Finite Group Databasebeta

← C2×#20298766 #28399898

Subgroups — C2×#20298766 #28399898

Order 14464.

844 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16813647C212yesmaximal, complemented
2C2×#16813618C211yesmaximal, characteristic, complemented
3#20298766C218yesmaximal, direct product factor
4C2×#16813549C212yesmaximal, complemented
5C2×#16813539C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C113C211yesmaximal, characteristic, complemented
7C22×D452C2211yescharacteristic, complemented
8C22×C22⋊D113—22no—
9#16813647—24no—
10#16813647C2218yescomplemented
11C2×#14268634—22no—
12C2×#14268632—22no—
13C2×#14268631C2212yescomplemented
14#16813618—22no—
15#16813618C2214yescomplemented
16C2×C4×Dic113C2211yescharacteristic, complemented
17#16813549C2218yescomplemented
18#16813539C2214yescomplemented
19H(5,𝔽2)×C113—24no—
20H(5,𝔽2)×C113—22no—
21H(5,𝔽2)×C113C2214yescomplemented
22C2×#41×C113—22no—
23C2×#41×C113C2212yes—
24C22×D4×C113—22no—
25C22×D4×C113—22no—
26C22×D4×C113C2211yescharacteristic
27C23×D113—21no—
28C2×D452—22no—
29C2×D452—22no—
30C2×D452—21no—
31C2×D452C2312yescomplemented
32C2×C22⋊D113—28no—
33C2×C22⋊D113—48no—
34C22×Dic113—21no—
35#14268634—28no—
36#14268632—28no—
37#14268631—22no—
38#14268631C2314yescomplemented
39C2×#11744751—22no—
40C4×Dic113—21no—
41C4×Dic113C2312yescomplemented
42C24×C113—22no—
43C24×C113D412yes—
44#41×C113—28no—
45#41×C113—28no—
46#41×C113—22no—
47#41×C113—22no—
48#41×C113D414yescomplemented
49#41×C113C2314yes—
50C2×Q8×C113D412yescomplemented
51C2×D4×C113—24no—
52C2×D4×C113—28no—
53C2×D4×C113—28no—
54C2×D4×C113—22no—
55C2×D4×C113—28no—
56C2×D4×C113—28no—
57C2×D4×C113—22no—
58C2×D4×C113—22no—
59C2×D4×C113D412yes—
60C2×D4×C113D412yescomplemented
61C2×D4×C113C2312yes—
62C22×C4×C113—22no—
63C22×C4×C113—22no—
64C22×C4×C113C2311yescharacteristic
65C23×C113—28no—
66C23×C113—22no—
67C23×C113—22no—
68C23×C113—22no—
69C23×C113—44no—
70C23×C113—84no—
71C23×C113C2×D411yescharacteristic
72C23×C113C2×D414yes—
73C22×D113—22no—
74C22×D113—44no—
75C22×D113—41no—
76C22×D113—41no—
77Q8×C113—22no—
78Q8×C113C2×D414yescomplemented
79D4×C113—216no—
80D4×C113—24no—
81D4×C113—24no—
82D4×C113—24no—
83D4×C113—22no—
84D4×C113—24no—
85D4×C113C2×D414yescomplemented
86C2×C4×C113—28no—
87C2×C4×C113—22no—
88C2×C4×C113—24no—
89C2×C4×C113—24no—
90C2×C4×C113C2×D411yescharacteristic
91C2×C4×C113C2×D412yes—
92C2×C4×C113C2×D412yes—
93C2×C4×C113C2411yescharacteristic
94C22⋊D113—416no—
95C2×Dic113—24no—
96C2×Dic113—22no—
97D452—24no—
98D452—42no—
99#11744751—24no—
100C22×C113—22no—
101C22×C113—24no—
102C22×C113—24no—
103C22×C113—44no—
104C22×C113—44no—
105C22×C113—44no—
106C22×C113—82no—
107C22×C113—88no—
108C22×C113#12011yescharacteristic
109C22×C113#12011yescharacteristic
110C22×C113C22×D411yescharacteristic
111C2×D113—44no—
112C2×D113—42no—
113C2×D113—82no—
114C2×D113—82no—
115C4×C113—24no—
116C4×C113#12012yes—
117C4×C113C22×D412yes—
118Dic113—24no—
119C2×C113—21no—
120C2×C113—21no—
121C2×C113—44no—
122C2×C113—44no—
123C2×C113#45312yescomplemented
124C2×C113C2×#12011yescharacteristic
125D113—82no—
126C2×#453—1131nomaximal, Sylow
127C113C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)D11311yescharacteristic, complemented
129C2×AGL(1,ℤ/8)—1132no—
130C2×#127—1131no—
131#453—1138no—
132C2×#104—1132no—
133C2×#100—1131no—
