Finite Group Databasebeta

← #22586676

Subgroups — #22586676

Order 9088.

316 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 77 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×C4×C22⋊D71C211yesmaximal, characteristic
2#17985963C211yesmaximal, characteristic, complemented
3#17985724C212yesmaximal, complemented
4#17985722C212yesmaximal, complemented
5#407×C71C211yesmaximal, characteristic, complemented
6C22×C22⋊D71C411yescharacteristic
7C22×C4×D71C411yescharacteristic
8C2×#11765849—21no—
9C4×C22⋊D71—21no—
10C4×C22⋊D71—21no—
11C4×C22⋊D71—21no—
12C4×C22⋊D71—21no—
13C2×Dic71⋊C4C2211yescharacteristic
14C2×C22⋊Dic71C411yescharacteristic
15C2×(C2×D71)⋊C4C2211yescharacteristic
16C2×C4×Dic71C411yescharacteristic
17C22⋊(C71⋊C8)—21no—
18C22⋊(C71⋊C8)C2212yes—
19C23×C4×C71C2211yescharacteristic
20C2×M16×C71—21no—
21C22⋊C8×C71—21no—
22C22⋊C8×C71C2212yes—
23C23×D71C2×C411yescharacteristic
24C2×C22⋊D71—21no—
25C2×C22⋊D71—21no—
26C2×C22⋊D71—22no—
27C2×C22⋊D71—22no—
28C22×Dic71C2×C411yescharacteristic
29C22×Dic71C2×C411yescharacteristic
30C22×Dic71C2×C411yescharacteristic
31C2×C4×D71—21no—
32C2×C4×D71—21no—
33C2×C4×D71—42no—
34#11765849—42no—
35C2×C71⋊C8—22no—
36Dic71⋊C4—21no—
37Dic71⋊C4—21no—
38C22⋊Dic71—21no—
39C22⋊Dic71—21no—
40(C2×D71)⋊C4—21no—
41(C2×D71)⋊C4—21no—
42C4×Dic71—21no—
43C4×Dic71—21no—
44C24×C71C2×C411yescharacteristic
45C22×C4×C71—42no—
46C22×C4×C71C2×C411yescharacteristic
47C22×C4×C71D411yescharacteristic
48C22×C4×C71D411yescharacteristic
49C22×C4×C71D411yescharacteristic
50C22×C4×C71D411yescharacteristic
51C22×C4×C71C2311yescharacteristic
52M16×C71—42no—
53C2×C8×C71—22no—
54C23×C71—21no—
55C23×C71—21no—
56C23×C71—41no—
57C23×C71—41no—
58C23×C71C22⋊C411yescharacteristic
59C23×C71C22⋊C411yescharacteristic
60C23×C71C22×C411yescharacteristic
61C22×D71—21no—
62C22×D71—21no—
63C22×D71—41no—
64C22×D71—41no—
65C22×D71M1612yescomplemented
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—41no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71C22⋊C411yescharacteristic
73C2×C4×C71C22⋊C411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C22⋊D71—24no—
77C22⋊D71—24no—
78C2×Dic71—21no—
79C2×Dic71—21no—
80C2×Dic71—21no—
81C2×Dic71—21no—
82C2×Dic71—41no—
83C2×Dic71—41no—
84C2×Dic71M1612yescomplemented
85C4×D71—42no—
86C4×D71—42no—
87C8×C71—42no—
88C71⋊C8—42no—
89C22×C71—21no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71—21no—
93C22×C71—21no—
94C22×C71—21no—
95C22×C71—21no—
96C22×C71—21no—
97C22×C71—42no—
98C22×C71—42no—
99C22×C71C2×C22⋊C411yescharacteristic
100C22×C71C2×M1611yescharacteristic
101C22×C71C2×M1611yescharacteristic
102C2×D71—22no—
103C2×D71—42no—
104C2×D71—41no—
105C2×D71—41no—
106C2×D71—42no—
107C4×C71—21no—
108C4×C71—21no—
109C4×C71—21no—
110C4×C71#9912yes—
111Dic71—22no—
112Dic71—41no—
113Dic71—41no—
114C2×C71—21no—
115C2×C71—21no—
116C2×C71—41no—
