Finite Group Databasebeta

← C2×#20260905 #27151062

Subgroups — C2×#20260905 #27151062

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×#15690838C212yesmaximal, complemented
3C2×#15690806C211yesmaximal, characteristic, complemented
4#20260905C218yesmaximal, direct product factor
5C2×#15690736C212yesmaximal, complemented
6C2×#100×C97C211yesmaximal, characteristic, complemented
7#15690855—24no—
8#15690855—22no—
9#15690855C2214yescomplemented
10C2×#13095511C2212yes—
11C2×#13095510—22no—
12C22×D388C2211yescharacteristic, complemented
13C22×C22⋊D97—22no—
14C22×C22⋊D97—22no—
15C2×D4×D97—22no—
16#15690838—24no—
17#15690838C2218yescomplemented
18C2×#13095500—22no—
19C2×D776—22no—
20#15690806—22no—
21#15690806C2214yescomplemented
22C2×C4×Dic97C2211yescharacteristic, complemented
23#15690736C2218yescomplemented
24C22×D4×C97C2211yescharacteristic
25C2×M16×C97C2212yescomplemented
26#100×C97C2214yescomplemented
27C23×D97—22no—
28C23×D97—21no—
29#13095511—22no—
30#13095511—22no—
31#13095511D414yescomplemented
32#13095511C2314yes—
33#13095510—28no—
34#13095510—28no—
35C2×Dic194D412yescomplemented
36C2×D388—22no—
37C2×D388—22no—
38C2×D388—21no—
39C2×D388D412yescomplemented
40C2×D388C2312yescomplemented
41C2×C22⋊D97—24no—
42C2×C22⋊D97—28no—
43C2×C22⋊D97—22no—
44C2×C22⋊D97—28no—
45C2×C22⋊D97—28no—
46C2×C22⋊D97—48no—
47D4×D97—28no—
48D4×D97—28no—
49C22×Dic97—22no—
50C22×Dic97—21no—
51C2×C4×D97—22no—
52#13095500—28no—
53D776—28no—
54C4×Dic97—21no—
55C4×Dic97C2312yescomplemented
56C24×C97D412yes—
57C2×D4×C97—22no—
58C2×D4×C97—22no—
59C2×D4×C97D412yes—
60C2×D4×C97C2312yes—
61C22×C4×C97C2311yescharacteristic
62M16×C97—22no—
63M16×C97C2314yescomplemented
64C2×C8×C97—22no—
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—44no—
68C23×C97C2×D411yescharacteristic
69C23×C97C2×D414yes—
70C22×D97—22no—
71C22×D97—28no—
72C22×D97—22no—
73C22×D97—44no—
74C22×D97—41no—
75C22×D97—41no—
76C22×D97—84no—
77D4×C97—24no—
78D4×C97—24no—
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D412yes—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2411yescharacteristic
83C22⋊D97—216no—
84C22⋊D97—24no—
85C22⋊D97—24no—
86C22⋊D97—416no—
87C2×Dic97—28no—
88C2×Dic97—24no—
89C2×Dic97—22no—
90C2×Dic97—22no—
91D388—22no—
92D388—24no—
93D388—42no—
94D388C2×D414yescomplemented
95C4×D97—24no—
96C4×D97—24no—
97Dic194—22no—
98Dic194C2×D414yescomplemented
99C8×C97—24no—
100C22×C97—22no—
101C22×C97—24no—
102C22×C97—44no—
103C22×C97—44no—
104C22×C97—82no—
105C22×C97#12011yescharacteristic
106C22×C97#12011yescharacteristic
107C22×C97C22×D411yescharacteristic
108C2×D97—24no—
109C2×D97—44no—
110C2×D97—44no—
111C2×D97—42no—
112C2×D97—88no—
113C2×D97—82no—
114C2×D97—82no—
115C4×C97#12012yes—
116C4×C97C22×D412yes—
117Dic97—24no—
118Dic97—24no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—44no—
122C2×C97#45312yescomplemented
123C2×C97C2×#12011yescharacteristic
124D97—44no—
125D97—82no—
126C2×#453—971nomaximal, Sylow
127C97C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)—971no—
129C2×AGL(1,ℤ/8)—972no—
130C2×#127—971no—
131#453—978no—
132C2×#104—972no—
133C2×#100D9711yescharacteristic, complemented
134H(5,𝔽2)—974no—
135H(5,𝔽2)—1944no—
