Finite Group Databasebeta

← C2×#17986362 #22598673

Subgroups — C2×#17986362 #22598673

Order 9088.

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

normal subgroups only

This group has 488 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×#14556124C211yesmaximal, characteristic
2C2×#14556119C211yesmaximal, characteristic, complemented
3#17986362C218yesmaximal, direct product factor
4C2×#14556050C211yesmaximal, characteristic, complemented
5C2×#14556049C211yesmaximal, characteristic, complemented
6C2×#14556048C211yesmaximal, characteristic, complemented
7C2×#14556038C211yesmaximal, characteristic, complemented
8C2×#102×C71C211yesmaximal, characteristic, complemented
9C22×D284C2211yescharacteristic
10C22×C22⋊D71—21no—
11C22×C22⋊D71—21no—
12C22×C71⋊C8C2211yescharacteristic
13#14556124—22no—
14#14556124C2214yes—
15#14556119—22no—
16#14556119C2214yescomplemented
17C2×C4⋊Dic71—21no—
18C2×C4⋊Dic71C2211yescharacteristic
19C2×Dic71⋊C4—21no—
20C2×C4×Dic71C2211yescharacteristic
21#14556050C2214yescomplemented
22#14556049C2214yescomplemented
23#14556048C2214yescomplemented
24#14556038C2214yescomplemented
25C22×D4×C71C2211yescharacteristic
26C22×C8×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28#102×C71C2214yescomplemented
29C23×D71—21no—
30C2×D284—22no—
31C2×D284—22no—
32C2×D284—21no—
33C2×D284C2312yes—
34C2×C22⋊D71—24no—
35C2×C22⋊D71—24no—
36C2×C22⋊D71—44no—
37C2×C22⋊D71—44no—
38C22×Dic71—21no—
39C22×Dic71D411yescharacteristic
40C22×Dic71D411yescharacteristic
41C2×C71⋊C8—22no—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71—21no—
44C4⋊Dic71—24no—
45C4⋊Dic71C2312yes—
46Dic71⋊C4—24no—
47C4×Dic71D412yes—
48C4×Dic71C2312yes—
49C24×C71—21no—
50C2×D4×C71—21no—
51C2×D4×C71—22no—
52C2×D4×C71—22no—
53C2×D4×C71C2312yes—
54C22×C4×C71—21no—
55C22×C4×C71C2311yescharacteristic
56C2×C8×C71—22no—
57C2×C8×C71C2312yes—
58C4⋊C4×C71—21no—
59C4⋊C4×C71C2312yes—
60C23×C71—22no—
61C23×C71—42no—
62C23×C71—42no—
63C23×C71—42no—
64C23×C71C2×D411yescharacteristic
65C22×D71—22no—
66C22×D71—42no—
67C22×D71—42no—
68C22×D71—42no—
69D4×C71—22no—
70D4×C71—22no—
71D4×C71—42no—
72C2×C4×C71—22no—
73C2×C4×C71—42no—
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71#4112yes—
76C2×C4×C71#4112yes—
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—48no—
79C22⋊D71—48no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—42no—
84C2×Dic71C2×D412yes—
85C2×Dic71C2×D412yes—
86D284—22no—
87D284—22no—
88D284—42no—
89C8×C71—22no—
90C71⋊C8—22no—
91C22×C71—42no—
92C22×C71—42no—
93C22×C71—42no—
94C22×C71—84no—
95C22×C71#12412yes—
96C22×C71AGL(1,ℤ/8)12yes—
97C22×C71AGL(1,ℤ/8)12yes—
98C22×C71C22×D411yescharacteristic
99C2×D71—42no—
100C2×D71—42no—
101C2×D71—42no—
102C2×D71—84no—
103C4×C71—42no—
104C4×C71#12412yes—
105C4×C71C2×#4111yescharacteristic
106C4×C71C2×#4111yescharacteristic
107Dic71—22no—
108Dic71—22no—
109Dic71—42no—
110C2×C71—82no—
111C2×C71#49014yescomplemented
112C2×C71C2×#12411yescharacteristic
113C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
