Finite Group Databasebeta

← #22593693

Subgroups — #22593693

Order 9088.

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

normal subgroups only

This group has 564 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×Q8×Dic71C211yesmaximal, characteristic, complemented
2C2×(C22⋊D71)⋊C4C211yesmaximal, characteristic
3#17985936C211yesmaximal, characteristic, complemented
4#17985935C211yesmaximal, characteristic, complemented
5#17985910C211yesmaximal, characteristic, complemented
6#17985899C212yesmaximal
7#17985896C212yesmaximal, complemented
8#17985874C212yesmaximal, complemented
9#17985873C212yesmaximal, complemented
10#17985863C211yesmaximal, characteristic, complemented
11#393×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13C23×Dic71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15Q8×Dic71—24no—
16(C22⋊D71)⋊C4—24no—
17C2×C4⋊Dic71C2212yes—
18C2×C4⋊Dic71C2211yescharacteristic
19C2×C4⋊Dic71C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×Dic71⋊C4C2212yes—
23C2×C22⋊Dic71C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×C4×Dic71C2212yes—
28C2×C4×Dic71C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2211yescharacteristic
31C71⋊#95C2212yes—
32C22×Q8×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2212yes—
34C2×C22⋊C4×C71C2211yescharacteristic
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—22no—
39C2×C22⋊D71—22no—
40C2×C22⋊D71Q814yescomplemented
41C22×Dic71—22no—
42C22×Dic71—42no—
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2312yes—
45C22×Dic71C2312yes—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2311yescharacteristic
48C2×C4×D71—22no—
49C2×C4×D71—42no—
50C4⋊Dic71—22no—
51C4⋊Dic71—24no—
52C4⋊Dic71—42no—
53Dic71⋊C4—22no—
54Dic71⋊C4—24no—
55Dic71⋊C4—42no—
56C22⋊Dic71—42no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—42no—
60C4×Dic71—24no—
61C4×Dic71D414yes—
62C24×C71C2311yescharacteristic
63C2×Q8×C71—22no—
64C2×Q8×C71—24no—
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2312yes—
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2312yes—
69C22⋊C4×C71—22no—
70C22⋊C4×C71—24no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—21no—
74C23×C71—22no—
75C23×C71C2×Q812yes—
76C23×C71C2411yescharacteristic
77C22×D71—21no—
78C22×D71—22no—
79C22×D71—21no—
80C22×D71—22no—
81C22×D71C2×Q812yes—
82Q8×C71—28no—
83C2×C4×C71—22no—
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×C71C2×D412yes—
92C2×C4×C71#4114yes—
93C22⋊D71—28no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—21no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—44no—
104C2×Dic71C2×D412yes—
105C2×Dic71C2×D412yes—
106C2×Dic71C2×Q812yes—
107C2×Dic71#4112yes—
108C4×D71—44no—
109C22×C71—22no—
110C22×C71—24no—
111C22×C71—41no—
112C22×C71—41no—
113C22×C71—42no—
114C22×C71C22×D411yescharacteristic
115C22×C71C22×Q811yescharacteristic
116C22×C71C2×#4111yescharacteristic
117C22×C71C2×#4112yes—
118C22×C71H(5,𝔽2)12yes—
119C2×D71—22no—
120C2×D71—24no—
121C2×D71—41no—
122C2×D71—41no—
123C2×D71—42no—
124C4×C71—22no—
125C4×C71—24no—
126C4×C71—41no—
127C4×C71—42no—
128Dic71—22no—
129Dic71—42no—
130Dic71—41no—
131Dic71—42no—
