Finite Group Databasebeta

← C2×#20298922 #28400458

Subgroups — C2×#20298922 #28400458

Order 14464.

468 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16813596C211yesmaximal, characteristic
2#20298922C218yesmaximal, direct product factor
3C2×#16813555C211yesmaximal, characteristic, complemented
4C2×#16813553C211yesmaximal, characteristic, complemented
5C2×#16813542C211yesmaximal, characteristic, complemented
6C2×#16813542C211yesmaximal, characteristic, complemented
7C2×C22⋊#11744751C211yesmaximal, characteristic, complemented
8C2×#117×C113C211yesmaximal, characteristic, complemented
9C22×D452C2211yescharacteristic
10C22×C22⋊D113—21no—
11C22×#11744751C2211yescharacteristic
12C22×#11744751C2211yescharacteristic
13#16813596—22no—
14#16813596C2214yes—
15C2×C4⋊Dic113C2211yescharacteristic
16C2×#14268624—21no—
17#16813555C2214yescomplemented
18#16813553C2214yescomplemented
19#16813542C2214yescomplemented
20#16813542C2214yescomplemented
21C22⋊#11744751C2214yescomplemented
22C23×C4×C113C2211yescharacteristic
23#117×C113—22no—
24#117×C113C2214yescomplemented
25C2×C4⋊C4×C113C2211yescharacteristic
26C2×C4⋊C4×C113C2211yescharacteristic
27C2×C22⋊C4×C113—21no—
28C2×C42×C113—21no—
29C23×D113—21no—
30C2×D452—22no—
31C2×D452—22no—
32C2×D452—21no—
33C2×D452C2312yes—
34C2×C22⋊D113—24no—
35C2×C22⋊D113—44no—
36C22×Dic113—21no—
37C2×#11744751—22no—
38C2×#11744751—22no—
39C2×#11744751C2312yes—
40C2×#11744751C2312yes—
41C4⋊Dic113—21no—
42C4⋊Dic113C2312yes—
43#14268624—24no—
44C24×C113D411yescharacteristic
45C22×C4×C113—22no—
46C22×C4×C113—22no—
47C22×C4×C113—21no—
48C22×C4×C113—21no—
49C22×C4×C113D411yescharacteristic
50C22×C4×C113D412yes—
51C22×C4×C113C2312yes—
52C22×C4×C113C2311yescharacteristic
53C4⋊C4×C113—21no—
54C4⋊C4×C113—21no—
55C4⋊C4×C113C2312yes—
56C4⋊C4×C113C2312yes—
57C22⋊C4×C113—24no—
58C42×C113—24no—
59C23×C113—22no—
60C23×C113—22no—
61C23×C113—22no—
62C23×C113C2×D412yes—
63C23×C113C2×D411yescharacteristic
64C22×D113—22no—
65C22×D113—42no—
66C22×D113—42no—
67C22×D113—42no—
68C2×C4×C113—22no—
69C2×C4×C113—24no—
70C2×C4×C113—22no—
71C2×C4×C113—22no—
72C2×C4×C113—22no—
73C2×C4×C113—22no—
74C2×C4×C113—42no—
75C2×C4×C113—42no—
76C2×C4×C113C2×D412yes—
77C2×C4×C113C2×D411yescharacteristic
78C2×C4×C113#4112yes—
79C2×C4×C113#4112yes—
80C2×C4×C113C2411yescharacteristic
81C22⋊D113—48no—
82C2×Dic113—22no—
83C2×Dic113—42no—
84D452—22no—
85D452—22no—
86D452—42no—
87#11744751—22no—
88#11744751—22no—
89C22×C113—22no—
90C22×C113—22no—
91C22×C113—22no—
92C22×C113—44no—
93C22×C113#12312yes—
94C22×C113#13512yes—
95C22×C113AGL(1,ℤ/8)12yes—
96C22×C113C22×D411yescharacteristic
97C2×D113—42no—
98C2×D113—42no—
99C2×D113—42no—
100C2×D113—84no—
101C4×C113—22no—
102C4×C113—42no—
103C4×C113—42no—
104C4×C113#12312yes—
105C4×C113C2×#4111yescharacteristic
106C4×C113C2×#4111yescharacteristic
107Dic113—42no—
108C2×C113—42no—
109C2×C113#48214yescomplemented
110C2×C113C2×#12311yescharacteristic
111C2×C113C2×#13511yescharacteristic
112C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
113D113—82no—
114C2×#482—1131nomaximal, Sylow
