Finite Group Databasebeta

← #22594207

Subgroups — #22594207

Order 9088.

512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556118C211yesmaximal, characteristic, complemented
2C2×D4×Dic71C211yesmaximal, characteristic
3#17985935C212yesmaximal, complemented
4#17985931C211yesmaximal, characteristic, complemented
5#17985929C211yesmaximal, characteristic, complemented
6#17985927C212yesmaximal, complemented
7#17985913C211yesmaximal, characteristic
8#17985874C212yesmaximal, complemented
9#17985871C212yesmaximal, complemented
10#17985866C211yesmaximal, characteristic, complemented
11#394×C71C211yesmaximal, characteristic
12C23×Dic71C2212yes—
13#14556118—24no—
14D4×Dic71—24no—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×Dic71⋊C4C2212yes—
23C2×C22⋊Dic71C2212yes—
24C2×C22⋊Dic71C2212yes—
25C2×C22⋊Dic71C2212yes—
26C2×C4×Dic71C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C22×D4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35C2×C22⋊C4×C71C2212yes—
36#95×C71C2211yescharacteristic
37C22×Dic71—24no—
38C22×Dic71—42no—
39C22×Dic71—42no—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2311yescharacteristic
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—42no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—42no—
54C22⋊Dic71—24no—
55C22⋊Dic71—24no—
56C22⋊Dic71—42no—
57C4×Dic71D414yes—
58C24×C71C2312yes—
59C2×D4×C71—24no—
60C2×D4×C71—22no—
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C22⋊C4×C71—24no—
67C22⋊C4×C71—42no—
68C23×C71—22no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71#4114yes—
75C23×C71C2411yescharacteristic
76D4×C71—28no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71C2×D412yes—
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—21no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—48no—
102C2×Dic71C2×D412yes—
103C2×Dic71C2×D412yes—
104C22×C71—24no—
105C22×C71—24no—
106C22×C71—24no—
107C22×C71—42no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71—42no—
111C22×C71C22×D411yescharacteristic
112C22×C71C2×#4112yes—
113C22×C71H(5,𝔽2)11yescharacteristic
114C22×C71H(5,𝔽2)11yescharacteristic
115C22×C712−1+412yes—
116C4×C71—22no—
117C4×C71—41no—
118C4×C71—41no—
119C4×C71—42no—
120Dic71—22no—
121Dic71—22no—
122Dic71—41no—
123Dic71—41no—
124Dic71—42no—
125Dic71—42no—
126C2×C71—44no—
127C2×C71#52511yescharacteristic
128C2×C71#53511yescharacteristic
129C2×C71#53611yescharacteristic
130C2×C71#53912yes—
131C2×C71#53912yes—
132#2633—711nomaximal, Sylow
133C71#263311yescharacteristic, complemented, Sylow
134C2×#125—711no—
135C2×C4×D4—711no—
136#399—712no—
137#398—711no—
138#398—711no—
139#397—712no—
140#394—711no—
141#394D7111yescharacteristic
142#385—712no—
143#385—712no—
144#383—711no—
145C22×D4C2×D7111yescharacteristic
146C23×C4—712no—
147#125—1424no—
148C4×D4—1424no—
149C2×C4⋊C4—712no—
150C2×C4⋊C4—712no—
151C2×C4⋊C4—711no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—711no—
154C2×C4⋊C4—711no—
155C2×C4⋊C4—711no—
156C2×C4⋊C4—711no—
157C2×C4⋊C4C2×D7111yescharacteristic
158C2×C22⋊C4—712no—
159C2×C22⋊C4—712no—
160C2×C22⋊C4—712no—
161C2×C22⋊C4C2×D7112yes—
162C2×C22⋊C4C2×D7112yes—
163C2×C42—711no—
164#95—712no—
165#95—711no—
166#95—712no—
167#95—712no—
168#95—712no—
169#95C2×D7111yescharacteristic
170C24C22×D7112yes—
171C2×D4—24no—
172C2×D4—22no—
173C22×C4—2842no—
174C22×C4—2842no—
175C22×C4—711no—
176C22×C4—711no—
177C22×C4—712no—
178C22×C4—711no—
179C22×C4—712no—
180C22×C4—711no—
181C22×C4—1424no—
182C22×C4C22×D7112yes—
183C22×C4C22×D7111yescharacteristic
184C22×C4C22×D7111yescharacteristic
185C22×C4C22×D7111yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—2842no—
188C4⋊C4—2842no—
189C4⋊C4—1422no—
190C4⋊C4—1422no—
191C4⋊C4—1422no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—2842no—
198C22⋊C4—1424no—
199C22⋊C4—1424no—
200C42—714no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23#1176586414yes—
208C23C23×D7111yescharacteristic
209D4—28no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—2848no—
220C2×C4—712no—
221C2×C4—712no—
222C2×C4—1422no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1421no—
228C2×C4—1422no—
229C2×C4—1421no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1422no—
233C2×C4—1421no—
234C2×C4—1421no—
235C2×C4—1422no—
236C2×C4D4×D7112yes—
237C22—24no—
238C22—24no—
239C22—24no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D7111yescharacteristic
245C22C2×#1176586412yes—
246C22#1455617311yescharacteristic
247C22#1455617411yescharacteristic
248C22#1455617512yes—
249C4—22no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—2842no—
254C4—2841no—
255C4—2842no—
256C4—2841no—
257C4—1422no—
258C4—1422no—
259C2—44no—
260C2#1798650511yescharacteristic
261C2#1798656511yescharacteristic
262C2#1798656911yescharacteristic
263C2#1798659312yes—
264C2#1798659412yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#394×C71C2
hypercenter#394D71
hypocenterC71#2633
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14556118C2semidirect
3. #17985935C2semidirect
4. #17985931C2semidirect
5. #17985929C2semidirect
6. #17985927C2semidirect
8. #17985874C2semidirect
9. #17985871C2semidirect
10. #17985866C2semidirect
133. C71#2633semidirect