Finite Group Databasebeta

← #22592223

Subgroups — #22592223

Order 9088.

660 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14556113C212yesmaximal, complemented
2C2×(C22⋊D71)⋊C4C212yesmaximal, complemented
3#17985915C212yesmaximal, complemented
4#17985908C211yesmaximal, characteristic
5#17985895C212yesmaximal, complemented
6#17985891C212yesmaximal, complemented
7#17985871C211yesmaximal, characteristic, complemented
8#17985870C212yesmaximal, complemented
9#399×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C23×Dic71C2211yescharacteristic
13C22×C4×D71C2212yes—
14#14556113—28no—
15(C22⋊D71)⋊C4—28no—
16C2×C4⋊Dic71C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×C22⋊Dic71C2211yescharacteristic
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×C4×Dic71C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2211yescharacteristic
28C2×C4⋊C4×C71C2212yes—
29C2×C4⋊C4×C71C2211yescharacteristic
30C2×C22⋊C4×C71C2212yes—
31C2×C22⋊C4×C71C2211yescharacteristic
32#95×C71C2211yescharacteristic
33C23×D71C2312yes—
34C2×D284—24no—
35C2×D284—41no—
36C2×C22⋊D71—24no—
37C2×C22⋊D71—24no—
38C2×C22⋊D71—24no—
39C22×Dic71—24no—
40C22×Dic71—41no—
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C22×Dic71C2312yes—
44C2×C4×D71—24no—
45C2×C4×D71—24no—
46C2×C4×D71—42no—
47C4⋊Dic71—24no—
48Dic71⋊C4—24no—
49Dic71⋊C4—24no—
50C22⋊Dic71—22no—
51(C2×D71)⋊C4—24no—
52(C2×D71)⋊C4—24no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4D418yescomplemented
55C4×Dic71—24no—
56C24×C71C2311yescharacteristic
57C22×C4×C71C2312yes—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2312yes—
61C4⋊C4×C71—24no—
62C4⋊C4×C71—41no—
63C22⋊C4×C71—24no—
64C22⋊C4×C71—41no—
65C23×C71—22no—
66C23×C71—22no—
67C23×C71—21no—
68C23×C71—21no—
69C23×C71#4112yes—
70C23×C71C2411yescharacteristic
71C22×D71—22no—
72C22×D71—22no—
73C22×D71—22no—
74C22×D71—22no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71C2×D414yes—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71C2×D414yes—
88C22⋊D71—216no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—44no—
97C2×Dic71C2×D414yes—
98C2×Dic71#4114yes—
99D284—44no—
100C4×D71—48no—
101C22×C71—24no—
102C22×C71—22no—
103C22×C71—41no—
104C22×C71—42no—
105C22×C71—41no—
106C22×C71C22×D412yes—
107C22×C71C2×#4112yes—
108C22×C71C2×#4111yescharacteristic
109C22×C71H(5,𝔽2)11yescharacteristic
110C22×C712−1+411yescharacteristic
111C2×D71—24no—
112C2×D71—24no—
113C2×D71—24no—
114C2×D71—42no—
115C2×D71—42no—
116C2×D71—42no—
117C2×D71—42no—
118C4×C71—24no—
119C4×C71—42no—
120C4×C71—41no—
121C4×C71—41no—
122Dic71—24no—
123Dic71—24no—
124Dic71—42no—
125C2×C71—42no—
126C2×C71#52512yes—
127C2×C71#53911yescharacteristic
128C2×C71D4211yescharacteristic
129C2×C71#54712yes—
130C2×C71#55911yescharacteristic
131D71—44no—
132#2455—711nomaximal, Sylow
133C71#245511yescharacteristic, complemented, Sylow
134C2×#123—712no—
135C2×C4×D4—712no—
136#399D7111yescharacteristic, complemented
137#394—712no—
138#392—711no—
139#389—712no—
140#388—712no—
141#385—712no—
142#385—711no—
143C22×D4—712no—
144C22×D4—711no—
145C23×C4—711no—
146C23×C4—712no—
147#123—1428no—
148C4×D4—1428no—
149C2×C4⋊C4—712no—
150C2×C4⋊C4—712no—
151C2×C4⋊C4—712no—
152C2×C4⋊C4C2×D7112yes—
153C2×C4⋊C4C2×D7111yescharacteristic
154C2×C22⋊C4—712no—
155C2×C22⋊C4—711no—
156C2×C22⋊C4—712no—
157C2×C22⋊C4—712no—
158C2×C22⋊C4—712no—
159C2×C22⋊C4C2×D7112yes—
160C2×C22⋊C4C2×D7111yescharacteristic
161C2×C42—712no—
162#95—712no—
163#95—712no—
164#95—711no—
165#95C2×D7111yescharacteristic
166C24—712no—
167C24C22×D7111yescharacteristic
168C2×D4—2841no—
169C2×D4—1424no—
170C2×D4—1424no—
171C2×D4—1424no—
172C2×D4—1424no—
173C22×C4—2842no—
174C22×C4—2841no—
175C22×C4—712no—
176C22×C4—712no—
177C22×C4—712no—
178C22×C4—1424no—
179C22×C4—1424no—
180C22×C4—1424no—
181C22×C4C22×D7111yescharacteristic
182C22×C4C22×D7112yes—
183C22×C4C22×D7111yescharacteristic
184C22×C4C22×D7112yes—
185C4⋊C4—24no—
186C4⋊C4—41no—
187C4⋊C4—1424no—
188C4⋊C4—1424no—
189C4⋊C4—1424no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—718no—
193C22⋊C4—1424no—
194C22⋊C4—1424no—
195C22⋊C4—1422no—
196C22⋊C4—1424no—
197C42—1424no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—714no—
203C23—1422no—
204C23—1422no—
205C23—1422no—
206C23—1422no—
207C23—1422no—
208C23—1422no—
209C23#1176586612yes—
210C23C23×D7111yescharacteristic
211D4—2844no—
212D4—14216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—2848no—
223C2×C4—2844no—
224C2×C4—714no—
225C2×C4—714no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1422no—
233C2×C4D4×D7114yes—
234C22—22no—
235C22—24no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—2842no—
240C22—2842no—
241C22—2842no—
242C22—2842no—
243C22—1424no—
244C22—1424no—
245C22—1424no—
246C22C2×D4×D7112yes—
247C22#41×D7112yes—
248C22C2×#1176586611yescharacteristic
249C22#1455617311yescharacteristic
250C22#1455617511yescharacteristic
251C4—24no—
252C4—42no—
253C4—41no—
254C4—41no—
255C4—2842no—
256C4—1424no—
257C4—1424no—
258C2—42no—
259C2—2844no—
260C2#1798651212yes—
261C2#1798659811yescharacteristic
262C2#1798663411yescharacteristic
263C2#1798664712yes—
264C2#1798673311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2455
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14556113C2semidirect
2. C2×(C22⋊D71)⋊C4C2semidirect
3. #17985915C2semidirect
5. #17985895C2semidirect
6. #17985891C2semidirect
7. #17985871C2semidirect
8. #17985870C2semidirect
9. #399×C71C2semidirect
54. (C2×D71)⋊C4D4semidirect
133. C71#2455semidirect
136. #399D71semidirect