Finite Group Databasebeta

← C2×#20260902 #27151059

Subgroups — C2×#20260902 #27151059

Order 12416.

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

normal subgroups only

This group has 844 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×#15690855C211yesmaximal, characteristic, complemented
2C2×#15690836C212yesmaximal, complemented
3#20260902C218yesmaximal, direct product factor
4C2×#15690738C212yesmaximal, complemented
5C2×#15690727C211yesmaximal, characteristic, complemented
6C2×#127×C97C211yesmaximal, characteristic, complemented
7#15690855—24no—
8#15690855—22no—
9#15690855C2214yescomplemented
10C2×#13095511C2212yes—
11C2×#13095510—22no—
12C22×C22⋊D97—22no—
13C2×D4×D97—22no—
14#15690836—24no—
15#15690836C2218yescomplemented
16C2×#13095499—22no—
17C2×#13095496—22no—
18C2×#13095495C2212yescomplemented
19#15690738C2218yescomplemented
20#15690727C2214yescomplemented
21C22×D4×C97—22no—
22C22×D4×C97C2211yescharacteristic
23C22×D4×C97C2211yescharacteristic, complemented
24#127×C97—22no—
25#127×C97C2214yescomplemented
26C2×C42×C97C2211yescharacteristic, complemented
27C23×D97—22no—
28#13095511—22no—
29#13095511—22no—
30#13095511D414yescomplemented
31#13095511C2314yes—
32#13095510—28no—
33#13095510—28no—
34C2×Dic194D412yescomplemented
35C2×D388D412yescomplemented
36C2×C22⋊D97—24no—
37C2×C22⋊D97—28no—
38C2×C22⋊D97—22no—
39C2×C22⋊D97—28no—
40D4×D97—28no—
41D4×D97—28no—
42C22×Dic97—22no—
43C2×C4×D97—22no—
44#13095499—28no—
45#13095496—28no—
46#13095495—22no—
47#13095495C2314yescomplemented
48C2×#11715895—22no—
49C24×C97—21no—
50C24×C97D412yes—
51C2×D4×C97—28no—
52C2×D4×C97—22no—
53C2×D4×C97—22no—
54C2×D4×C97—22no—
55C2×D4×C97—22no—
56C2×D4×C97—21no—
57C2×D4×C97—48no—
58C2×D4×C97D412yes—
59C2×D4×C97C2312yes—
60C2×D4×C97C2312yescomplemented
61C22×C4×C97—21no—
62C22×C4×C97C2311yescharacteristic
63C42×C97—21no—
64C42×C97C2312yescomplemented
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—44no—
69C23×C97—44no—
70C23×C97—41no—
71C23×C97—41no—
72C23×C97C2×D411yescharacteristic
73C23×C97C2×D414yes—
74C22×D97—22no—
75C22×D97—28no—
76C22×D97—84no—
77D4×C97—24no—
78D4×C97—24no—
79D4×C97—24no—
80D4×C97—416no—
81D4×C97—42no—
82C2×C4×C97—24no—
83C2×C4×C97—22no—
84C2×C4×C97C2×D412yes—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2411yescharacteristic
88C22⋊D97—216no—
89C22⋊D97—24no—
90C22⋊D97—24no—
91C2×Dic97—28no—
92C2×Dic97—22no—
93D388—22no—
94D388C2×D414yescomplemented
95C4×D97—24no—
96C4×D97—24no—
97Dic194—22no—
98Dic194C2×D414yescomplemented
99#11715895—24no—
100C22×C97—22no—
101C22×C97—24no—
102C22×C97—44no—
103C22×C97—44no—
104C22×C97—44no—
105C22×C97—42no—
106C22×C97—82no—
107C22×C97—82no—
108C22×C97—82no—
109C22×C97#12011yescharacteristic
110C22×C97#12011yescharacteristic
111C22×C97C22×D411yescharacteristic
112C2×D97—24no—
113C2×D97—44no—
114C2×D97—88no—
115C4×C97—24no—
116C4×C97#12012yes—
117C4×C97C22×D412yes—
118Dic97—24no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—44no—
122C2×C97—82no—
123C2×C97#45312yescomplemented
124C2×C97C2×#12011yescharacteristic
125D97—44no—
126C2×#453—971nomaximal, Sylow
127C97C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)—971no—
129C2×AGL(1,ℤ/8)—972no—
130C2×#127D9711yescharacteristic, complemented
131#453—978no—
132C2×#104—972no—
133C2×#100—971no—
