Finite Group Databasebeta

← #22592619

Subgroups — #22592619

Order 9088.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×(C2×D71)⋊C4C211yesmaximal, characteristic
2C2×C22⋊C4×D71C211yesmaximal, characteristic
3#17985928C212yesmaximal, complemented
4#17985926C212yesmaximal, complemented
5#17985874C211yesmaximal, characteristic, complemented
6#17985873C212yesmaximal, complemented
7#17985871C211yesmaximal, characteristic, complemented
8#17985862C211yesmaximal, characteristic, complemented
9#17985861C212yesmaximal, complemented
10#17985854C211yesmaximal, characteristic
11#381×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C23×Dic71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16C2×C4⋊Dic71C2212yescomplemented
17C2×Dic71⋊C4C2212yescomplemented
18C2×C22⋊Dic71C2211yescharacteristic
19C2×C22⋊Dic71C2211yescharacteristic
20C2×(C2×D71)⋊C4—24no—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2211yescharacteristic
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C22⋊C4×D71—24no—
28C71⋊#95C2212yescomplemented
29C71⋊#95C2212yescomplemented
30C71⋊#95C2212yescomplemented
31C71⋊#95C2212yescomplemented
32C23×C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2211yescharacteristic
35#95×C71C2212yescomplemented
36#95×C71C2212yescomplemented
37C23×D71—22no—
38C23×D71—22no—
39C23×D71—24no—
40C23×D71—42no—
41C23×D71D414yes—
42C23×D71C2311yescharacteristic
43C23×D71C2311yescharacteristic
44C22×Dic71—22no—
45C22×Dic71—42no—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2312yes—
49C22×Dic71C2312yes—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—24no—
53C2×C4×D71—42no—
54C4⋊Dic71—42no—
55Dic71⋊C4—42no—
56C22⋊Dic71—22no—
57C22⋊Dic71—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—24no—
63(C2×D71)⋊C4—42no—
64C24×C71C2311yescharacteristic
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2311yescharacteristic
70C22×C4×C71C2312yes—
71C22×C4×C71C2312yes—
72C22⋊C4×C71—22no—
73C22⋊C4×C71—22no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71C2×D412yes—
77C23×C71#4112yes—
78C23×C71#4112yes—
79C23×C71C2411yescharacteristic
80C22×D71—21no—
81C22×D71—21no—
82C22×D71—22no—
83C22×D71—28no—
84C22×D71—22no—
85C22×D71—21no—
86C22×D71—21no—
87C22×D71—22no—
88C22×D71—28no—
89C22×D71—44no—
90C22×D71—48no—
91C22×D71—48no—
92C22×D71C2×D412yes—
93C22×D71C2×D412yes—
94C22×D71#4114yes—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—44no—
106C2×Dic71—22no—
107C2×Dic71—21no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—22no—
114C2×Dic71—22no—
115C2×Dic71—22no—
116C2×Dic71—44no—
117C4×D71—44no—
118C4×D71—44no—
119C22×C71—22no—
120C22×C71—22no—
121C22×C71—24no—
122C22×C71—24no—
123C22×C71#12314yes—
124C22×C71C22×D411yescharacteristic
125C22×C71C2×#4112yes—
126C22×C71C2×#4111yescharacteristic
127C22×C71C2×#4111yescharacteristic
128C22×C71H(5,𝔽2)12yes—
129C2×D71—22no—
130C2×D71—22no—
131C2×D71—24no—
132C2×D71—24no—
133C2×D71—22no—
134C2×D71—22no—
135C2×D71—22no—
136C2×D71—416no—
137C2×D71—42no—
138C2×D71—42no—
139C2×D71#12314yes—
140C4×C71—41no—
141C4×C71—42no—
142C4×C71—41no—
143C4×C71—42no—
144Dic71—41no—
145Dic71—41no—
146Dic71—42no—
147Dic71—42no—
148C2×C71—24no—
149C2×C71C2×#12311yescharacteristic
150C2×C71C2×#12311yescharacteristic
151C2×C71#53411yescharacteristic
152C2×C71#55112yes—
153C2×C71#55112yes—
154D71—24no—
155D71—42no—
156#2483—711nomaximal, Sylow
157C71#248311yescharacteristic, complemented, Sylow
