Finite Group Databasebeta

← C2×#20260903 #27151060

Subgroups — C2×#20260903 #27151060

Order 12416.

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×#15690835C212yesmaximal, complemented
2C2×#15690806C211yesmaximal, characteristic, complemented
3#20260903C218yesmaximal, direct product factor
4C2×#15690737C212yesmaximal, complemented
5C2×#15690727C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
7C22×D388C2211yescharacteristic, complemented
8C22×C22⋊D97—22no—
9#15690835—24no—
10#15690835C2218yescomplemented
11C2×#13095498—22no—
12C2×#13095496—22no—
13C2×#13095495C2212yescomplemented
14#15690806—22no—
15#15690806C2214yescomplemented
16C2×C4×Dic97C2211yescharacteristic, complemented
17#15690737C2218yescomplemented
18#15690727C2214yescomplemented
19H(5,𝔽2)×C97—24no—
20H(5,𝔽2)×C97—22no—
21H(5,𝔽2)×C97C2214yescomplemented
22C2×#41×C97—22no—
23C2×#41×C97C2212yes—
24C22×D4×C97—22no—
25C22×D4×C97—22no—
26C22×D4×C97C2211yescharacteristic
27C23×D97—21no—
28C2×D388—22no—
29C2×D388—22no—
30C2×D388—21no—
31C2×D388C2312yescomplemented
32C2×C22⋊D97—28no—
33C2×C22⋊D97—48no—
34C22×Dic97—21no—
35#13095498—28no—
36#13095496—28no—
37#13095495—22no—
38#13095495C2314yescomplemented
39C2×#11715895—22no—
40C4×Dic97—21no—
41C4×Dic97C2312yescomplemented
42C24×C97—22no—
43C24×C97D412yes—
44#41×C97—28no—
45#41×C97—28no—
46#41×C97—22no—
47#41×C97—22no—
48#41×C97D414yescomplemented
49#41×C97C2314yes—
50C2×Q8×C97D412yescomplemented
51C2×D4×C97—24no—
52C2×D4×C97—28no—
53C2×D4×C97—28no—
54C2×D4×C97—22no—
55C2×D4×C97—28no—
56C2×D4×C97—28no—
57C2×D4×C97—22no—
58C2×D4×C97—22no—
59C2×D4×C97D412yes—
60C2×D4×C97D412yescomplemented
61C2×D4×C97C2312yes—
62C22×C4×C97—22no—
63C22×C4×C97—22no—
64C22×C4×C97C2311yescharacteristic
65C23×C97—28no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—22no—
69C23×C97—44no—
70C23×C97—84no—
71C23×C97C2×D411yescharacteristic
72C23×C97C2×D414yes—
73C22×D97—22no—
74C22×D97—44no—
75C22×D97—41no—
76C22×D97—41no—
77Q8×C97—22no—
78Q8×C97C2×D414yescomplemented
79D4×C97—216no—
80D4×C97—24no—
81D4×C97—24no—
82D4×C97—24no—
83D4×C97—22no—
84D4×C97—24no—
85D4×C97C2×D414yescomplemented
86C2×C4×C97—28no—
87C2×C4×C97—22no—
88C2×C4×C97—24no—
89C2×C4×C97—24no—
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2×D412yes—
92C2×C4×C97C2×D412yes—
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—416no—
95C2×Dic97—24no—
96C2×Dic97—22no—
97D388—24no—
98D388—42no—
99#11715895—24no—
100C22×C97—22no—
101C22×C97—24no—
102C22×C97—24no—
103C22×C97—44no—
104C22×C97—44no—
105C22×C97—44no—
106C22×C97—82no—
107C22×C97—88no—
108C22×C97#12011yescharacteristic
109C22×C97#12011yescharacteristic
110C22×C97C22×D411yescharacteristic
111C2×D97—44no—
112C2×D97—42no—
113C2×D97—82no—
114C2×D97—82no—
115C4×C97—24no—
116C4×C97#12012yes—
117C4×C97C22×D412yes—
118Dic97—24no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—44no—
122C2×C97—44no—
123C2×C97#45312yescomplemented
124C2×C97C2×#12011yescharacteristic
125D97—82no—
126C2×#453—971nomaximal, Sylow
127C97C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)D9711yescharacteristic, complemented
129C2×AGL(1,ℤ/8)—972no—
130C2×#127—971no—
131#453—978no—
132C2×#104—972no—
133C2×#100—971no—