115C113C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—1131no—
117C2×#117D11311yescharacteristic, complemented
118#482—1138no—
119C2×#107—1131no—
120C2×#106—1131no—
121C2×#102—1131no—
122C2×#102—1131no—
123C2×C22⋊C8—1131no—
124C22×D4—1131no—
125C22×D4—2261no—
126C23×C4C2×D11311yescharacteristic
127C22×C8—1131no—
128C22×C8—1131no—
129#121—1134no—
130#121—2262no—
131#117—22no—
132#117C2×D11314yescomplemented
133C2×C4⋊C4—1131no—
134C2×C4⋊C4C2×D11311yescharacteristic
135C2×C4⋊C4C2×D11311yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—2261no—
138C2×C42—21no—
139#107—1134no—
140#106—1134no—
141#102—1134no—
142#102—1134no—
143C22⋊C8—1134no—
144C24—2261no—
145C24C22⋊D11311yescharacteristic
146C2×D4—1132no—
147C2×D4—4524no—
148C2×D4—2264no—
149C2×D4—2261no—
150C2×D4—2262no—
151C2×D4—2262no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—2261no—
157C22×C4C22⋊D11312yes—
158C22×C4C22⋊D11311yescharacteristic
159C22×C4C22×D11312yes—
160C22×C4C22×D11311yescharacteristic
161C2×C8—1132no—
162C2×C8—1132no—
163C2×C8—2262no—
164C2×C8—2262no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1132no—
168C4⋊C4—2261no—
169C4⋊C4C22×D11312yes—
170C4⋊C4C22×D11312yes—
171C22⋊C4—24no—
172C22⋊C4—2264no—
173C42—24no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—4522no—
178C23—4522no—
179C23—4522no—
180C23—2262no—
181C23C2×C22⋊D11312yes—
182C23C2×C22⋊D11311yescharacteristic
183D4—4528no—
184D4—4522no—
185D4—2262no—
186D4—2262no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—4522no—
196C2×C4—2262no—
197C2×C4C2×C22⋊D11312yes—
198C2×C4C2×C22⋊D11311yescharacteristic
199C2×C4#1426864712yes—
200C2×C4#1426864712yes—
201C2×C4C23×D11311yescharacteristic
202C8—2262no—
203C8—2262no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—9044no—
209C22—4522no—
210C22—4522no—
211C22—4522no—
212C22#1681360512yes—
213C22#1681364212yes—
214C22#1681364712yes—
215C22C22×C22⋊D11311yescharacteristic
216C4—22no—
217C4—42no—
218C4—42no—
219C4—4522no—
220C4#1681360512yes—
221C4C2×#1426864711yescharacteristic
222C4C2×#1426864711yescharacteristic
223C2—42no—
224C2—9042no—
225C2#2029892214yesdirect product factor
226C2C2×#1681360511yescharacteristic
227C2C2×#1681364211yescharacteristic
228C2C2×#1681364711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×#117×C113C2
socleC23×C113C2×D4
prime radicalC2×#16813596C2
hypercenterC2×#117D113
hypocenterC113C2×#482
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20298922C2direct (N × H)
3. C2×#16813555C2semidirect
4. C2×#16813553C2semidirect
5. C2×#16813542C2semidirect
6. C2×#16813542C2semidirect
7. C2×C22⋊#11744751C2semidirect
8. C2×#117×C113C2semidirect
17. #16813555C22semidirect
18. #16813553C22semidirect
19. #16813542C22semidirect
20. #16813542C22semidirect
21. C22⋊#11744751C22semidirect
24. #117×C113C22semidirect
109. C2×C113#482semidirect
115. C113C2×#482semidirect
117. C2×#117D113semidirect
132. #117C2×D113semidirect
225. C2#20298922direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168135962
#202989222
C2×#168135552
C2×#168135532
C2×#168135422
C2×C22⋊#117447512
C2×#117×C1132
C2×#482113