Finite Group Databasebeta

← #22591515

Subgroups — #22591515

Order 9088.

648 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14556124C211yesmaximal, characteristic
2#17985879C211yesmaximal, characteristic, complemented
3#17985878C212yesmaximal, complemented
4#17985877C212yesmaximal, complemented
5#17985876C212yesmaximal, complemented
6#17985875C212yesmaximal, complemented
7#17985875C212yesmaximal, complemented
8#17985866C212yesmaximal, complemented
9#386×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C22×C22⋊D71C2212yes—
13C23×Dic71C2211yescharacteristic
14C22×C4×D71C2212yes—
15#14556124C418yescomplemented
16C2×C22⋊Dic71C2211yescharacteristic
17C2×C22⋊Dic71C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2211yescharacteristic
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C22×D4×C71C2211yescharacteristic
28C23×C4×C71C2211yescharacteristic
29C2×C22⋊C4×C71C2211yescharacteristic
30C2×C22⋊C4×C71C2212yes—
31#95×C71C2212yes—
32C23×D71C2312yes—
33C2×D284—41no—
34C2×D284—41no—
35C2×D284C2×C414yescomplemented
36C2×C22⋊D71—42no—
37C2×C22⋊D71—42no—
38C2×C22⋊D71—42no—
39C2×C22⋊D71—42no—
40C2×C22⋊D71C2×C418yescomplemented
41C2×C22⋊D71C2×C418yescomplemented
42C22×Dic71—41no—
43C22×Dic71—42no—
44C22×Dic71C2312yes—
45C22×Dic71C2312yes—
46C22×Dic71C2312yes—
47C2×C4×D71—42no—
48C2×C4×D71—42no—
49C2×C4×D71—42no—
50C22⋊Dic71—41no—
51C22⋊Dic71—42no—
52(C2×D71)⋊C4—42no—
53(C2×D71)⋊C4—42no—
54(C2×D71)⋊C4—42no—
55C4×Dic71C2×C414yescomplemented
56C24×C71C2311yescharacteristic
57C24×C71C2311yescharacteristic
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71C2×C414yescomplemented
61C22×C4×C71—42no—
62C22×C4×C71—41no—
63C22×C4×C71C2312yes—
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C22⋊C4×C71—42no—
67C22⋊C4×C71—41no—
68C23×C71—21no—
69C23×C71—22no—
70C23×C71—21no—
71C23×C71—22no—
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71C22×C412yes—
77C23×C71C22×C412yes—
78C23×C71C2411yescharacteristic
79C22×D71—22no—
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71C22×C414yes—
86D4×C71—44no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—44no—
96C2×C4×C71C22×C412yes—
97C22⋊D71—48no—
98C22⋊D71—48no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—44no—
108C2×Dic71C22×C414yes—
109D284—44no—
110C4×D71—48no—
111C22×C71—24no—
112C22×C71—24no—
113C22×C71—22no—
114C22×C71—22no—
115C22×C71—41no—
116C22×C71—41no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71C23×C411yescharacteristic
122C22×C71H(5,𝔽2)12yes—
123C22×C71H(5,𝔽2)11yescharacteristic
124C22×C71H(5,𝔽2)11yescharacteristic
125C22×C71H(5,𝔽2)12yes—
126C2×D71—24no—
127C2×D71—24no—
128C2×D71—24no—
129C2×D71—42no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71—42no—
133C4×C71—22no—
134C4×C71—42no—
135C4×C71—42no—
136Dic71—24no—
137Dic71—42no—
138Dic71—42no—
139C2×C71—42no—
140C2×C71—42no—
141C2×C71#51812yes—
142C2×C71#51811yescharacteristic
143C2×C71#56012yes—
144C2×C71#56011yescharacteristic
145C2×C71#56011yescharacteristic
146D71—44no—
147#2380—711nomaximal, Sylow
148C71#238011yescharacteristic, complemented, Sylow