132Dic71C22⋊Q814yes—
133C2×C71—42no—
134C2×C71C2×C22⋊Q811yescharacteristic
135C2×C71#52511yescharacteristic
136C2×C71#53511yescharacteristic
137C2×C71#55412yes—
138C2×C71#55912yes—
139D71—42no—
140#2580—711nomaximal, Sylow
141C71#258011yescharacteristic, complemented, Sylow
142C2×C4×Q8—711no—
143C2×C4×D4—711no—
144#399—711no—
145#399—711no—
146#393—711no—
147#393D7111yescharacteristic, complemented
148#390—712no—
149#389—712no—
150#385—712no—
151#385—712no—
152#382—711no—
153C22×Q8C2×D7111yescharacteristic
154C22×D4—711no—
155C23×C4—711no—
156C23×C4—711no—
157C4×Q8—1424no—
158C4×D4—1424no—
159C2×C4⋊C4—712no—
160C2×C4⋊C4—712no—
161C2×C4⋊C4—711no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—711no—
164C2×C4⋊C4—712no—
165C2×C22⋊C4—712no—
166C2×C22⋊C4—712no—
167C2×C22⋊C4—711no—
168C2×C22⋊C4—712no—
169C2×C22⋊C4C2×D7111yescharacteristic
170C2×C22⋊C4C2×D7112yes—
171C2×C42—711no—
172C2×C42—712no—
173#95—712no—
174#95—711no—
175#95—712no—
176#95C2×D7111yescharacteristic
177#95C2×D7112yes—
178C24—711no—
179C24C22×D7111yescharacteristic
180C2×Q8—22no—
181C2×Q8—24no—
182C2×D4—714no—
183C2×D4—1422no—
184C2×D4—1422no—
185C22×C4—2842no—
186C22×C4—2842no—
187C22×C4—711no—
188C22×C4—711no—
189C22×C4—712no—
190C22×C4—711no—
191C22×C4—712no—
192C22×C4—1422no—
193C22×C4—1422no—
194C22×C4C22×D7112yes—
195C22×C4C22×D7111yescharacteristic
196C22×C4C22×D7111yescharacteristic
197C22×C4C22×D7112yes—
198C4⋊C4—2842no—
199C4⋊C4—2842no—
200C4⋊C4—1424no—
201C4⋊C4—1422no—
202C4⋊C4—1424no—
203C4⋊C4—1422no—
204C22⋊C4—22no—
205C22⋊C4—24no—
206C22⋊C4—2842no—
207C22⋊C4—2842no—
208C22⋊C4—1422no—
209C22⋊C4—1424no—
210C42—714no—
211C42—1424no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23—712no—
217C23—1421no—
218C23—1421no—
219C23—1422no—
220C23—1422no—
221C23Q8×D7112yes—
222C23C23×D7111yescharacteristic
223Q8—28no—
224D4—1428no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—2844no—
234C2×C4—2844no—
235C2×C4—712no—
236C2×C4—712no—
237C2×C4—712no—
238C2×C4—712no—
239C2×C4—1422no—
240C2×C4—1421no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4D4×D7112yes—
249C2×C4#1176586414yes—
250C22—24no—
251C22—22no—
252C22—41no—
253C22—42no—
254C22—41no—
255C22—2841no—
256C22—2842no—
257C22—2841no—
258C22—1424no—
259C22—1422no—
260C22C2×D4×D7111yescharacteristic
261C22C2×Q8×D7111yescharacteristic
262C22#41×D7111yescharacteristic
263C22C2×#1176586412yes—
264C22#1455617412yes—
265C4—22no—
266C4—24no—
267C4—41no—
268C4—42no—
269C4—2841no—
270C4—2842no—
271C4—2842no—
272C4—714no—
273C4—1422no—
274C2—42no—
275C2—2842no—
276C2C22⋊Q8×D7111yescharacteristic
277C2#1798650511yescharacteristic
278C2#1798656711yescharacteristic
279C2#1798670712yes—
280C2#1798673112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#393×C71C2
hypercenter#393D71
hypocenterC71#2580
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic71C2semidirect
3. #17985936C2semidirect
4. #17985935C2semidirect
5. #17985910C2semidirect
7. #17985896C2semidirect
8. #17985874C2semidirect
9. #17985873C2semidirect
10. #17985863C2semidirect
11. #393×C71C2semidirect
40. C2×C22⋊D71Q8semidirect
141. C71#2580semidirect
147. #393D71semidirect