134H(5,𝔽2)—24no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)C2×D11314yescomplemented
137C2×#41—22no—
138C2×#41C2×D11312yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—1131no—
142C22×D4—2262no—
143C22×D4C2×D11311yescharacteristic
144AGL(1,ℤ/8)—1138no—
145AGL(1,ℤ/8)—2264no—
146C2×SD16—2262no—
147C2×D8—2262no—
148C2×M16—1132no—
149#127—1134no—
150#127—2262no—
151C2×C42—1131no—
152#104—1138no—
153#100—1134no—
154C24—22no—
155C24—2261no—
156C24C22⋊D11312yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41C22⋊D11314yescomplemented
162#41C22×D11314yes—
163C2×Q8C22⋊D11312yescomplemented
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—1132no—
173C2×D4—4528no—
174C2×D4—2268no—
175C2×D4—2261no—
176C2×D4—2262no—
177C2×D4—2262no—
178C2×D4C22⋊D11312yes—
179C2×D4C22⋊D11312yescomplemented
180C2×D4C22×D11312yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—2261no—
184C22×C4C22×D11311yescharacteristic
185SD16—2268no—
186D8—2268no—
187M16—1134no—
188M16—2262no—
189C2×C8—2262no—
190C42—1132no—
191C42—2261no—
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—44no—
197C23—84no—
198C23—4524no—
199C23—4521no—
200C23—4521no—
201C23—2262no—
202C23C2×C22⋊D11311yescharacteristic
203C23C2×C22⋊D11314yes—
204Q8—22no—
205Q8C2×C22⋊D11314yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4—24no—
212D4—45216no—
213D4—4522no—
214D4—2264no—
215D4C2×C22⋊D11314yescomplemented
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—2264no—
221C2×C4—2262no—
222C2×C4C2×C22⋊D11311yescharacteristic
223C2×C4C2×C22⋊D11312yes—
224C2×C4C2×C22⋊D11312yes—
225C2×C4C23×D11311yescharacteristic
226C8—2264no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—82no—
234C22—88no—
235C22—9042no—
236C22—9042no—
237C22—4524no—
238C22—4522no—
239C22#1681359111yescharacteristic
240C22#1681359111yescharacteristic
241C22C22×C22⋊D11311yescharacteristic
242C4—24no—
243C4—2264no—
244C4#1681359112yes—
245C4C22×C22⋊D11312yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—9042no—
251C2#2029876612yesdirect product factor
252C2C2×#1681359111yescharacteristic
subgroupG/H
centerC22#16813591
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×H(5,𝔽2)×C113C2
socleC22×C113#120
hypercenterC2×H(5,𝔽2)D113
hypocenterC113C2×#453
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16813647C2semidirect
2. C2×#16813618C2semidirect
3. #20298766C2direct (N × H)
4. C2×#16813549C2semidirect
5. C2×#16813539C2semidirect
6. C2×H(5,𝔽2)×C113C2semidirect
7. C22×D452C22semidirect
10. #16813647C22semidirect
13. C2×#14268631C22semidirect
15. #16813618C22semidirect
16. C2×C4×Dic113C22semidirect
17. #16813549C22semidirect
18. #16813539C22semidirect
21. H(5,𝔽2)×C113C22semidirect
31. C2×D452C23semidirect
38. #14268631C23semidirect
41. C4×Dic113C23semidirect
48. #41×C113D4semidirect
50. C2×Q8×C113D4semidirect
60. C2×D4×C113D4semidirect
78. Q8×C113C2×D4semidirect
85. D4×C113C2×D4semidirect
123. C2×C113#453semidirect
127. C113C2×#453semidirect
128. C2×H(5,𝔽2)D113semidirect
136. H(5,𝔽2)C2×D113semidirect
161. #41C22⋊D113semidirect
163. C2×Q8C22⋊D113semidirect
179. C2×D4C22⋊D113semidirect
205. Q8C2×C22⋊D113semidirect
215. D4C2×C22⋊D113semidirect
251. C2#20298766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136472
C2×#168136182
#202987662
C2×#168135492
C2×#168135392
C2×H(5,𝔽2)×C1132
C2×#453113