117C2×C71—41no—
118C2×C71#40711yescharacteristic
119C2×C71C2×#9911yescharacteristic
120C2×C71#41311yescharacteristic
121D71—42no—
122#1465—711nomaximal, Sylow
123C71#146511yescharacteristic, complemented, Sylow
124C2×C4×D4—711no—
125#407—711no—
126#407D7111yescharacteristic, complemented
127#324—712no—
128#323—712no—
129C22×D4—711no—
130C23×C4—711no—
131C23×C4C2×D7111yescharacteristic
132C2×M16—21no—
133C2×M16—1421no—
134C4×D4—1421no—
135C4×D4—1421no—
136C4×D4—1421no—
137C4×D4—1421no—
138C2×C4⋊C4—711no—
139C2×C22⋊C4—711no—
140C2×C22⋊C4—711no—
141C2×C42—711no—
142C22⋊C8—21no—
143C22⋊C8—712no—
144C22⋊C8—1421no—
145C22⋊C8C2×D7112yes—
146C24—711no—
147C24C4×D7111yescharacteristic
148C2×D4—1422no—
149C2×D4—1422no—
150C2×D4—1421no—
151C2×D4—1421no—
152C22×C4—42no—
153C22×C4—2842no—
154C22×C4—711no—
155C22×C4—711no—
156C22×C4—711no—
157C22×C4—1421no—
158C22×C4—1421no—
159C22×C4C4×D7111yescharacteristic
160C22×C4D28411yescharacteristic
161C22×C4D28411yescharacteristic
162C22×C4C22⋊D7111yescharacteristic
163C22×C4C22⋊D7111yescharacteristic
164C22×C4C22×D7111yescharacteristic
165M16—42no—
166M16—2842no—
167C2×C8—22no—
168C2×C8—1422no—
169C4⋊C4—1421no—
170C4⋊C4—1421no—
171C22⋊C4—1421no—
172C22⋊C4—1421no—
173C22⋊C4—1421no—
174C22⋊C4—1421no—
175C42—1421no—
176C42—1421no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—2841no—
182C23—2841no—
183C23—712no—
184C23—1421no—
185C23—1421no—
186C23(C2×D71)⋊C411yescharacteristic
187C23(C2×D71)⋊C411yescharacteristic
188C23C2×C4×D7111yescharacteristic
189D4—1424no—
190D4—1424no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—2842no—
198C2×C4—2842no—
199C2×C4—2841no—
200C2×C4—2841no—
201C2×C4—712no—
202C2×C4—1421no—
203C2×C4—1421no—
204C2×C4—1421no—
205C2×C4—1421no—
206C2×C4(C2×D71)⋊C411yescharacteristic
207C2×C4(C2×D71)⋊C411yescharacteristic
208C2×C4C2×C22⋊D7111yescharacteristic
209C2×C4C2×D28411yescharacteristic
210C8—42no—
211C8—2842no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—2842no—
223C22—2842no—
224C22—2841no—
225C22—2841no—
226C22—1422no—
227C22C2×(C2×D71)⋊C411yescharacteristic
228C22M16×D7111yescharacteristic
229C22M16×D7111yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—2841no—
234C4—2841no—
235C4—1422no—
236C4#1455604212yes—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2—2842no—
242C2#1798596211yescharacteristic
243C2C2×#1455604211yescharacteristic
244C2#1798598111yescharacteristic
subgroupG/H
centerC22C2×(C2×D71)⋊C4
commutatorC23×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#407×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#407D71
hypocenterC71#1465
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #17985963C2semidirect
3. #17985724C2semidirect
4. #17985722C2semidirect
5. #407×C71C2semidirect
65. C22×D71M16semidirect
84. C2×Dic71M16semidirect
123. C71#1465semidirect
126. #407D71semidirect