Finite Group Databasebeta

← #22593522

Subgroups — #22593522

Order 9088.

584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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×C4×C22⋊D71C211yesmaximal, characteristic
2#17985937C211yesmaximal, characteristic, complemented
3#17985934C211yesmaximal, characteristic, complemented
4#17985918C211yesmaximal, characteristic, complemented
5#17985910C211yesmaximal, characteristic, complemented
6#17985880C212yesmaximal, complemented
7#17985876C212yesmaximal
8#17985870C212yesmaximal, complemented
9#17985857C212yesmaximal, complemented
10C2×C71⋊#95C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13C23×Dic71C2212yes—
14C22×C4×D71C2211yescharacteristic
15C4×C22⋊D71—24no—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2211yescharacteristic
18C2×Dic71⋊C4C2212yes—
19C2×C22⋊Dic71C2211yescharacteristic
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2212yes—
23C2×C4×Dic71C2211yescharacteristic
24C71⋊#95C2212yes—
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C22×Q8×C71C2211yescharacteristic
32C23×C4×C71C2211yescharacteristic
33C22⋊Q8×C71—24no—
34C2×C4⋊C4×C71C2212yes—
35C2×C4⋊C4×C71C2211yescharacteristic
36C2×C22⋊C4×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—22no—
39C2×C22⋊D71—22no—
40C2×C22⋊D71Q814yescomplemented
41C22×Dic71—24no—
42C22×Dic71—24no—
43C22×Dic71—42no—
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2312yes—
49C2×C4×D71—22no—
50C2×C4×D71—42no—
51Dic71⋊C4—22no—
52Dic71⋊C4—24no—
53Dic71⋊C4—42no—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—42no—
58C4×Dic71—22no—
59C24×C71C2311yescharacteristic
60C2×Q8×C71—22no—
61C2×Q8×C71—42no—
62C22×C4×C71—42no—
63C22×C4×C71D414yes—
64C22×C4×C71C2312yes—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68C4⋊C4×C71—22no—
69C4⋊C4×C71—24no—
70C22⋊C4×C71—24no—
71C23×C71—22no—
72C23×C71—22no—
73C23×C71C2×D412yes—
74C23×C71C2×Q812yes—
75C23×C71#4112yes—
76C23×C71C2411yescharacteristic
77C22×D71—21no—
78C22×D71—22no—
79C22×D71—21no—
80C22×D71—22no—
81C22×D71C2×Q812yes—
82Q8×C71—44no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—44no—
92C2×C4×C71C2×D412yes—
93C2×C4×C71C2×D412yes—
94C22⋊D71—28no—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—48no—
107C2×Dic71C2×Q812yes—
108C2×Dic71#4114yes—
109C4×D71—44no—
110C22×C71—22no—
111C22×C71—22no—
112C22×C71—24no—
113C22×C71—24no—
114C22×C71C22⋊Q814yes—
115C22×C71C22×D411yescharacteristic
116C22×C71C22×Q811yescharacteristic
117C22×C71C2×#4112yes—
118C22×C71C2×#4111yescharacteristic
119C22×C71H(5,𝔽2)12yes—
120C2×D71—22no—
121C2×D71—24no—
122C2×D71—41no—
123C2×D71—41no—
124C2×D71—42no—
125C4×C71—22no—
126C4×C71—22no—
127C4×C71—42no—
128C4×C71—42no—
129Dic71—22no—
130Dic71—24no—
131Dic71—41no—
132Dic71—41no—
133Dic71—42no—
134C2×C71—24no—
135C2×C71C2×C22⋊Q811yescharacteristic
136C2×C71#53211yescharacteristic
137C2×C71#53411yescharacteristic
138C2×C71#55112yes—
139C2×C71#55412yes—
140D71—42no—
141#2564—711nomaximal, Sylow
