Finite Group Databasebeta

← #22593523

Subgroups — #22593523

Order 9088.

584 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 147 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×#14556108C211yesmaximal, characteristic, complemented
2C2×#14556090C211yesmaximal, characteristic
3#17985934C211yesmaximal, characteristic, complemented
4#17985919C211yesmaximal, characteristic, complemented
5#17985912C211yesmaximal, characteristic, complemented
6#17985886C212yesmaximal, complemented
7#17985877C212yesmaximal
8#17985873C212yesmaximal, complemented
9#17985858C212yesmaximal, complemented
10#95×D71C211yesmaximal, characteristic, complemented
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C4×D71C2212yes—
15C22×C4×D71C2211yescharacteristic
16C22×C4×D71C2211yescharacteristic
17#14556108—24no—
18#14556090—24no—
19C2×C4⋊Dic71C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×C4×Dic71C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C23×D71C2311yescharacteristic
39C2×Dic142—22no—
40C2×Dic142—42no—
41C2×D284—22no—
42C2×D284—22no—
43C2×D284Q814yescomplemented
44C22×Dic71C2312yes—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2312yes—
48C2×C4×D71—24no—
49C2×C4×D71—22no—
50C2×C4×D71—24no—
51C2×C4×D71—42no—
52C2×C4×D71—42no—
53C2×C4×D71—42no—
54C2×C4×D71D414yes—
55C4⋊Dic71—24no—
56Dic71⋊C4—24no—
57Dic71⋊C4—42no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—42no—
63C4×Dic71—22no—
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2312yes—
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C23×C71C2411yescharacteristic
72C22×D71—22no—
73C22×D71—22no—
74C22×D71—21no—
75C22×D71—22no—
76C22×D71—21no—
77C22×D71—22no—
78C22×D71C2×D412yes—
79C22×D71C2×Q812yes—
80C22×D71C2×Q812yes—
81C22×D71#4112yes—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71C2×D412yes—
94C2×C4×C71C2×Q812yes—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—21no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71C2×D412yes—
104C2×Dic71#4114yes—
105D284—28no—
106C4×D71—44no—
107C4×D71—48no—
108C4×D71—44no—
109Dic142—44no—
110C22×C71C22×D411yescharacteristic
111C22×C71C22×Q811yescharacteristic
112C22×C71C2×#4111yescharacteristic
113C22×C71C2×#4112yes—
114C22×C71H(5,𝔽2)12yes—
115C2×D71—22no—
116C2×D71—22no—
117C2×D71—24no—
118C2×D71—24no—
119C2×D71—22no—
120C2×D71—24no—
121C2×D71—41no—
122C2×D71—41no—
123C2×D71—42no—
124C2×D71C22⋊Q814yes—
125C4×C71—22no—
126C4×C71—22no—
127C4×C71—42no—
128C4×C71—41no—
129C4×C71—42no—
130Dic71—22no—
131Dic71—24no—
132Dic71—41no—
133Dic71—42no—
134C2×C71C2×C22⋊Q811yescharacteristic
135C2×C71#53211yescharacteristic
136C2×C71#53411yescharacteristic
137C2×C71#55112yes—
138C2×C71#55412yes—
139D71—24no—
140D71—42no—
141#2564—711nomaximal, Sylow
142C71#256411yescharacteristic, complemented, Sylow
143C2×C22⋊Q8—711no—
144C2×C4×D4—711no—
145#400D7111yescharacteristic, complemented
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×Q8—711no—
155C22×D4—711no—
156C23×C4—711no—
157C23×C4—711no—
158C23×C4—712no—
159C22⋊Q8—1424no—
160C4×D4—1424no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—712no—
164C2×C4⋊C4C2×D7111yescharacteristic
165C2×C4⋊C4C2×D7111yescharacteristic
166C2×C22⋊C4—712no—
167C2×C22⋊C4—711no—
168C2×C22⋊C4—711no—
169C2×C22⋊C4—712no—
170C2×C22⋊C4—712no—
171C2×C42—711no—
172#95—712no—
173#95—711no—
174#95—712no—
175#95—712no—
176#95C2×D7111yescharacteristic
177#95C2×D7112yes—
178#95C2×D7112yes—
179C24—711no—
180C24—711no—
181C2×Q8—2842no—
182C2×Q8—1422no—
183C2×D4—714no—
184C2×D4—1422no—
185C2×D4—1422no—
186C22×C4—2842no—
187C22×C4—2842no—
188C22×C4—2842no—
189C22×C4—714no—
190C22×C4—712no—
191C22×C4—711no—
192C22×C4—711no—
193C22×C4—712no—
194C22×C4—1422no—
195C22×C4—1424no—
196C22×C4—1424no—
197C22×C4C22×D7112yes—
198C22×C4C22×D7112yes—
199C22×C4C22×D7111yescharacteristic
200C22×C4C22×D7111yescharacteristic
201C22×C4C22×D7111yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—2842no—
205C4⋊C4—1424no—
206C4⋊C4—1424no—
207C22⋊C4—2842no—
208C22⋊C4—1424no—
209C22⋊C4—1422no—
210C22⋊C4—1424no—
211C22⋊C4—1422no—
212C42—1422no—
213C23—712no—
214C23—712no—
215C23—712no—
216C23—712no—
217C23—1421no—
218C23—1422no—
219C23—1421no—
220C23—1422no—
221C23—1422no—
222C23—1422no—
223C23C23×D7111yescharacteristic
224Q8—2844no—
225D4—1428no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—2844no—
238C2×C4—2844no—
239C2×C4—2848no—
240C2×C4—712no—
241C2×C4—714no—
242C2×C4—1421no—
243C2×C4—1422no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4—1422no—
249C2×C4—1422no—
250C2×C4D4×D7112yes—
251C2×C4Q8×D7112yes—
252C22—2841no—
253C22—2842no—
254C22—2841no—
255C22—714no—
256C22—1424no—
257C22—1422no—
258C22—1422no—
259C22—1424no—
260C22—1424no—
261C22—1422no—
262C22C2×D4×D7111yescharacteristic
263C22C2×Q8×D7111yescharacteristic
264C22#41×D7111yescharacteristic
265C22#41×D7112yes—
266C22#1455617412yes—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—2842no—
273C4—2841no—
274C4—1422no—
275C4—1424no—
276C2—2842no—
277C2—1424no—
278C2C22⋊Q8×D7111yescharacteristic
279C2#1798654811yescharacteristic
280C2#1798656011yescharacteristic
281C2#1798667412yes—
282C2#1798670512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2564
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#14556108C2semidirect
3. #17985934C2semidirect
4. #17985919C2semidirect
5. #17985912C2semidirect
6. #17985886C2semidirect
8. #17985873C2semidirect
9. #17985858C2semidirect
10. #95×D71C2semidirect
11. #400×C71C2semidirect
43. C2×D284Q8semidirect
142. C71#2564semidirect
145. #400D71semidirect