Finite Group Databasebeta

← #22592749

Subgroups — #22592749

Order 9088.

412 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#17985937C212yesmaximal, complemented
2#17985928C212yesmaximal
3#17985895C211yesmaximal, characteristic, complemented
4#17985889C212yesmaximal
5#17985866C212yesmaximal, complemented
6#17985865C211yesmaximal, characteristic, complemented
7#17985856C211yesmaximal, characteristic
8#17985856C211yesmaximal, characteristic
9C4⋊C4×Dic71C211yesmaximal, characteristic, complemented
10C4×(C2×D71)⋊C4C211yesmaximal, characteristic
11#389×C71C211yesmaximal, characteristic, complemented
12C22×C4×D71C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×C4⋊Dic71C2211yescharacteristic
15C2×Dic71⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2211yescharacteristic
17C2×(C2×D71)⋊C4C2211yescharacteristic
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×C4×Dic71C2211yescharacteristic
21C2×C4×Dic71C2212yes—
22C71⋊#95C2211yescharacteristic
23C71⋊#95C2211yescharacteristic
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C2×C4⋊C4×C71C2211yescharacteristic
31C2×C4⋊C4×C71C2211yescharacteristic
32C2×C4⋊C4×C71C2212yes—
33C2×C42×C71C2211yescharacteristic
34#95×C71C2212yes—
35C23×D71C2311yescharacteristic
36C22×Dic71C2311yescharacteristic
37C22×Dic71C2312yes—
38C22×Dic71C2312yes—
39C22×Dic71C2311yescharacteristic
40C22×Dic71C2311yescharacteristic
41C2×C4×D71—22no—
42C2×C4×D71—22no—
43C2×C4×D71—22no—
44C4⋊Dic71—22no—
45C4⋊Dic71—22no—
46Dic71⋊C4—42no—
47(C2×D71)⋊C4—22no—
48(C2×D71)⋊C4—24no—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—42no—
51C4×Dic71—22no—
52C4×Dic71—24no—
53C22×C4×C71C2311yescharacteristic
54C22×C4×C71C2312yes—
55C22×C4×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57C22×C4×C71C2311yescharacteristic
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—42no—
61C42×C71—22no—
62C23×C71C2411yescharacteristic
63C22×D71—21no—
64C22×D71—21no—
65C22×D71—21no—
66C22×D71—21no—
67C22×D71—21no—
68C22×D71—21no—
69C22×D71#4112yes—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71#4112yes—
83C2×C4×C71#4112yes—
84C2×C4×C71#4112yes—
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—22no—
88C2×Dic71—21no—
89C2×Dic71—22no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—21no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—22no—
99C2×Dic71#4112yes—
100C4×D71—44no—
101C22×C71C2×#4111yescharacteristic
102C22×C71C2×#4111yescharacteristic
103C22×C71C2×#4111yescharacteristic
104C22×C71C2×#4111yescharacteristic
105C22×C71C2×#4111yescharacteristic
106C22×C71H(5,𝔽2)11yescharacteristic
107C22×C712−1+411yescharacteristic
108C2×D71—22no—
109C2×D71—22no—
110C2×D71—22no—
111C2×D71—41no—
112C2×D71—41no—
113C2×D71—41no—
114C2×D71—41no—
115C4×C71—41no—
116C4×C71—42no—
117C4×C71—41no—
118C4×C71—42no—
119C4×C71#12614yes—
120Dic71—22no—
121Dic71—41no—
122Dic71—42no—
123Dic71—42no—
124Dic71—41no—
125C2×C71C2×#12611yescharacteristic
126C2×C71#52911yescharacteristic
127C2×C71#54211yescharacteristic
128C2×C71#55211yescharacteristic
129C2×C71#55211yescharacteristic
130C2×C71#55311yescharacteristic
131C2×C71#55511yescharacteristic
132D71—42no—
133#2495—711nomaximal, Sylow
134C71#249511yescharacteristic, complemented, Sylow
135#400—712no—
136#397—712no—
137#389—711no—
138#389D7111yescharacteristic, complemented
139#388—712no—
140#383—712no—
141#382—711no—
142#380—711no—
143#380—711no—
144C4×C4⋊C4—711no—
145C4×C22⋊C4—711no—
146C23×C4—711no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—711no—
149C2×C4⋊C4—711no—
150C2×C4⋊C4C2×D7111yescharacteristic
151C2×C4⋊C4C2×D7111yescharacteristic
152C2×C4⋊C4C2×D7112yes—
153C2×C22⋊C4—712no—
154C2×C22⋊C4—711no—
155C2×C22⋊C4—711no—
156C2×C22⋊C4—712no—
157C2×C42—711no—
158C2×C42—712no—
159C2×C42C2×D7111yescharacteristic
160#95—711no—
161#95—711no—
162#95—712no—
163#95—712no—
164#95—712no—
165#95—712no—
166#95—712no—
167#95—712no—
168#95C2×D7112yes—
169C24—711no—
170C22×C4—711no—
171C22×C4—711no—
172C22×C4—712no—
173C22×C4—712no—
174C22×C4—711no—
175C22×C4—1422no—
176C22×C4—1422no—
177C22×C4—1422no—
178C22×C4C22×D7111yescharacteristic
179C22×C4C22×D7111yescharacteristic
180C22×C4C22×D7112yes—
181C22×C4C22×D7112yes—
182C22×C4C22×D7111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—2842no—
187C4⋊C4—1422no—
188C4⋊C4—1422no—
189C22⋊C4—2842no—
190C22⋊C4—1422no—
191C22⋊C4—1422no—
192C22⋊C4—1424no—
193C42—22no—
194C42—1424no—
195C42—1422no—
196C23—712no—
197C23—1421no—
198C23—1421no—
199C23—1421no—
200C23—1421no—
201C23—1421no—
202C23—1421no—
203C23C23×D7111yescharacteristic
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—2844no—
217C2×C4—712no—
218C2×C4—1422no—
219C2×C4—1421no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4—1421no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1421no—
227C2×C4—1422no—
228C2×C4—1421no—
229C2×C4—1422no—
230C2×C4—1421no—
231C2×C4—1421no—
232C2×C4#1176586412yes—
233C2×C4#1176586512yes—
234C2×C4#1176586612yes—
235C22—2841no—
236C22—2841no—
237C22—2841no—
238C22—2841no—
239C22—1422no—
240C22—1422no—
241C22—1422no—
242C22#41×D7111yescharacteristic
243C22#41×D7111yescharacteristic
244C22C2×#1176586411yescharacteristic
245C22C2×#1176586511yescharacteristic
246C22C2×#1176586611yescharacteristic
247C22#1455617311yescharacteristic
248C22#1455617611yescharacteristic
249C4—41no—
250C4—42no—
251C4—42no—
252C4—41no—
253C4—2842no—
254C4—2841no—
255C4—2841no—
256C4—2842no—
257C4—1422no—
258C4#1455612314yes—
259C2—2842no—
260C2C2×#1455612311yescharacteristic
261C2#1798653411yescharacteristic
262C2#1798662311yescharacteristic
263C2#1798668311yescharacteristic
264C2#1798669011yescharacteristic
265C2#1798669911yescharacteristic
266C2#1798671311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#389×C71C2
hypercenter#389D71
hypocenterC71#2495
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #17985937C2semidirect
3. #17985895C2semidirect
5. #17985866C2semidirect
6. #17985865C2semidirect
9. C4⋊C4×Dic71C2semidirect
11. #389×C71C2semidirect
134. C71#2495semidirect
138. #389D71semidirect