Finite Group Databasebeta

← #323×D71 #22586645

Subgroups — #323×D71 #22586645

Order 9088.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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×C22⋊C4×D71C211yesmaximal, characteristic
2C22⋊C8×D71C212yesmaximal, complemented
3#17985723C211yesmaximal, characteristic, complemented
4#17985722C212yesmaximal, complemented
5#323×C71C211yesmaximal, characteristic, complemented
6C24×D71C411yescharacteristic
7C22×C4×D71C411yescharacteristic
8C22×C4×D71C2211yescharacteristic
9C2×C8×D71—22no—
10C2×C22⋊Dic71C411yescharacteristic
11C2×(C2×D71)⋊C4C411yescharacteristic
12C2×(C2×D71)⋊C4C2211yescharacteristic
13C22⋊C4×D71—22no—
14C22⋊C4×D71—22no—
15C22⋊(C71⋊C8)C2212yescomplemented
16(C2×D71)⋊C8—22no—
17C2×C22⋊C4×C71C2211yescharacteristic
18C22⋊C8×C71C2212yescomplemented
19C23×D71—21no—
20C23×D71—21no—
21C23×D71—22no—
22C23×D71—42no—
23C23×D71—42no—
24C23×D71C814yescomplemented
25C23×D71C2×C411yescharacteristic
26C23×D71C2×C411yescharacteristic
27C22×Dic71C2×C411yescharacteristic
28C22×Dic71C2×C411yescharacteristic
29C2×C4×D71—21no—
30C2×C4×D71—21no—
31C2×C4×D71—42no—
32C2×C4×D71—42no—
33C2×C4×D71D412yes—
34C2×C4×D71D412yes—
35C2×C71⋊C8—22no—
36C8×D71—44no—
37C22⋊Dic71—22no—
38(C2×D71)⋊C4—22no—
39(C2×D71)⋊C4—22no—
40C24×C71C2×C411yescharacteristic
41C22×C4×C71C2×C411yescharacteristic
42C22×C4×C71C2311yescharacteristic
43C2×C8×C71—22no—
44C22⋊C4×C71—22no—
45C23×C71—21no—
46C23×C71—21no—
47C23×C71—41no—
48C23×C71—41no—
49C23×C71C2×C812yescomplemented
50C23×C71C22×C411yescharacteristic
51C22×D71—22no—
52C22×D71—22no—
53C22×D71—21no—
54C22×D71—21no—
55C22×D71—24no—
56C22×D71—24no—
57C22×D71—42no—
58C22×D71—42no—
59C22×D71—48no—
60C22×D71—48no—
61C22×D71—42no—
62C22×D71—42no—
63C22×D71—41no—
64C22×D71—41no—
65C22×D71C22⋊C411yescharacteristic
66C22×D71C22⋊C411yescharacteristic
67C22×D71C2×C812yes—
68C22×D71C2×C812yescomplemented
69C22×D71M1612yes—
70C2×C4×C71—21no—
71C2×C4×C71—41no—
72C2×C4×C71—41no—
73C2×C4×C71C2×D412yes—
74C2×Dic71—21no—
75C2×Dic71—41no—
76C2×Dic71—41no—
77C2×Dic71C22⋊C412yes—
78C4×D71—24no—
79C4×D71—42no—
80C4×D71—42no—
81C4×D71—42no—
82C8×C71—42no—
83C71⋊C8—42no—
84C22×C71—21no—
85C22×C71—21no—
86C22×C71—22no—
87C22×C71—42no—
88C22×C71—42no—
89C22×C71—41no—
90C22×C71—41no—
91C22×C71C2×C22⋊C411yescharacteristic
92C22×C71C22×C811yescharacteristic
93C22×C71C2×M1611yescharacteristic
94C2×D71—21no—
95C2×D71—21no—
96C2×D71—24no—
97C2×D71—24no—
98C2×D71—22no—
99C2×D71—48no—
100C2×D71—42no—
101C2×D71—41no—
102C2×D71—41no—
103C2×D71—48no—
104C2×D71—42no—
105C2×D71C22⋊C812yes—
106C2×D71C22⋊C812yes—
107C2×D71#9912yes—
108C2×D71#10012yes—
109C4×C71—22no—
110C4×C71—41no—
111Dic71—22no—
112Dic71—41no—
113C2×C71—21no—
114C2×C71—21no—
115C2×C71—42no—
116C2×C71C2×C22⋊C811yescharacteristic
117C2×C71C2×#9911yescharacteristic
118C2×C71C2×#10011yescharacteristic
