Finite Group Databasebeta

← C2×#17986301 #22598099

Subgroups — C2×#17986301 #22598099

Order 9088.

448 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14556119C211yesmaximal, characteristic, complemented
2C2×D4×Dic71C211yesmaximal, characteristic
3#17986301C218yesmaximal, direct product factor
4C2×#14556061C211yesmaximal, characteristic, complemented
5C2×#14556060C211yesmaximal, characteristic, complemented
6C2×Dic71⋊C8C211yesmaximal, characteristic, complemented
7C2×#14556049C211yesmaximal, characteristic
8C2×#102×C71C211yesmaximal, characteristic
9C23×Dic71—21no—
10C22×C71⋊C8C2211yescharacteristic
11#14556119—22no—
12#14556119C2214yescomplemented
13D4×Dic71—22no—
14D4×Dic71C2214yes—
15C2×C4⋊Dic71—21no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2211yescharacteristic
18C2×Dic71⋊C4—21no—
19C2×C22⋊Dic71—21no—
20C2×C4×Dic71C2211yescharacteristic
21#14556061C2214yescomplemented
22#14556060C2214yescomplemented
23Dic71⋊C8C2214yescomplemented
24#14556049C2214yes—
25C22×D4×C71C2211yescharacteristic
26C22×C8×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28#102×C71C2214yes—
29C22×Dic71—21no—
30C22×Dic71—21no—
31C22×Dic71—24no—
32C22×Dic71—44no—
33C22×Dic71D411yescharacteristic
34C22×Dic71D411yescharacteristic
35C2×C71⋊C8—22no—
36C2×C71⋊C8C2312yes—
37C4⋊Dic71—21no—
38C4⋊Dic71—21no—
39C4⋊Dic71—24no—
40C4⋊Dic71C2312yes—
41C4⋊Dic71C2312yes—
42Dic71⋊C4—24no—
43C22⋊Dic71—24no—
44C4×Dic71D412yes—
45C4×Dic71C2312yes—
46C24×C71—21no—
47C2×D4×C71—21no—
48C2×D4×C71—22no—
49C2×D4×C71Q814yes—
50C2×D4×C71C2312yes—
51C22×C4×C71—21no—
52C22×C4×C71C2311yescharacteristic
53C2×C8×C71—22no—
54C2×C8×C71C2312yes—
55C4⋊C4×C71—21no—
56C4⋊C4×C71C2312yes—
57C23×C71—24no—
58C23×C71—22no—
59C23×C71—42no—
60C23×C71—42no—
61C23×C71C2×D411yescharacteristic
62D4×C71—24no—
63D4×C71—22no—
64D4×C71C2×Q814yes—
65C2×C4×C71—22no—
66C2×C4×C71—42no—
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2×Q812yes—
69C2×C4×C71#4112yes—
70C2×C4×C71C2411yescharacteristic
71C2×Dic71—22no—
72C2×Dic71—22no—
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71—48no—
76C2×Dic71—42no—
77C2×Dic71—42no—
78C2×Dic71C2×D412yes—
79C2×Dic71C2×D412yes—
80C8×C71—22no—
81C71⋊C8—22no—
82C22×C71—24no—
83C22×C71—42no—
84C22×C71—42no—
85C22×C71—48no—
86C22×C71C22⋊Q812yes—
87C22×C71#13512yes—
88C22×C71AGL(1,ℤ/8)12yes—
89C22×C71C22×D411yescharacteristic
90C4×C71—42no—
91C4×C71C22⋊Q812yes—
92C4×C71C22×Q811yescharacteristic
93C4×C71C2×#4111yescharacteristic
94Dic71—22no—
95Dic71—22no—
96Dic71—42no—
97Dic71—42no—
98C2×C71—44no—
99C2×C71#47814yescomplemented
100C2×C71C2×C22⋊Q811yescharacteristic
101C2×C71C2×#13511yescharacteristic
102C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
103C2×#478—711nomaximal, Sylow
104C71C2×#47811yescharacteristic, complemented, Sylow
105C2×#125—711no—
106C2×C4×D4—711no—
107#478—718no—
108C2×#107—711no—
109C2×#106—711no—
110C2×C4⋊C8—711no—
111C2×#102—711no—
112C2×#102D7111yescharacteristic
113C22×D4C2×D7111yescharacteristic
114C23×C4—1421no—
115C22×C8—711no—
116C22×C8C2×D7111yescharacteristic
117#125—714no—
118#125—1422no—
119C4×D4—714no—
120C4×D4—1422no—
121C2×C4⋊C4—711no—
122C2×C4⋊C4—711no—
123C2×C4⋊C4—1421no—
124C2×C4⋊C4—1421no—
125C2×C4⋊C4C2×D7111yescharacteristic
126C2×C22⋊C4—1421no—
127C2×C42—711no—
128#107—714no—
129#106—714no—
130C4⋊C8—714no—
131#102—714no—
132#102C2×D7114yes—
133C24—21no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4Dic14214yes—
137C2×D4C22×D7112yes—
138C22×C4—21no—
139C22×C4—2844no—
140C22×C4—711no—
141C22×C4—711no—
142C22×C4—1424no—
143C22×C4—1421no—
144C22×C4—1421no—
145C22×C4C22×D7111yescharacteristic
146C2×C8—22no—
147C2×C8—712no—
148C2×C8—1422no—
149C2×C8C22×D7112yes—
150C4⋊C4—21no—
151C4⋊C4—712no—
152C4⋊C4—712no—
153C4⋊C4—1424no—
154C4⋊C4—1424no—
155C4⋊C4—1421no—
156C4⋊C4—1421no—
157C4⋊C4C22×D7112yes—
158C22⋊C4—1424no—
159C42—712no—
160C42—712no—
161C23—22no—
162C23—24no—
163C23—42no—
164C23—42no—
165C23D4×D7111yescharacteristic
166D4—22no—
167D4—24no—
168D4C2×Dic14214yes—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—2842no—
172C2×C4—2842no—
173C2×C4—2848no—
174C2×C4—712no—
175C2×C4—712no—
176C2×C4—1422no—
177C2×C4—1422no—
178C2×C4—1422no—
179C2×C4—1422no—
180C2×C4D4×D7111yescharacteristic
181C2×C4C2×Dic14212yes—
182C2×C4#1176586412yes—
183C2×C4C23×D7111yescharacteristic
184C8—22no—
185C8—1422no—
186C22—24no—
187C22—42no—
188C22—42no—
189C22—48no—
190C22#1455610512yes—
191C22#1455614912yes—
192C22#1455615212yes—
193C22C2×D4×D7111yescharacteristic
194C4—42no—
195C4—2842no—
196C4—2842no—
197C4—1422no—
198C4—1422no—
199C4#1455610512yes—
200C4C22×Dic14211yescharacteristic
201C4C2×#1176586411yescharacteristic
202C2—44no—
203C2#1798630114yesdirect product factor
204C2C2×#1455610511yescharacteristic
205C2C2×#1455614911yescharacteristic
206C2C2×#1455615211yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#102×C71C2
socleC23×C71C2×D4
hypercenterC2×#102D71
hypocenterC71C2×#478
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
1. C2×#14556119C2semidirect
3. #17986301C2direct (N × H)
4. C2×#14556061C2semidirect
5. C2×#14556060C2semidirect
6. C2×Dic71⋊C8C2semidirect
12. #14556119C22semidirect
21. #14556061C22semidirect
22. #14556060C22semidirect
23. Dic71⋊C8C22semidirect
99. C2×C71#478semidirect
104. C71C2×#478semidirect
203. C2#17986301direct (N × H)