134H(5,𝔽2)—974no—
135H(5,𝔽2)—1944no—
136H(5,𝔽2)—1942no—
137C2×#41—972no—
138C2×#41—1942no—
139C22×D4—22no—
140C22×D4—1942no—
141C22×D4—1942no—
142C22×D4C2×D9711yescharacteristic
143C22×D4C2×D9711yescharacteristic, complemented
144AGL(1,ℤ/8)—978no—
145AGL(1,ℤ/8)—1944no—
146C2×SD16—1942no—
147C2×D8—1942no—
148C2×M16—972no—
149#127—22no—
150#127C2×D9714yescomplemented
151C2×C42C2×D9711yescharacteristic, complemented
152#104—978no—
153#100—974no—
154C24—21no—
155C24—1942no—
156C24C22⋊D9712yes—
157#41—974no—
158#41—974no—
159#41—1948no—
160#41—1948no—
161#41—1942no—
162#41—1942no—
163C2×Q8—972no—
164C2×D4—28no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—21no—
170C2×D4—48no—
171C2×D4—972no—
172C2×D4—1948no—
173C2×D4—1948no—
174C2×D4—1944no—
175C2×D4—1948no—
176C2×D4—1942no—
177C2×D4—1948no—
178C2×D4C22⋊D9712yes—
179C2×D4C22×D9712yes—
180C2×D4C22×D9712yescomplemented
181C22×C4—21no—
182C22×C4—1942no—
183C22×C4—1942no—
184C22×C4C22×D9711yescharacteristic
185SD16—1948no—
186D8—1948no—
187M16—974no—
188M16—1942no—
189C2×C8—1942no—
190C42—21no—
191C42C22×D9712yescomplemented
192C23—22no—
193C23—22no—
194C23—22no—
195C23—44no—
196C23—44no—
197C23—41no—
198C23—41no—
199C23—7764no—
200C23—1948no—
201C23—1942no—
202C23C2×C22⋊D9711yescharacteristic
203C23C2×C22⋊D9714yes—
204Q8—974no—
205Q8—1942no—
206D4—24no—
207D4—24no—
208D4—24no—
209D4—416no—
210D4—42no—
211D4—974no—
212D4—19416no—
213D4—1944no—
214D4—1944no—
215D4—1942no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—1948no—
219C2×C4—1942no—
220C2×C4—1944no—
221C2×C4—1944no—
222C2×C4D4×D9712yes—
223C2×C4D4×D9712yes—
224C2×C4C2×C22⋊D9711yescharacteristic
225C2×C4C23×D9711yescharacteristic
226C8—1944no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—42no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—7768no—
237C22—3884no—
238C22—1944no—
239C22#1569077811yescharacteristic
240C22#1569077811yescharacteristic
241C22C22×C22⋊D9711yescharacteristic
242C4—24no—
243C4—1944no—
244C4#1569077812yes—
245C4C2×D4×D9712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—82no—
250C2—3884no—
251C2#2026090212yesdirect product factor
252C2C2×#1569077811yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#127×C97C2
socleC22×C97#120
hypercenterC2×#127D97
hypocenterC97C2×#453
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#15690855C2semidirect
2. C2×#15690836C2semidirect
3. #20260902C2direct (N × H)
4. C2×#15690738C2semidirect
5. C2×#15690727C2semidirect
6. C2×#127×C97C2semidirect
9. #15690855C22semidirect
15. #15690836C22semidirect
18. C2×#13095495C22semidirect
19. #15690738C22semidirect
20. #15690727C22semidirect
23. C22×D4×C97C22semidirect
25. #127×C97C22semidirect
26. C2×C42×C97C22semidirect
30. #13095511D4semidirect
34. C2×Dic194D4semidirect
35. C2×D388D4semidirect
47. #13095495C23semidirect
60. C2×D4×C97C23semidirect
64. C42×C97C23semidirect
94. D388C2×D4semidirect
98. Dic194C2×D4semidirect
123. C2×C97#453semidirect
127. C97C2×#453semidirect
130. C2×#127D97semidirect
143. C22×D4C2×D97semidirect
150. #127C2×D97semidirect
151. C2×C42C2×D97semidirect
180. C2×D4C22×D97semidirect
191. C42C22×D97semidirect
251. C2#20260902direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908552
C2×#156908362
#202609022
C2×#156907382
C2×#156907272
C2×#127×C972
C2×#45397