115D71—82no—
116C2×#490—711nomaximal, Sylow
117C71C2×#49011yescharacteristic, complemented, Sylow
118C2×#127—711no—
119C2×#125—711no—
120#490—718no—
121C2×#102—711no—
122C2×#102—711no—
123C2×#102—711no—
124C2×#102D7111yescharacteristic, complemented
125C2×#97—711no—
126C22×D4—711no—
127C22×D4—1421no—
128C22×D4—1421no—
129C22×D4C2×D7111yescharacteristic
130C22×C8—711no—
131C22×C8C2×D7111yescharacteristic
132#127—714no—
133#127—1422no—
134#125—714no—
135#125—1422no—
136C2×C4⋊C4—711no—
137C2×C4⋊C4—1421no—
138C2×C4⋊C4—1421no—
139C2×C4⋊C4C2×D7111yescharacteristic
140C2×C42—711no—
141#102—714no—
142#102—714no—
143#102—714no—
144#102C2×D7114yescomplemented
145#97—714no—
146C24—21no—
147C24—1421no—
148C2×D4—21no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—2844no—
152C2×D4—2844no—
153C2×D4—712no—
154C2×D4—1424no—
155C2×D4—1424no—
156C2×D4—1421no—
157C2×D4—1422no—
158C2×D4—1422no—
159C2×D4C22×D7112yes—
160C22×C4—21no—
161C22×C4—711no—
162C22×C4—711no—
163C22×C4—1421no—
164C22×C4C22×D7111yescharacteristic
165C2×C8—22no—
166C2×C8—712no—
167C2×C8—1422no—
168C2×C8C22×D7112yes—
169C4⋊C4—21no—
170C4⋊C4—712no—
171C4⋊C4—1424no—
172C4⋊C4—1424no—
173C4⋊C4—1421no—
174C4⋊C4C22×D7112yes—
175C42—712no—
176C42—712no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—2842no—
182C23—2842no—
183C23—2842no—
184C23—1422no—
185C23D4×D7111yescharacteristic
186D4—22no—
187D4—22no—
188D4—42no—
189D4—2848no—
190D4—2848no—
191D4—2842no—
192D4—1422no—
193D4—1422no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—2842no—
197C2×C4—712no—
198C2×C4—712no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4D4×D7111yescharacteristic
203C2×C4#1176586412yes—
204C2×C4#1176586512yes—
205C2×C4C23×D7111yescharacteristic
206C8—22no—
207C8—1422no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—84no—
212C22—2842no—
213C22—2842no—
214C22—2842no—
215C22—5684no—
216C22#1455611412yes—
217C22#1455615212yes—
218C22#1455615512yes—
219C22C2×D4×D7111yescharacteristic
220C4—42no—
221C4—2842no—
222C4—1422no—
223C4—1422no—
224C4#1455611412yes—
225C4C2×#1176586411yescharacteristic
226C4C2×#1176586511yescharacteristic
227C2—82no—
228C2—5682no—
229C2#1798636214yesdirect product factor
230C2C2×#1455611411yescharacteristic
231C2C2×#1455615211yescharacteristic
232C2C2×#1455615511yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#102×C71C2
socleC23×C71C2×D4
hypercenterC2×#102D71
hypocenterC71C2×#490
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#14556119C2semidirect
3. #17986362C2direct (N × H)
4. C2×#14556050C2semidirect
5. C2×#14556049C2semidirect
6. C2×#14556048C2semidirect
7. C2×#14556038C2semidirect
8. C2×#102×C71C2semidirect
16. #14556119C22semidirect
21. #14556050C22semidirect
22. #14556049C22semidirect
23. #14556048C22semidirect
24. #14556038C22semidirect
28. #102×C71C22semidirect
111. C2×C71#490semidirect
117. C71C2×#490semidirect
124. C2×#102D71semidirect
144. #102C2×D71semidirect
229. C2#17986362direct (N × H)