Finite Group Databasebeta

← C2×#20261547 #27156497

Subgroups — C2×#20261547 #27156497

Order 12416.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690857C211yesmaximal, characteristic, complemented
2C2×#15690856C211yesmaximal, characteristic, complemented
3#20261547C2116yesmaximal, direct product factor
4C2×#15690843C213yesmaximal, complemented
5C2×#15690838C213yesmaximal, complemented
6C2×#15690832C213yesmaximal, complemented
7C22×#13095500C213yesmaximal, complemented
8C2×#131×C97C211yesmaximal, characteristic, complemented
9#15690857—24no—
10#15690857C2218yescomplemented
11#15690856—24no—
12#15690856C2218yescomplemented
13C2×#13095512—21no—
14C2×#13095511C2213yes—
15C2×#13095511C2213yescomplemented
16C2×#13095510—23no—
17C22×Dic194C2213yescomplemented
18C2×Q8×D97—21no—
19C22×D388C2213yes—
20C2×D4×D97—23no—
21#15690843C22124yescomplemented
22#15690838C22124yescomplemented
23#15690832C22124yescomplemented
24C2×Dic388C2213yescomplemented
25C2×#13095500C2213yescomplemented
26C2×#13095500C2216yescomplemented
27C2×#13095500C2211yescharacteristic, complemented
28C2×#13095500C22124yescomplemented
29C2×D776C2213yescomplemented
30C2×#41×C97C2211yescharacteristic
31#131×C97C2218yescomplemented
32C2×M16×C97C2213yescomplemented
33C22×C8×C97C2213yescomplemented
34C23×D97—23no—
35#13095512—28no—
36#13095511—26no—
37#13095511—26no—
38#13095511C23112yes—
39#13095511C23112yescomplemented
40#13095510—224no—
41C2×Dic194—23no—
42C2×Dic194—26no—
43C2×Dic194C2313yes—
44C2×Dic194C2311yescharacteristic
45C2×Dic194C23112yescomplemented
46Q8×D97—28no—
47C2×D388—23no—
48C2×D388—26no—
49C2×D388C2311yescharacteristic
50C2×D388C2313yes—
51C2×D388C23112yes—
52C2×C22⋊D97—23no—
53C2×C22⋊D97—23no—
54D4×D97—224no—
55C22×Dic97—23no—
56C2×C4×D97—23no—
57C2×C4×D97—23no—
58Dic388C23112yescomplemented
59#13095500C23112yescomplemented
60#13095500C23124yescomplemented
61#13095500C2314yescomplemented
62D776C23112yescomplemented
63#41×C97D414yes—
64#41×C97C2314yes—
65C2×Q8×C97D411yescharacteristic
66C2×D4×C97D413yes—
67C22×C4×C97C2313yes—
68M16×C97C23112yescomplemented
69C2×C8×C97C2313yes—
70C2×C8×C97C2311yescharacteristic
71C2×C8×C97C23112yescomplemented
72C23×C97C2×D413yes—
73C22×D97—21no—
74C22×D97—23no—
75C22×D97—212no—
76C22×D97—86no—
77Q8×C97C2×D414yes—
78D4×C97C2×D4112yes—
79C2×C4×C97C2×D413yes—
80C2×C4×C97C2×D416yes—
81C2×C4×C97C2411yescharacteristic
82C2×C4×C97C2416yes—
83C22⋊D97—212no—
84C22⋊D97—212no—
85C2×Dic97—21no—
86C2×Dic97—23no—
87C2×Dic97—212no—
88D388—212no—
89D388—23no—
90D388—21no—
91D388C2412yes—
92D388C2416yes—
93C4×D97—212no—
94C4×D97—212no—
95Dic194—212no—
96Dic194—23no—
97Dic194—21no—
98Dic194C2416yes—
99Dic194C2412yes—
100C8×C97C2412yes—
101C8×C97C2416yes—
102C22×C97—26no—
103C22×C97C22×D416yes—
104C22×C97C22×D411yescharacteristic
105C2×D97—22no—
106C2×D97—26no—
107C2×D97—42no—
108C2×D97—46no—
109C2×D97—812no—
110C4×C97C22×D411yescharacteristic
111C4×C97C22×D416yes—
112C4×C97C2511yescharacteristic
113Dic97—26no—
114Dic97—22no—
115C2×C97—26no—
116C2×C97#57712yescomplemented
117C2×C97C23×D411yescharacteristic
118D97—42no—
119D97—46no—
120C2×#577—971nomaximal, Sylow
121C97C2×#57711yescharacteristic, complemented, Sylow
122C2×2−1+4—971no—
123C2×H(5,𝔽2)—971no—
124#577—9716no—
125C2×#137—973no—
126C2×AGL(1,ℤ/8)—973no—
127C2×#135—973no—
128C22×SD16—973no—
129C2×#131D9711yescharacteristic, complemented
1302−1+4—978no—
1312−1+4—1944no—
132H(5,𝔽2)—978no—
133H(5,𝔽2)—1944no—
134C2×#41—973no—
135C2×#41—973no—
