Finite Group Databasebeta

← #22595178

Subgroups — #22595178

Order 9088.

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

normal subgroups only

This group has 512 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×#14556115C211yesmaximal, characteristic
2C2×#14556109C211yesmaximal, characteristic
3#17985924C212yesmaximal, complemented
4#17985924C212yesmaximal, complemented
5#17985920C212yesmaximal, complemented
6#17985884C212yesmaximal, complemented
7#17985874C212yesmaximal, complemented
8#17985859C212yesmaximal, complemented
9#386×C71C211yesmaximal, characteristic
10C22×Dic142C2211yescharacteristic
11C23×Dic71C2211yescharacteristic
12#14556115—24no—
13#14556109—24no—
14C2×C4⋊Dic71C2212yes—
15C2×C4⋊Dic71C2212yes—
16C2×C4⋊Dic71C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×C22⋊Dic71C2212yes—
19C2×C22⋊Dic71C2212yes—
20C2×C22⋊Dic71C2212yes—
21C2×C22⋊Dic71C2211yescharacteristic
22C2×C4×Dic71C2211yescharacteristic
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C22×D4×C71C2211yescharacteristic
29C23×C4×C71C2211yescharacteristic
30C2×C22⋊C4×C71C2212yes—
31C2×C22⋊C4×C71C2211yescharacteristic
32#95×C71C2212yes—
33C2×Dic142—22no—
34C2×Dic142—41no—
35C2×Dic142—41no—
36C22×Dic71—22no—
37C22×Dic71—41no—
38C22×Dic71—41no—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C4⋊Dic71—24no—
44C4⋊Dic71—42no—
45C4⋊Dic71—42no—
46Dic71⋊C4—24no—
47C22⋊Dic71—22no—
48C22⋊Dic71—24no—
49C22⋊Dic71—24no—
50C22⋊Dic71—24no—
51C4×Dic71—22no—
52C24×C71C2311yescharacteristic
53C24×C71C2311yescharacteristic
54C2×D4×C71—41no—
55C2×D4×C71—41no—
56C2×D4×C71D414yes—
57C22×C4×C71—24no—
58C22×C4×C71—41no—
59C22×C4×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2312yes—
62C22⋊C4×C71—24no—
63C22⋊C4×C71—41no—
64C23×C71—22no—
65C23×C71—22no—
66C23×C71—21no—
67C23×C71—21no—
68C23×C71—21no—
69C23×C71—21no—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71C2×D412yes—
73C23×C71C2×D412yes—
74C23×C71C2411yescharacteristic
75D4×C71—44no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—44no—
84C2×C4×C71C2×D412yes—
85C2×C4×C71#4114yes—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—44no—
99Dic142—44no—
100C22×C71—22no—
101C22×C71—22no—
102C22×C71—24no—
103C22×C71—24no—
104C22×C71—42no—
105C22×C71—42no—
106C22×C71—41no—
107C22×C71—41no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71C22×D411yescharacteristic
111C22×C71C2×#4112yes—
112C22×C71H(5,𝔽2)11yescharacteristic
113C22×C71H(5,𝔽2)11yescharacteristic
114C22×C712−1+412yes—
115C4×C71—22no—
116C4×C71—24no—
117C4×C71—42no—
118Dic71—42no—
119Dic71—42no—
120Dic71—42no—
121Dic71—42no—
122C2×C71—42no—
123C2×C71—42no—
124C2×C71#53411yescharacteristic
125C2×C71#54212yes—
126C2×C71#54712yes—
127C2×C71#55111yescharacteristic
128C2×C71#56311yescharacteristic
129#2717—711nomaximal, Sylow
130C71#271711yescharacteristic, complemented, Sylow
131C2×#124—711no—
132C2×#123—711no—
133#397—712no—
134#397—712no—
135#395—712no—
136#387—712no—
137#386D7111yescharacteristic
138#385—712no—
139#380—712no—
140C22×Q8—711no—
141C22×D4C2×D7111yescharacteristic
142C23×C4—711no—
143C23×C4C2×D7111yescharacteristic
144#124—1424no—
145#123—1424no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—712no—
149C2×C4⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—711no—
152C2×C22⋊C4—712no—
153C2×C22⋊C4—712no—
154C2×C22⋊C4C2×D7112yes—
155C2×C22⋊C4C2×D7111yescharacteristic
156C2×C42—711no—
157#95—712no—
158#95—712no—
159#95—712no—
160#95—712no—
161#95—712no—
162#95C2×D7112yes—
163C24C22×D7111yescharacteristic
164C24C22×D7111yescharacteristic
165C2×Q8—2841no—
166C2×Q8—2841no—
167C2×Q8—1422no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4D28414yes—
171C22×C4—24no—
172C22×C4—41no—
173C22×C4—2841no—
174C22×C4—2841no—
175C22×C4—712no—
176C22×C4—712no—
177C22×C4—712no—
178C22×C4—712no—
179C22×C4—1422no—
180C22×C4C22×D7112yes—
181C22×C4C22×D7111yescharacteristic
182C22×C4C22×D7112yes—
183C4⋊C4—2842no—
184C4⋊C4—2842no—
185C4⋊C4—1424no—
186C4⋊C4—1424no—
187C22⋊C4—24no—
188C22⋊C4—41no—
189C22⋊C4—1422no—
190C22⋊C4—1424no—
191C22⋊C4—1424no—
192C22⋊C4—1424no—
193C42—1422no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×D28412yes—
203C23C2×D28412yes—
204C23C23×D7111yescharacteristic
205Q8—2844no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—2844no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4C2×D28412yes—
229C2×C4#1176586414yes—
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—42no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22C22×D28411yescharacteristic
241C22C2×#1176586412yes—
242C22#1455617311yescharacteristic
243C22#1455617311yescharacteristic
244C22#1455617512yes—
245C4—22no—
246C4—24no—
247C4—42no—
248C4—2842no—
249C4—2842no—
250C4—2842no—
251C4—2842no—
252C2—42no—
253C2—42no—
254C2#1798656111yescharacteristic
255C2#1798661612yes—
256C2#1798665312yes—
257C2#1798668011yescharacteristic
258C2#1798675211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#386×C71C2
hypercenter#386D71
hypocenterC71#2717
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #17985924C2semidirect
4. #17985924C2semidirect
5. #17985920C2semidirect
6. #17985884C2semidirect
7. #17985874C2semidirect
8. #17985859C2semidirect
130. C71#2717semidirect