Finite Group Databasebeta

← #22592641

Subgroups — #22592641

Order 9088.

624 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14556110C212yesmaximal
2C2×(C22⋊D71)⋊C4C211yesmaximal, characteristic
3C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#17985931C211yesmaximal, characteristic, complemented
5#17985930C211yesmaximal, characteristic, complemented
6#17985924C211yesmaximal, characteristic, complemented
7#17985896C212yesmaximal, complemented
8#17985861C212yesmaximal, complemented
9#17985858C211yesmaximal, characteristic, complemented
10#17985857C212yesmaximal, complemented
11#397×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13C23×Dic71C2211yescharacteristic
14C22×C4×D71C2212yes—
15C22×C4×D71C2211yescharacteristic
16#14556110—28no—
17(C22⋊D71)⋊C4—24no—
18C2×C4⋊Dic71C2212yes—
19C2×C4⋊Dic71C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2212yes—
22C2×Dic71⋊C4C2212yes—
23C2×Dic71⋊C4C2211yescharacteristic
24C4⋊C4×D71—24no—
25C2×C22⋊Dic71C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2211yescharacteristic
28C2×C4×Dic71C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2211yescharacteristic
32C71⋊#95C2212yes—
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35C2×C22⋊C4×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37#95×C71C2212yes—
38C23×D71C2311yescharacteristic
39C2×C22⋊D71—22no—
40C2×C22⋊D71—22no—
41C2×C22⋊D71D414yes—
42C22×Dic71—22no—
43C22×Dic71—24no—
44C22×Dic71C2312yes—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C2×C4×D71—24no—
50C2×C4×D71—24no—
51C2×C4×D71—24no—
52C2×C4×D71—24no—
53C2×C4×D71D414yes—
54C4⋊Dic71—22no—
55C4⋊Dic71—24no—
56Dic71⋊C4—22no—
57Dic71⋊C4—24no—
58Dic71⋊C4—24no—
59Dic71⋊C4—24no—
60C22⋊Dic71—24no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—24no—
63C4×Dic71—22no—
64C24×C71C2311yescharacteristic
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2312yes—
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—22no—
70C22⋊C4×C71—24no—
71C22⋊C4×C71—22no—
72C23×C71—21no—
73C23×C71—22no—
74C23×C71—21no—
75C23×C71—22no—
76C23×C71C2×D412yes—
77C23×C71C2411yescharacteristic
78C22×D71—22no—
79C22×D71—22no—
80C22×D71C2×D412yes—
81C22×D71C2×D412yes—
82C22×D71#4112yes—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
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—22no—
92C2×C4×C71C2×D412yes—
93C22⋊D71—28no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—44no—
105C2×Dic71C2×D412yes—
106C2×Dic71C2×D412yes—
107C2×Dic71#4114yes—
108C4×D71—28no—
109C4×D71—48no—
110C22×C71—22no—
111C22×C71—24no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71—42no—
115C22×C71C22×D411yescharacteristic
116C22×C71C22×D411yescharacteristic
117C22×C71C2×#4111yescharacteristic
118C22×C71C2×#4112yes—
119C22×C712−1+412yes—
120C2×D71—22no—
121C2×D71—22no—
122C2×D71—24no—
123C2×D71—24no—
124C2×D71#12114yes—
125C4×C71—22no—
126C4×C71—42no—
127C4×C71—41no—
128C4×C71—42no—
129Dic71—22no—
130Dic71—42no—
131Dic71—41no—
132Dic71—42no—
133Dic71#12314yescomplemented
134C2×C71—42no—
135C2×C71C2×#12111yescharacteristic
136C2×C71C2×#12311yescharacteristic
