Finite Group Databasebeta

← #22592820

Subgroups — #22592820

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×#14556106C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic
3#17985939C211yesmaximal, characteristic, complemented
4#17985938C211yesmaximal, characteristic, complemented
5#17985937C211yesmaximal, characteristic, complemented
6#17985903C212yesmaximal, complemented
7#17985863C212yesmaximal, complemented
8#17985861C212yesmaximal
9#17985861C212yesmaximal
10#17985846C211yesmaximal, characteristic
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C4×D71C2212yes—
14C22×C4×D71C2211yescharacteristic
15#14556106—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×Dic71⋊C4C2211yescharacteristic
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C4⋊C4×D71—24no—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2211yescharacteristic
24C2×C4×Dic71C2212yes—
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—24no—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2312yes—
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C2×C4×D71—24no—
46C2×C4×D71—24no—
47C2×C4×D71—28no—
48C2×C4×D71D414yes—
49C4⋊Dic71—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—24no—
52Dic71⋊C4—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—24no—
56C4×Dic71—24no—
57C22×C4×C71C2312yes—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C23×C71C2411yescharacteristic
65C22×D71—22no—
66C22×D71—21no—
67C22×D71—21no—
68C22×D71C2×D412yes—
69C22×D71#4114yes—
70C2×C4×C71—22no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—24no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71C2×D412yes—
82C2×Dic71—21no—
83C2×Dic71—21no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—24no—
88C2×Dic71—22no—
89C2×Dic71—21no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71C2×D412yes—
93C2×Dic71#4114yes—
94C4×D71—28no—
95C4×D71—48no—
96Dic142—44no—
97C22×C71C22×D411yescharacteristic
98C22×C71C2×#4112yes—
99C22×C71C2×#4112yes—
100C22×C712−1+411yescharacteristic
101C22×C712−1+411yescharacteristic
102C2×D71—22no—
103C2×D71—22no—
104C2×D71—24no—
105C2×D71—24no—
106C2×D71#12414yes—
107C4×C71—22no—
108C4×C71—41no—
109C4×C71—42no—
110C4×C71—41no—
111C4×C71—42no—
112Dic71—41no—
113Dic71—42no—
114Dic71—41no—
115Dic71—42no—
116Dic71#12314yes—
117C2×C71C2×#12311yescharacteristic
118C2×C71C2×#12411yescharacteristic
119C2×C71#53611yescharacteristic
120C2×C71#55212yes—
121C2×C71#55212yes—
122D71—24no—
123#2503—711nomaximal, Sylow
124C71#250311yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—711no—
126C22×C4⋊C4—711no—
127#400—711no—
128#400—711no—
129#400—711no—
130#400D7111yescharacteristic, complemented
131#391—712no—
132#382—712no—
133#381—712no—
134#381—712no—
135C4×C22⋊C4—711no—
136C22×Q8—711no—
137C23×C4—711no—
138C23×C4—712no—
139C22⋊Q8—1424no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4—711no—
142C2×C4⋊C4—711no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—1424no—
145C2×C4⋊C4C2×D7111yescharacteristic
146C2×C4⋊C4C2×D7111yescharacteristic
147C2×C22⋊C4—711no—
148C2×C22⋊C4—711no—
149C2×C22⋊C4—712no—
150C2×C42—712no—
151#95—711no—
152#95—712no—
153#95—712no—
154#95—712no—
155#95—712no—
156#95—712no—
157#95—712no—
158#95C2×D7111yescharacteristic
159#95C2×D7112yes—
160#95C2×D7112yes—
161C24—711no—
162C2×Q8—1422no—
163C2×Q8—1424no—
164C22×C4—711no—
165C22×C4—714no—
166C22×C4—712no—
167C22×C4—711no—
168C22×C4—711no—
169C22×C4—712no—
170C22×C4—1424no—
171C22×C4—1428no—
172C22×C4—1424no—
173C22×C4C22×D7112yes—
174C22×C4C22×D7111yescharacteristic
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7112yes—
177C22×C4C22×D7111yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—1424no—
181C4⋊C4—1422no—
182C4⋊C4—1422no—
183C4⋊C4—1422no—
184C22⋊C4—1424no—
185C22⋊C4—1422no—
186C22⋊C4—1422no—
187C42—1424no—
188C23—712no—
189C23—714no—
190C23—1422no—
191C23—1421no—
192C23—1421no—
193C23C23×D7111yescharacteristic
194Q8—2844no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—2848no—
207C2×C4—712no—
208C2×C4—714no—
209C2×C4—1428no—
210C2×C4—1421no—
211C2×C4—1421no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1424no—
216C2×C4—1422no—
217C2×C4—1421no—
218C2×C4—1421no—
219C2×C4—1422no—
220C2×C4D4×D7112yes—
221C22—714no—
222C22—1422no—
223C22—1424no—
224C22—1424no—
225C22—1422no—
226C22C2×D4×D7111yescharacteristic
227C22#41×D7112yes—
228C22#41×D7112yes—
229C22#1455617511yescharacteristic
230C22#1455617611yescharacteristic
231C4—22no—
232C4—41no—
233C4—42no—
234C4—41no—
235C4—42no—
236C4—2841no—
237C4—2842no—
238C4—2841no—
239C4—2842no—
240C4—714no—
241C2—1424no—
242C2#123×D7111yescharacteristic
243C2#124×D7111yescharacteristic
244C2#1798657211yescharacteristic
245C2#1798668112yes—
246C2#1798668512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2503
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #17985939C2semidirect
4. #17985938C2semidirect
5. #17985937C2semidirect
6. #17985903C2semidirect
7. #17985863C2semidirect
11. #400×C71C2semidirect
124. C71#2503semidirect
130. #400D71semidirect