149C2×#127—711no—
150#386—711no—
151#386—712no—
152#386—712no—
153#386—712no—
154#386—712no—
155#386—712no—
156#386D7111yescharacteristic, complemented
157#383—712no—
158C22×D4—712no—
159C22×D4—712no—
160C22×D4—711no—
161C22×D4C2×D7111yescharacteristic
162C23×C4—711no—
163C23×C4—712no—
164C23×C4C2×D7111yescharacteristic
165#127—718no—
166C2×C22⋊C4—712no—
167C2×C22⋊C4—711no—
168C2×C22⋊C4—712no—
169C2×C22⋊C4—712no—
170C2×C22⋊C4—712no—
171C2×C22⋊C4C2×D7112yes—
172C2×C22⋊C4C2×D7111yescharacteristic
173C2×C42—711no—
174#95—712no—
175#95—712no—
176#95—712no—
177#95—712no—
178#95—712no—
179#95C2×D7112yes—
180C24—712no—
181C24C22×D7111yescharacteristic
182C24C22×D7111yescharacteristic
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—2841no—
186C2×D4—2841no—
187C2×D4—2842no—
188C2×D4—2842no—
189C2×D4—2842no—
190C2×D4—2842no—
191C2×D4—718no—
192C2×D4—718no—
193C2×D4—714no—
194C2×D4C4×D7114yescomplemented
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—2842no—
198C22×C4—2842no—
199C22×C4—2842no—
200C22×C4—2841no—
201C22×C4—2842no—
202C22×C4—712no—
203C22×C4—712no—
204C22×C4—712no—
205C22×C4C22×D7112yes—
206C22×C4C22×D7112yes—
207C22×C4C22×D7111yescharacteristic
208C22⋊C4—42no—
209C22⋊C4—41no—
210C22⋊C4—2842no—
211C22⋊C4—2842no—
212C22⋊C4—2842no—
213C22⋊C4—2841no—
214C22⋊C4—2842no—
215C42—714no—
216C23—21no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—714no—
225C23—1422no—
226C23—1422no—
227C23—1422no—
228C23—1422no—
229C23—1422no—
230C23—1422no—
231C23C2×C4×D7112yes—
232C23C2×C4×D7112yes—
233C23C23×D7111yescharacteristic
234D4—44no—
235D4—2844no—
236D4—2848no—
237D4—2848no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—44no—
247C2×C4—2848no—
248C2×C4—2844no—
249C2×C4—714no—
250C2×C4—1422no—
251C2×C4—1422no—
252C2×C4—1422no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4C2×C4×D7112yes—
259C22—24no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—2842no—
270C22—2842no—
271C22—2842no—
272C22—2842no—
273C22—1424no—
274C22—1424no—
275C22—1424no—
276C22C22×C4×D7111yescharacteristic
277C22#1455617312yes—
278C22#1455617311yescharacteristic
279C22#1455617311yescharacteristic
280C22#1455617412yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—2842no—
285C4—2842no—
286C4—1424no—
287C2—42no—
288C2—42no—
289C2—2844no—
290C2#1798647112yes—
291C2#1798647211yescharacteristic
292C2#1798673712yes—
293C2#1798673811yescharacteristic
294C2#1798673911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutatorC22×C71C23×C4
Fitting#386×C71C2
socleC23×C71C24
hypercenter#386D71
hypocenterC71#2380
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #17985879C2semidirect
3. #17985878C2semidirect
4. #17985877C2semidirect
5. #17985876C2semidirect
6. #17985875C2semidirect
7. #17985875C2semidirect
8. #17985866C2semidirect
9. #386×C71C2semidirect
15. #14556124C4semidirect
35. C2×D284C2×C4semidirect
40. C2×C22⋊D71C2×C4semidirect
41. C2×C22⋊D71C2×C4semidirect
55. C4×Dic71C2×C4semidirect
60. C2×D4×C71C2×C4semidirect
148. C71#2380semidirect
156. #386D71semidirect
194. C2×D4C4×D71semidirect