142C71#256411yescharacteristic, complemented, Sylow
143C2×C22⋊Q8D7111yescharacteristic, complemented
144C2×C4×D4—711no—
145#400—711no—
146#399—711no—
147#395—711no—
148#393—711no—
149#387—712no—
150#386—712no—
151#385—712no—
152#380—712no—
153C2×#95—711no—
154C22×Q8C2×D7111yescharacteristic
155C22×D4—711no—
156C23×C4—711no—
157C23×C4—712no—
158C23×C4C2×D7111yescharacteristic
159C22⋊Q8—24no—
160C4×D4—1424no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4—711no—
163C2×C4⋊C4—712no—
164C2×C4⋊C4C2×D7112yes—
165C2×C4⋊C4C2×D7111yescharacteristic
166C2×C22⋊C4—712no—
167C2×C22⋊C4—711no—
168C2×C22⋊C4—711no—
169C2×C22⋊C4—712no—
170C2×C22⋊C4C2×D7112yes—
171C2×C42—711no—
172#95—712no—
173#95—711no—
174#95—711no—
175#95—712no—
176#95—712no—
177#95—712no—
178#95—712no—
179C24—711no—
180C24C22×D7111yescharacteristic
181C2×Q8—22no—
182C2×Q8—42no—
183C2×D4—714no—
184C2×D4—1422no—
185C2×D4—1422no—
186C22×C4—42no—
187C22×C4—2842no—
188C22×C4—2842no—
189C22×C4—711no—
190C22×C4—712no—
191C22×C4—712no—
192C22×C4—711no—
193C22×C4—711no—
194C22×C4—1424no—
195C22×C4—1424no—
196C22×C4—1422no—
197C22×C4C22⋊D7114yes—
198C22×C4C22×D7112yes—
199C22×C4C22×D7112yes—
200C22×C4C22×D7111yescharacteristic
201C22×C4C22×D7111yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—2842no—
205C4⋊C4—1424no—
206C4⋊C4—1422no—
207C22⋊C4—24no—
208C22⋊C4—2842no—
209C22⋊C4—1424no—
210C22⋊C4—1422no—
211C22⋊C4—1422no—
212C42—1422no—
213C23—22no—
214C23—22no—
215C23—712no—
216C23—1421no—
217C23—1422no—
218C23—1421no—
219C23—1422no—
220C23C2×C22⋊D7112yes—
221C23Q8×D7112yes—
222C23#1176586612yes—
223C23C23×D7111yescharacteristic
224Q8—44no—
225D4—1428no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—2844no—
236C2×C4—2848no—
237C2×C4—712no—
238C2×C4—714no—
239C2×C4—1421no—
240C2×C4—1421no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1421no—
247C2×C4—1422no—
248C2×C4—1421no—
249C2×C4—1422no—
250C2×C4C2×C22⋊D7112yes—
251C2×C4C2×C22⋊D7112yes—
252C22—24no—
253C22—24no—
254C22—22no—
255C22—22no—
256C22—2842no—
257C22—2841no—
258C22—2841no—
259C22—1424no—
260C22—1422no—
261C22#1455610414yes—
262C22C22×C22⋊D7111yescharacteristic
263C22C2×Q8×D7111yescharacteristic
264C22#41×D7112yes—
265C22C2×#1176586611yescharacteristic
266C22#1455617312yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—2842no—
272C4—2841no—
273C4—2841no—
274C4—1424no—
275C4—1422no—
276C2—24no—
277C2—2842no—
278C2C2×#1455610411yescharacteristic
279C2#1798654511yescharacteristic
280C2#1798655811yescharacteristic
281C2#1798667612yes—
282C2#1798670312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×C22⋊Q8×C71C2
hypercenterC2×C22⋊Q8D71
hypocenterC71#2564
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #17985937C2semidirect
3. #17985934C2semidirect
4. #17985918C2semidirect
5. #17985910C2semidirect
6. #17985880C2semidirect
8. #17985870C2semidirect
9. #17985857C2semidirect
10. C2×C71⋊#95C2semidirect
11. C2×C22⋊Q8×C71C2semidirect
40. C2×C22⋊D71Q8semidirect
142. C71#2564semidirect
143. C2×C22⋊Q8D71semidirect