134H(5,𝔽2)—24no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)C2×D9714yescomplemented
137C2×#41—22no—
138C2×#41C2×D9712yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—971no—
142C22×D4—1942no—
143C22×D4C2×D9711yescharacteristic
144AGL(1,ℤ/8)—978no—
145AGL(1,ℤ/8)—1944no—
146C2×SD16—1942no—
147C2×D8—1942no—
148C2×M16—972no—
149#127—974no—
150#127—1942no—
151C2×C42—971no—
152#104—978no—
153#100—974no—
154C24—22no—
155C24—1941no—
156C24C22⋊D9712yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41C22⋊D9714yescomplemented
162#41C22×D9714yes—
163C2×Q8C22⋊D9712yescomplemented
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—972no—
173C2×D4—3888no—
174C2×D4—1948no—
175C2×D4—1941no—
176C2×D4—1942no—
177C2×D4—1942no—
178C2×D4C22⋊D9712yes—
179C2×D4C22⋊D9712yescomplemented
180C2×D4C22×D9712yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—1941no—
184C22×C4C22×D9711yescharacteristic
185SD16—1948no—
186D8—1948no—
187M16—974no—
188M16—1942no—
189C2×C8—1942no—
190C42—972no—
191C42—1941no—
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—44no—
197C23—84no—
198C23—3884no—
199C23—3881no—
200C23—3881no—
201C23—1942no—
202C23C2×C22⋊D9711yescharacteristic
203C23C2×C22⋊D9714yes—
204Q8—22no—
205Q8C2×C22⋊D9714yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4—24no—
212D4—38816no—
213D4—3882no—
214D4—1944no—
215D4C2×C22⋊D9714yescomplemented
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—1944no—
221C2×C4—1942no—
222C2×C4C2×C22⋊D9711yescharacteristic
223C2×C4C2×C22⋊D9712yes—
224C2×C4C2×C22⋊D9712yes—
225C2×C4C23×D9711yescharacteristic
226C8—1944no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—82no—
234C22—88no—
235C22—7762no—
236C22—7762no—
237C22—3884no—
238C22—3882no—
239C22#1569077911yescharacteristic
240C22#1569077911yescharacteristic
241C22C22×C22⋊D9711yescharacteristic
242C4—24no—
243C4—1944no—
244C4#1569077912yes—
245C4C22×C22⋊D9712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—7762no—
251C2#2026090312yesdirect product factor
252C2C2×#1569077911yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×H(5,𝔽2)×C97C2
socleC22×C97#120
hypercenterC2×H(5,𝔽2)D97
hypocenterC97C2×#453
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690806C2semidirect
3. #20260903C2direct (N × H)
4. C2×#15690737C2semidirect
5. C2×#15690727C2semidirect
6. C2×H(5,𝔽2)×C97C2semidirect
7. C22×D388C22semidirect
10. #15690835C22semidirect
13. C2×#13095495C22semidirect
15. #15690806C22semidirect
16. C2×C4×Dic97C22semidirect
17. #15690737C22semidirect
18. #15690727C22semidirect
21. H(5,𝔽2)×C97C22semidirect
31. C2×D388C23semidirect
38. #13095495C23semidirect
41. C4×Dic97C23semidirect
48. #41×C97D4semidirect
50. C2×Q8×C97D4semidirect
60. C2×D4×C97D4semidirect
78. Q8×C97C2×D4semidirect
85. D4×C97C2×D4semidirect
123. C2×C97#453semidirect
127. C97C2×#453semidirect
128. C2×H(5,𝔽2)D97semidirect
136. H(5,𝔽2)C2×D97semidirect
161. #41C22⋊D97semidirect
163. C2×Q8C22⋊D97semidirect
179. C2×D4C22⋊D97semidirect
205. Q8C2×C22⋊D97semidirect
215. D4C2×C22⋊D97semidirect
251. C2#20260903direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156908062
#202609032
C2×#156907372
C2×#156907272
C2×H(5,𝔽2)×C972
C2×#45397