119D71—22no—
120D71—42no—
121D71#32314yesdirect product factor
122C2×#323—711nomaximal, Sylow
123C71C2×#32311yescharacteristic, complemented, Sylow
124C22×C22⋊C4—711no—
125C2×C22⋊C8—712no—
126#323—711no—
127#323—712no—
128#323D7111yescharacteristic, direct product factor
129C25—711no—
130C23×C4—711no—
131C23×C4—711no—
132C22×C8—1422no—
133C2×C22⋊C4—711no—
134C2×C22⋊C4—711no—
135C2×C22⋊C4—711no—
136C2×C22⋊C4—1422no—
137C2×C22⋊C4—1422no—
138C2×C22⋊C4C2×D7111yescharacteristic
139C22⋊C8—712no—
140C22⋊C8—1422no—
141C22⋊C8C2×D7112yescomplemented
142C24—2842no—
143C24—2842no—
144C24—714no—
145C24—711no—
146C24—711no—
147C24—1421no—
148C24—1421no—
149C24—1422no—
150C24C4×D7111yescharacteristic
151C22×C4—2842no—
152C22×C4—2842no—
153C22×C4—711no—
154C22×C4—712no—
155C22×C4—711no—
156C22×C4—712no—
157C22×C4—1421no—
158C22×C4—1421no—
159C22×C4C4×D7111yescharacteristic
160C22×C4C22×D7111yescharacteristic
161C2×C8—22no—
162C2×C8—2844no—
163C2×C8—1422no—
164C22⋊C4—22no—
165C22⋊C4—1422no—
166C22⋊C4—1422no—
167C22⋊C4—1422no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—2841no—
173C23—2841no—
174C23—2842no—
175C23—2842no—
176C23—2848no—
177C23—2842no—
178C23—2842no—
179C23—2848no—
180C23—712no—
181C23—711no—
182C23—711no—
183C23—712no—
184C23—712no—
185C23—1424no—
186C23—1424no—
187C23—1422no—
188C23—1422no—
189C23—1421no—
190C23—1421no—
191C23C8×D7112yescomplemented
192C23C2×C4×D7111yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—2842no—
197C2×C4—2842no—
198C2×C4—2842no—
199C2×C4—2841no—
200C2×C4—2841no—
201C2×C4—712no—
202C2×C4—1424no—
203C2×C4—1421no—
204C2×C4D4×D7112yes—
205C8—42no—
206C8—2842no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—2842no—
215C22—2842no—
216C22—2848no—
217C22—2848no—
218C22—2841no—
219C22—2841no—
220C22—712no—
221C22—712no—
222C22—712no—
223C22—712no—
224C22—1424no—
225C22—1424no—
226C22—1422no—
227C22—1421no—
228C22—1421no—
229C22C22⋊C4×D7111yescharacteristic
230C22C2×C8×D7111yescharacteristic
231C22M16×D7111yescharacteristic
232C4—22no—
233C4—41no—
234C4—2841no—
235C4—1422no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—2842no—
240C2—714no—
241C2—1422no—
242C2C22⋊C8×D7111yescharacteristic
243C2#99×D7111yescharacteristic
244C2#100×D7111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC22×C71C22×C8
FrattiniC22×C4C22×D71
Fitting#323×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#323D71
hypocenterC71C2×#323
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D71C2semidirect
3. #17985723C2semidirect
4. #17985722C2semidirect
5. #323×C71C2semidirect
15. C22⋊(C71⋊C8)C22semidirect
18. C22⋊C8×C71C22semidirect
24. C23×D71C8semidirect
49. C23×C71C2×C8semidirect
68. C22×D71C2×C8semidirect
121. D71#323direct (N × H)
123. C71C2×#323semidirect
128. #323D71direct (N × H)
141. C22⋊C8C2×D71semidirect
191. C23C8×D71semidirect