158C22×C22⋊C4—711no—
159C22×C22⋊C4—711no—
160#397—712no—
161#397—712no—
162#385—711no—
163#385—711no—
164#385—712no—
165#381—711no—
166#381—712no—
167#381D7111yescharacteristic, complemented
168#379—711no—
169C25—711no—
170C23×C4—711no—
171C23×C4—711no—
172C23×C4—711no—
173C23×C4C2×D7111yescharacteristic
174C2×C4⋊C4—712no—
175C2×C4⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C22⋊C4—711no—
178C2×C22⋊C4—711no—
179C2×C22⋊C4—712no—
180C2×C22⋊C4—711no—
181C2×C22⋊C4—712no—
182C2×C22⋊C4—711no—
183C2×C22⋊C4—712no—
184C2×C22⋊C4—1424no—
185C2×C22⋊C4—1424no—
186C2×C22⋊C4C2×D7111yescharacteristic
187C2×C22⋊C4C2×D7111yescharacteristic
188#95—712no—
189#95—712no—
190#95—712no—
191#95—712no—
192#95C2×D7112yescomplemented
193#95C2×D7112yescomplemented
194C24—2842no—
195C24—714no—
196C24—711no—
197C24—711no—
198C24—1424no—
199C24—1422no—
200C24—1422no—
201C24C22×D7111yescharacteristic
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—2842no—
206C22×C4—2842no—
207C22×C4—711no—
208C22×C4—712no—
209C22×C4—711no—
210C22×C4—712no—
211C22×C4—1422no—
212C22×C4—1422no—
213C22×C4—1424no—
214C22×C4—1422no—
215C22×C4C22×D7112yes—
216C22×C4C22×D7111yescharacteristic
217C22×C4C22×D7112yes—
218C22×C4C22×D7111yescharacteristic
219C4⋊C4—2842no—
220C4⋊C4—2842no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—2842no—
224C22⋊C4—1424no—
225C22⋊C4—1422no—
226C22⋊C4—1422no—
227C22⋊C4—1424no—
228C22⋊C4—1424no—
229C22⋊C4—1422no—
230C22⋊C4—1422no—
231C23—22no—
232C23—22no—
233C23—2848no—
234C23—2848no—
235C23—2844no—
236C23—712no—
237C23—714no—
238C23—712no—
239C23—1428no—
240C23—1422no—
241C23—1421no—
242C23—1421no—
243C23—1422no—
244C23—1428no—
245C23—1421no—
246C23—1421no—
247C23—1422no—
248C23D4×D7112yes—
249C23#1176586412yes—
250C23#1176586512yes—
251C23C23×D7111yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—2844no—
264C2×C4—2844no—
265C2×C4—2844no—
266C2×C4—1422no—
267C2×C4—1421no—
268C2×C4—1421no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4—1422no—
274C2×C4—1422no—
275C2×C4—1422no—
276C22—22no—
277C22—24no—
278C22—24no—
279C22—22no—
280C22—28416no—
281C22—2842no—
282C22—2842no—
283C22—714no—
284C22—1422no—
285C22—1422no—
286C22—1422no—
287C22—1422no—
288C22—1424no—
289C22—1424no—
290C22—1422no—
291C22#1455611014yes—
292C22C2×D4×D7111yescharacteristic
293C22#41×D7112yes—
294C22C2×#1176586411yescharacteristic
295C22C2×#1176586511yescharacteristic
296C22#1455617412yes—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—42no—
301C4—2841no—
302C4—2842no—
303C4—2842no—
304C4—2841no—
305C2—24no—
306C2—2842no—
307C2—1424no—
308C2#123×D7111yescharacteristic
309C2C2×#1455611011yescharacteristic
310C2#1798656011yescharacteristic
311C2#1798667212yes—
312C2#1798667312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#381×C71C2
hypercenter#381D71
hypocenterC71#2483
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #17985928C2semidirect
4. #17985926C2semidirect
5. #17985874C2semidirect
6. #17985873C2semidirect
7. #17985871C2semidirect
8. #17985862C2semidirect
9. #17985861C2semidirect
11. #381×C71C2semidirect
16. C2×C4⋊Dic71C22semidirect
17. C2×Dic71⋊C4C22semidirect
28. C71⋊#95C22semidirect
29. C71⋊#95C22semidirect
30. C71⋊#95C22semidirect
31. C71⋊#95C22semidirect
35. #95×C71C22semidirect
36. #95×C71C22semidirect
157. C71#2483semidirect
167. #381D71semidirect
192. #95C2×D71semidirect
193. #95C2×D71semidirect