136C2×#41—1941no—
137C2×#41—1943no—
138C2×#41C2×D9711yescharacteristic
139C22×Q8—973no—
140C22×Q8—1941no—
141C22×D4—973no—
142C22×D4—1943no—
143#137—9724no—
144AGL(1,ℤ/8)—9724no—
145#135—9724no—
146C2×Q16—973no—
147C2×SD16—973no—
148C2×SD16—976no—
149C2×SD16—971no—
150C2×SD16—9724no—
151C2×D8—973no—
152#131C2×D9718yescomplemented
153C2×M16C2×D9713yescomplemented
154C22×C8C2×D9713yescomplemented
155C24—1943no—
156#41—9712no—
157#41—9712no—
158#41—1948no—
159#41—19424no—
160#41—1946no—
161#41—1946no—
162#41D38814yes—
163#41C22×D9714yes—
164C2×Q8—973no—
165C2×Q8—971no—
166C2×Q8—9712no—
167C2×Q8—1948no—
168C2×Q8—1943no—
169C2×Q8—1946no—
170C2×Q8D38811yescharacteristic
171C2×D4—971no—
172C2×D4—973no—
173C2×D4—9712no—
174C2×D4—1943no—
175C2×D4—1943no—
176C2×D4—19424no—
177C2×D4—1943no—
178C2×D4—1946no—
179C2×D4D38813yes—
180C22×C4—1943no—
181C22×C4—1943no—
182C22×C4—1943no—
183C22×C4C22×D9713yes—
184Q16—9712no—
185SD16—9712no—
186SD16—9724no—
187SD16—974no—
188D8—9712no—
189M16C22×D97112yescomplemented
190C2×C8C22×D9711yescharacteristic
191C2×C8C22×D9713yes—
192C2×C8C22×D97112yescomplemented
193C23—7766no—
194C23—19412no—
195C23—1941no—
196C23—1943no—
197C23C2×D38813yes—
198Q8—976no—
199Q8—972no—
200Q8—19412no—
201Q8—1943no—
202Q8—1941no—
203Q8C2×D38814yes—
204D4—972no—
205D4—976no—
206D4—19412no—
207D4—19412no—
208D4—19412no—
209D4—1943no—
210D4—1941no—
211D4C2×D388112yes—
212C2×C4—19412no—
213C2×C4—1941no—
214C2×C4—1943no—
215C2×C4—19412no—
216C2×C4—19412no—
217C2×C4C2×D38816yes—
218C2×C4C2×D38813yes—
219C2×C4C23×D9716yes—
220C2×C4C23×D9711yescharacteristic
221C8C23×D9712yes—
222C8C23×D9716yes—
223C22—26no—
224C22—77612no—
225C22—3886no—
226C22—3882no—
227C22—1942no—
228C22—1946no—
229C22C22×D38816yes—
230C22C22×D38811yescharacteristic
231C4—1942no—
232C4—1946no—
233C4C22×D38816yes—
234C4C22×D38811yescharacteristic
235C4C24×D9711yescharacteristic
236C2—26no—
237C2—3886no—
238C2—3882no—
239C2#2026154712yesdirect product factor
240C2C23×D38811yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×#131×C97C2
socleC22×C97C22×D4
hypercenterC2×#131D97
hypocenterC97C2×#577
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#15690857C2semidirect
2. C2×#15690856C2semidirect
3. #20261547C2direct (N × H)
4. C2×#15690843C2semidirect
5. C2×#15690838C2semidirect
6. C2×#15690832C2semidirect
7. C22×#13095500C2semidirect
8. C2×#131×C97C2semidirect
10. #15690857C22semidirect
12. #15690856C22semidirect
15. C2×#13095511C22semidirect
17. C22×Dic194C22semidirect
21. #15690843C22semidirect
22. #15690838C22semidirect
23. #15690832C22semidirect
24. C2×Dic388C22semidirect
25. C2×#13095500C22semidirect
26. C2×#13095500C22semidirect
27. C2×#13095500C22semidirect
28. C2×#13095500C22semidirect
29. C2×D776C22semidirect
31. #131×C97C22semidirect
32. C2×M16×C97C22semidirect
33. C22×C8×C97C22semidirect
39. #13095511C23semidirect
45. C2×Dic194C23semidirect
58. Dic388C23semidirect
59. #13095500C23semidirect
60. #13095500C23semidirect
61. #13095500C23semidirect
62. D776C23semidirect
68. M16×C97C23semidirect
71. C2×C8×C97C23semidirect
116. C2×C97#577semidirect
121. C97C2×#577semidirect
129. C2×#131D97semidirect
152. #131C2×D97semidirect
153. C2×M16C2×D97semidirect
154. C22×C8C2×D97semidirect
189. M16C22×D97semidirect
192. C2×C8C22×D97semidirect
239. C2#20261547direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908572
C2×#156908562
#202615472
C2×#156908432
C2×#156908382
C2×#156908322
C22×#130955002
C2×#131×C972
C2×#57797