137C2×C71#53611yescharacteristic
138C2×C71#54712yes—
139C2×C71#55212yes—
140D71—24no—
141#2486—711nomaximal, Sylow
142C71#248611yescharacteristic, complemented, Sylow
143C2×#123—712no—
144C2×C4×D4—711no—
145C22×C4⋊C4—711no—
146#398—711no—
147#398—711no—
148#397—711no—
149#397D7111yescharacteristic, complemented
150#389—712no—
151#381—712no—
152#380—712no—
153#380—711no—
154C22×D4—711no—
155C23×C4—711no—
156C23×C4—711no—
157C23×C4—712no—
158#123—1428no—
159C4×D4—1424no—
160C2×C4⋊C4—712no—
161C2×C4⋊C4—711no—
162C2×C4⋊C4—711no—
163C2×C4⋊C4—712no—
164C2×C4⋊C4—712no—
165C2×C4⋊C4—712no—
166C2×C4⋊C4—1424no—
167C2×C4⋊C4C2×D7111yescharacteristic
168C2×C22⋊C4—712no—
169C2×C22⋊C4—711no—
170C2×C22⋊C4—712no—
171C2×C22⋊C4C2×D7111yescharacteristic
172C2×C22⋊C4C2×D7112yes—
173C2×C42—711no—
174#95—712no—
175#95—712no—
176#95—711no—
177#95—712no—
178#95C2×D7111yescharacteristic
179#95C2×D7112yes—
180C24—711no—
181C24C22×D7111yescharacteristic
182C2×D4—714no—
183C2×D4—1422no—
184C2×D4—1422no—
185C22×C4—711no—
186C22×C4—711no—
187C22×C4—714no—
188C22×C4—711no—
189C22×C4—712no—
190C22×C4—712no—
191C22×C4—1424no—
192C22×C4—1424no—
193C22×C4—1424no—
194C22×C4—1424no—
195C22×C4—1422no—
196C22×C4—1424no—
197C22×C4C22×D7111yescharacteristic
198C22×C4C22×D7112yes—
199C22×C4C22×D7112yes—
200C22×C4C22×D7111yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—1424no—
203C4⋊C4—1424no—
204C4⋊C4—1424no—
205C4⋊C4—1424no—
206C4⋊C4—1422no—
207C4⋊C4—1422no—
208C22⋊C4—24no—
209C22⋊C4—22no—
210C22⋊C4—1422no—
211C22⋊C4—1424no—
212C22⋊C4—1424no—
213C42—1422no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—712no—
219C23—712no—
220C23—712no—
221C23—1422no—
222C23—1422no—
223C23D4×D7112yes—
224C23C23×D7111yescharacteristic
225D4—1428no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—2848no—
236C2×C4—2844no—
237C2×C4—712no—
238C2×C4—714no—
239C2×C4—712no—
240C2×C4—1428no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1421no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4—1422no—
249C2×C4—1421no—
250C2×C4—1421no—
251C2×C4D4×D7112yes—
252C22—24no—
253C22—22no—
254C22—41no—
255C22—42no—
256C22—41no—
257C22—714no—
258C22—1422no—
259C22—1424no—
260C22—1424no—
261C22—1422no—
262C22C2×D4×D7111yescharacteristic
263C22C2×D4×D7111yescharacteristic
264C22#41×D7112yes—
265C22#41×D7111yescharacteristic
266C22#1455617512yes—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—42no—
271C4—2842no—
272C4—2842no—
273C4—2841no—
274C4—714no—
275C4—1422no—
276C2—42no—
277C2—1424no—
278C2#121×D7111yescharacteristic
279C2#123×D7111yescharacteristic
280C2#1798656911yescharacteristic
281C2#1798664712yes—
282C2#1798668512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#397×C71C2
hypercenter#397D71
hypocenterC71#2486
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D71C2semidirect
4. #17985931C2semidirect
5. #17985930C2semidirect
6. #17985924C2semidirect
7. #17985896C2semidirect
8. #17985861C2semidirect
9. #17985858C2semidirect
10. #17985857C2semidirect
11. #397×C71C2semidirect
133. Dic71#123semidirect
142. C71#2486semidirect
149. #397D71semidirect