×D71—24no—
132C2×D71—24no—
133C2×D71C22⋊Q814yescomplemented
134C4×C71—22no—
135C4×C71—24no—
136C4×C71—41no—
137C4×C71—42no—
138Dic71—24no—
139Dic71—22no—
140Dic71—41no—
141Dic71—42no—
142Dic71#12014yes—
143C2×C71—42no—
144C2×C71C2×#12011yescharacteristic
145C2×C71C2×C22⋊Q811yescharacteristic
146C2×C71#53611yescharacteristic
147C2×C71#54812yes—
148C2×C71D4×Q812yes—
149D71—24no—
150#2409—711nomaximal, Sylow
151C71#240911yescharacteristic, complemented, Sylow
152C23×Q8—711no—
153C2×C22⋊Q8—712no—
154C2×C4×D4—711no—
155#395—711no—
156#395—711no—
157#393—711no—
158#393D7111yescharacteristic, complemented
159#391—712no—
160#386—712no—
161#385—712no—
162#380—711no—
163C22×Q8—712no—
164C22×Q8—711no—
165C22×Q8—1424no—
166C22×Q8—1428no—
167C22×Q8C2×D7111yescharacteristic
168C22×D4—711no—
169C23×C4—711no—
170C23×C4—711no—
171C23×C4—712no—
172C22⋊Q8—1428no—
173C4×D4—1424no—
174C2×C4⋊C4—711no—
175C2×C4⋊C4—712no—
176C2×C4⋊C4—712no—
177C2×C4⋊C4—712no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4—711no—
180C2×C22⋊C4—712no—
181C2×C22⋊C4C2×D7111yescharacteristic
182C2×C22⋊C4C2×D7112yes—
183C2×C42—711no—
184#95—711no—
185#95—712no—
186#95—712no—
187#95—712no—
188#95C2×D7111yescharacteristic
189#95C2×D7112yes—
190C24—711no—
191C24C22×D7111yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—718no—
195C2×Q8—1424no—
196C2×Q8—1424no—
197C2×Q8—1424no—
198C2×Q8—14232no—
199C2×Q8—1422no—
200C2×D4—714no—
201C2×D4—1422no—
202C2×D4—1422no—
203C22×C4—711no—
204C22×C4—711no—
205C22×C4—712no—
206C22×C4—714no—
207C22×C4—712no—
208C22×C4—711no—
209C22×C4—1424no—
210C22×C4—1424no—
211C22×C4—1424no—
212C22×C4—1422no—
213C22×C4—1424no—
214C22×C4—1424no—
215C22×C4C22×D7111yescharacteristic
216C22×C4C22×D7112yes—
217C22×C4C22×D7111yescharacteristic
218C22×C4C22×D7112yes—
219C4⋊C4—1424no—
220C4⋊C4—1424no—
221C4⋊C4—1422no—
222C4⋊C4—1424no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1424no—
226C22⋊C4—1422no—
227C22⋊C4—1424no—
228C42—1422no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—712no—
234C23—712no—
235C23—712no—
236C23—1422no—
237C23—1422no—
238C23Q8×D7112yescomplemented
239C23C23×D7111yescharacteristic
240Q8—28no—
241Q8—14216no—
242Q8—1428no—
243D4—1428no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—2844no—
253C2×C4—712no—
254C2×C4—714no—
255C2×C4—714no—
256C2×C4—712no—
257C2×C4—1428no—
258C2×C4—14216no—
259C2×C4—1422no—
260C2×C4—1421no—
261C2×C4—1422no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1421no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4—1421no—
268C2×C4D4×D7112yes—
269C2×C4D4×D7114yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—714no—
276C22—1422no—
277C22—1424no—
278C22—1424no—
279C22—1422no—
280C22C2×D4×D7111yescharacteristic
281C22C2×D4×D7112yes—
282C22C2×Q8×D7111yescharacteristic
283C22#41×D7111yescharacteristic
284C22#1455617512yes—
285C4—22no—
286C4—24no—
287C4—41no—
288C4—42no—
289C4—2842no—
290C4—2841no—
291C4—714no—
292C4—1424no—
293C4—1422no—
294C2—42no—
295C2—1424no—
296C2#120×D7111yescharacteristic
297C2C22⋊Q8×D7111yescharacteristic
298C2#1798656911yescharacteristic
299C2#1798665712yes—
300C2#1798666212yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#393×C71C2
hypercenter#393D71
hypocenterC71#2409
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D71C2semidirect
2. C2×#14556105C2semidirect
4. #17985920C2semidirect
5. #17985918C2semidirect
6. #17985911C2semidirect
7. #17985903C2semidirect
9. #17985873C2semidirect
10. #17985858C2semidirect
11. #393×C71C2semidirect
44. C2×Dic142D4semidirect
48. C2×C22⋊D71Q8semidirect
82. C23×C71C2×Q8semidirect
114. C2×Dic71C2×Q8semidirect
133. C2×D71C22⋊Q8semidirect
151. C71#2409semidirect
158. #393D71semidirect
238. C23Q8×D71semidirect