Finite Group Databasebeta

← #22592622

Subgroups — #22592622

Order 9088.

512 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14556086C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic
3#17985939C211yesmaximal, characteristic, complemented
4#17985938C211yesmaximal, characteristic, complemented
5#17985930C211yesmaximal, characteristic, complemented
6#17985885C212yesmaximal, complemented
7#17985881C212yesmaximal, complemented
8#17985870C212yesmaximal
9#17985861C212yesmaximal
10#17985861C211yesmaximal, characteristic
11#398×C71C211yesmaximal, characteristic, complemented
12C22×C4×D71C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15#14556086—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2211yescharacteristic
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C4⋊C4×D71—24no—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×C4×Dic71C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2212yes—
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C4⋊C4×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C22×Dic71C2311yescharacteristic
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C2×C4×D71—22no—
44C2×C4×D71—24no—
45C2×C4×D71—22no—
46C2×C4×D71—22no—
47C2×C4×D71—22no—
48C2×C4×D71—24no—
49C2×C4×D71D414yes—
50C4⋊Dic71—22no—
51C4⋊Dic71—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—24no—
55(C2×D71)⋊C4—24no—
56(C2×D71)⋊C4—24no—
57C4×Dic71—24no—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2312yes—
60C22×C4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—24no—
66C23×C71C2411yescharacteristic
67C22×D71—21no—
68C22×D71—21no—
69C22×D71—22no—
70C22×D71C2×D412yes—
71C22×D71#4114yes—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71C2×D412yes—
84C2×C4×C71#4112yes—
85C2×Dic71—21no—
86C2×Dic71—22no—
87C2×Dic71—21no—
88C2×Dic71—22no—
89C2×Dic71—21no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—24no—
94C2×Dic71—22no—
95C2×Dic71C2×D412yes—
96C2×Dic71#4112yes—
97C4×D71—28no—
98C4×D71—28no—
99C4×D71—44no—
100C22×C71C22×D411yescharacteristic
101C22×C71C2×#4112yes—
102C22×C71C2×#4111yescharacteristic
103C22×C71C2×#4111yescharacteristic
104C22×C712−1+412yes—
105C2×D71—22no—
106C2×D71—22no—
107C2×D71—24no—
108C2×D71—24no—
109C2×D71#12314yes—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—41no—
113C4×C71—42no—
114C4×C71—42no—
115Dic71—22no—
116Dic71—22no—
117Dic71—41no—
118Dic71—42no—
119Dic71—42no—
120C2×C71C2×#12311yescharacteristic
121C2×C71#52511yescharacteristic
122C2×C71#53611yescharacteristic
123C2×C71#55212yes—
124C2×C71#55212yes—
125D71—24no—
126#2484—711nomaximal, Sylow
127C71#248411yescharacteristic, complemented, Sylow
128C2×#117—711no—
129C22×C4⋊C4—711no—
130#400—711no—
131#400—711no—
132#398—711no—
133#398D7111yescharacteristic, complemented
134#387—712no—
135#387—712no—
136#385—712no—
137#381—711no—
138#381—712no—
139C23×C4—711no—
140C23×C4—711no—
141C23×C4—711no—
142#117—1424no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—711no—
146C2×C4⋊C4—711no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—1424no—
149C2×C4⋊C4C2×D7111yescharacteristic
150C2×C4⋊C4C2×D7111yescharacteristic
151C2×C4⋊C4C2×D7112yes—
152C2×C22⋊C4—712no—
153C2×C22⋊C4—712no—
154C2×C42—712no—
155#95—712no—
156#95—712no—
157#95—712no—
158#95—712no—
159#95—711no—
160#95—712no—
161#95—712no—
162#95C2×D7112yes—
163#95C2×D7111yescharacteristic
164C24—711no—
165C22×C4—711no—
166C22×C4—711no—
167C22×C4—714no—
168C22×C4—712no—
169C22×C4—712no—
170C22×C4—711no—
171C22×C4—1422no—
172C22×C4—1424no—
173C22×C4—1422no—
174C22×C4—1422no—
175C22×C4—1424no—
176C22×C4—1422no—
177C22×C4C22×D7112yes—
178C22×C4C22×D7111yescharacteristic
179C22×C4C22×D7112yes—
180C22×C4C22×D7111yescharacteristic
181C22×C4C22×D7111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—1424no—
186C4⋊C4—1422no—
187C4⋊C4—1422no—
188C4⋊C4—1422no—
189C4⋊C4—1422no—
190C22⋊C4—1424no—
191C22⋊C4—1424no—
192C42—1424no—
193C23—712no—
194C23—714no—
195C23—1421no—
196C23—1421no—
197C23—1422no—
198C23C23×D7111yescharacteristic
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—2844no—
211C2×C4—712no—
212C2×C4—712no—
213C2×C4—1428no—
214C2×C4—1428no—
215C2×C4—1422no—
216C2×C4—1421no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4—1424no—
221C2×C4—1422no—
222C2×C4—1421no—
223C2×C4—1422no—
224C2×C4—1421no—
225C2×C4D4×D7112yes—
226C2×C4#1176586412yes—
227C22—714no—
228C22—1422no—
229C22—1424no—
230C22—1424no—
231C22—1422no—
232C22C2×D4×D7111yescharacteristic
233C22#41×D7111yescharacteristic
234C22#41×D7112yes—
235C22C2×#1176586411yescharacteristic
236C22#1455617612yes—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—41no—
241C4—42no—
242C4—2842no—
243C4—2842no—
244C4—2841no—
245C4—1422no—
246C4—1422no—
247C2—1424no—
248C2#123×D7111yescharacteristic
249C2#1798650911yescharacteristic
250C2#1798657411yescharacteristic
251C2#1798668112yes—
252C2#1798668312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2484
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #17985939C2semidirect
4. #17985938C2semidirect
5. #17985930C2semidirect
6. #17985885C2semidirect
7. #17985881C2semidirect
11. #398×C71C2semidirect
127. C71#2484semidirect
133. #398D71semidirect