Finite Group Databasebeta

← #22594029

Subgroups — #22594029

Order 9088.

576 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556085C211yesmaximal, characteristic, complemented
2#17985918C211yesmaximal, characteristic, complemented
3#17985913C211yesmaximal, characteristic, complemented
4#17985910C211yesmaximal, characteristic, complemented
5#17985907C211yesmaximal, characteristic
6#17985899C212yesmaximal, complemented
7#17985893C212yesmaximal, complemented
8#17985887C212yesmaximal, complemented
9#17985875C212yesmaximal, complemented
10#17985845C211yesmaximal, characteristic, complemented
11C2×#124×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2212yes—
13C22×C4×D71C2211yescharacteristic
14#14556085—24no—
15C2×Dic71⋊C4C2211yescharacteristic
16C2×Dic71⋊C4C2211yescharacteristic
17C2×Dic71⋊C4C2212yes—
18C2×C22⋊Dic71C2212yes—
19C2×C22⋊Dic71C2212yes—
20C2×C22⋊Dic71C2212yes—
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×C4×Dic71C2212yes—
25C2×C4×Dic71C2211yescharacteristic
26C71⋊#95C2212yes—
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C22×Q8×C71C2211yescharacteristic
32C22×D4×C71C2211yescharacteristic
33#124×C71—24no—
34C2×C22⋊C4×C71C2212yes—
35C2×C22⋊C4×C71C2212yes—
36C2×C42×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—24no—
39C2×C22⋊D71—24no—
40C2×C22⋊D71—42no—
41C22×Dic71C2312yes—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C2×C4×D71—22no—
47C2×C4×D71—22no—
48C2×C4×D71—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—42no—
52C22⋊Dic71—24no—
53C22⋊Dic71—24no—
54C22⋊Dic71—42no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—42no—
58C4×Dic71—22no—
59C4×Dic71—24no—
60C24×C71C2312yes—
61C2×Q8×C71—22no—
62C2×Q8×C71—42no—
63C2×D4×C71—22no—
64C2×D4×C71—42no—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2311yescharacteristic
69C22⋊C4×C71—24no—
70C22⋊C4×C71—24no—
71C42×C71D414yescomplemented
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71—22no—
78C23×C71—22no—
79C23×C71C2411yescharacteristic
80C22×D71—21no—
81C22×D71—21no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71#4112yes—
85Q8×C71—44no—
86D4×C71—44no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2×D412yes—
96C2×C4×C71C2×D412yes—
97C22⋊D71—48no—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—21no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71#4114yes—
110C2×Dic71#4112yes—
111C4×D71—28no—
112C22×C71—42no—
113C22×C71—42no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71—42no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71C22×D411yescharacteristic
120C22×C71C2×#4111yescharacteristic
121C22×C71C2×#4111yescharacteristic
122C22×C71C2×#4112yes—
123C22×C71H(5,𝔽2)12yes—
124C2×D71—22no—
125C2×D71—24no—
126C2×D71—41no—
127C2×D71—41no—
128C2×D71—42no—
129C4×C71—22no—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—42no—
133Dic71—22no—
134Dic71—24no—
135Dic71—41no—
136Dic71—41no—
137Dic71—42no—
138C2×C71—82no—
139C2×C71#52511yescharacteristic
140C2×C71#53211yescharacteristic
141C2×C71#53511yescharacteristic
142C2×C71#55112yes—
143C2×C71#55512yes—
144D71—42no—
145#2610—711nomaximal, Sylow
146C71#261011yescharacteristic, complemented, Sylow
147C2×#124D7111yescharacteristic, complemented
148C2×#117—711no—
149#395—711no—
150#394—711no—
151#393—711no—
152#392—711no—
153#390—712no—
154#388—712no—
155#388—712no—
156#386—712no—
157#376—711no—
158C22×Q8C2×D7111yescharacteristic
159C22×D4—712no—
160C22×D4C2×D7111yescharacteristic
161C23×C4—711no—
162#124—24no—
163#117—1424no—
164C2×C4⋊C4—712no—
165C2×C4⋊C4—711no—
166C2×C4⋊C4—711no—
167C2×C22⋊C4—712no—
168C2×C22⋊C4—712no—
169C2×C22⋊C4—712no—
170C2×C22⋊C4—712no—
171C2×C22⋊C4—711no—
172C2×C22⋊C4—711no—
173C2×C22⋊C4C2×D7112yes—
174C2×C22⋊C4C2×D7112yes—
175C2×C42—711no—
176C2×C42—712no—
177C2×C42C2×D7111yescharacteristic
178#95—712no—
179#95—711no—
180#95—711no—
181#95—712no—
182#95—712no—
183C24—711no—
184C24C22×D7112yes—
185C2×Q8—22no—
186C2×Q8—42no—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—2842no—
190C2×D4—1424no—
191C2×D4—1424no—
192C22×C4—712no—
193C22×C4—711no—
194C22×C4—712no—
195C22×C4—711no—
196C22×C4—711no—
197C22×C4—1422no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4C22×D7112yes—
201C22×C4C22×D7111yescharacteristic
202C22×C4C22×D7111yescharacteristic
203C22×C4C22×D7111yescharacteristic
204C4⋊C4—2842no—
205C4⋊C4—1422no—
206C4⋊C4—1422no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—2842no—
210C22⋊C4—2842no—
211C22⋊C4—1424no—
212C22⋊C4—1424no—
213C22⋊C4—1422no—
214C22⋊C4—1422no—
215C42—1424no—
216C42—1422no—
217C42C22⋊D7114yescomplemented
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—712no—
226C23—1421no—
227C23—1421no—
228C23—1422no—
229C23—1422no—
230C23C23×D7111yescharacteristic
231Q8—44no—
232D4—44no—
233D4—2848no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—714no—
242C2×C4—712no—
243C2×C4—1428no—
244C2×C4—1421no—
245C2×C4—1421no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4—1422no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4—1421no—
252C2×C4—1421no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4C2×C22⋊D7112yes—
256C2×C4C2×C22⋊D7112yes—
257C2×C4C2×C22⋊D7112yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—2841no—
266C22—2842no—
267C22—2841no—
268C22—1424no—
269C22—1422no—
270C22C22×C22⋊D7111yescharacteristic
271C22#41×D7111yescharacteristic
272C22#41×D7112yes—
273C22#41×D7111yescharacteristic
274C22#1455617312yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—2841no—
280C4—2842no—
281C4—2841no—
282C4—1424no—
283C4—1422no—
284C2—82no—
285C2—2842no—
286C2#1798650611yescharacteristic
287C2#1798654511yescharacteristic
288C2#1798656611yescharacteristic
289C2#1798667912yes—
290C2#1798671112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#124×C71C2
hypercenterC2×#124D71
hypocenterC71#2610
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14556085C2semidirect
2. #17985918C2semidirect
3. #17985913C2semidirect
4. #17985910C2semidirect
6. #17985899C2semidirect
7. #17985893C2semidirect
8. #17985887C2semidirect
9. #17985875C2semidirect
10. #17985845C2semidirect
11. C2×#124×C71C2semidirect
71. C42×C71D4semidirect
146. C71#2610semidirect
147. C2×#124D71semidirect
217. C42C22⋊D71semidirect