136H(5,𝔽2)—1942no—
137C2×#41—972no—
138C2×#41—1942no—
139C22×D4—971no—
140C22×D4—1942no—
141C22×D4—1942no—
142C22×D4—1942no—
143C22×D4C2×D9711yescharacteristic
144AGL(1,ℤ/8)—978no—
145AGL(1,ℤ/8)—1944no—
146C2×SD16—1942no—
147C2×D8—1942no—
148C2×M16C2×D9712yescomplemented
149#127—974no—
150#127—1942no—
151C2×C42—971no—
152#104—978no—
153#100C2×D9714yescomplemented
154C24—1942no—
155C24—1941no—
156C24D38812yes—
157#41—974no—
158#41—974no—
159#41—1948no—
160#41—1948no—
161#41—1942no—
162#41—1942no—
163C2×Q8—972no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—972no—
167C2×D4—972no—
168C2×D4—3888no—
169C2×D4—1948no—
170C2×D4—1948no—
171C2×D4—1948no—
172C2×D4—1944no—
173C2×D4—1948no—
174C2×D4—1942no—
175C2×D4—1948no—
176C2×D4—1942no—
177C2×D4—1942no—
178C2×D4—1941no—
179C2×D4D38812yes—
180C2×D4C22×D9712yes—
181C22×C4—1942no—
182C22×C4—1941no—
183C22×C4—1942no—
184C22×C4C22×D9711yescharacteristic
185SD16—1948no—
186D8—1948no—
187M16—22no—
188M16C22×D9714yescomplemented
189C2×C8—22no—
190C42—972no—
191C42—1941no—
192C23—22no—
193C23—22no—
194C23—44no—
195C23—7764no—
196C23—3884no—
197C23—3881no—
198C23—3881no—
199C23—1948no—
200C23—1942no—
201C23—1942no—
202C23C2×D38811yescharacteristic
203C23C2×D38814yes—
204Q8—974no—
205Q8—1942no—
206D4—24no—
207D4—24no—
208D4—974no—
209D4—38816no—
210D4—3882no—
211D4—19416no—
212D4—1944no—
213D4—1944no—
214D4—1942no—
215D4—1944no—
216C2×C4—1944no—
217C2×C4—1948no—
218C2×C4—1942no—
219C2×C4—1942no—
220C2×C4—1944no—
221C2×C4—1944no—
222C2×C4D4×D9712yes—
223C2×C4D4×D9712yes—
224C2×C4C2×D38811yescharacteristic
225C2×C4C23×D9711yescharacteristic
226C8—24no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—82no—
232C22—7762no—
233C22—7768no—
234C22—7762no—
235C22—3884no—
236C22—3882no—
237C22—3884no—
238C22—1944no—
239C22#1569078011yescharacteristic
240C22#1569078011yescharacteristic
241C22C22×D38811yescharacteristic
242C4—1944no—
243C4—1944no—
244C4#1569078012yes—
245C4C2×D4×D9712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—7762no—
250C2—3884no—
251C2#2026090512yesdirect product factor
252C2C2×#1569078011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#100×C97C2
socleC22×C97#120
hypercenterC2×#100D97
hypocenterC97C2×#453
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#15690855C2semidirect
2. C2×#15690838C2semidirect
3. C2×#15690806C2semidirect
4. #20260905C2direct (N × H)
5. C2×#15690736C2semidirect
6. C2×#100×C97C2semidirect
9. #15690855C22semidirect
12. C22×D388C22semidirect
17. #15690838C22semidirect
21. #15690806C22semidirect
22. C2×C4×Dic97C22semidirect
23. #15690736C22semidirect
25. C2×M16×C97C22semidirect
26. #100×C97C22semidirect
31. #13095511D4semidirect
35. C2×Dic194D4semidirect
39. C2×D388D4semidirect
40. C2×D388C23semidirect
55. C4×Dic97C23semidirect
63. M16×C97C23semidirect
94. D388C2×D4semidirect
98. Dic194C2×D4semidirect
122. C2×C97#453semidirect
127. C97C2×#453semidirect
133. C2×#100D97semidirect
148. C2×M16C2×D97semidirect
153. #100C2×D97semidirect
188. M16C22×D97semidirect
251. C2#20260905direct (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×#156908382
C2×#156908062
#202609052
C2×#156907362
C2×#100×C972
C2×#45397