Finite Group Databasebeta

← C2×#120×D97 #27154950

Subgroups — C2×#120×D97 #27154950

Order 12416.

4544 conjugacy classes of subgroups; only the 661 normal subgroups are shown.

normal subgroups only

This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 661 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
1C25×D97C211yesmaximal, characteristic, complemented
2C22×D4×D97C213yesmaximal, complemented
3C2×#15690780C213yesmaximal, complemented
4C2×#15690779C211yesmaximal, characteristic, complemented
5C2×#15690778C213yesmaximal, complemented
6#120×D97C2116yesmaximal, direct product factor
7C2×C22⋊C4×D97C213yesmaximal, complemented
8C2×#120×C97C211yesmaximal, characteristic, complemented
9C24×D97—224no—
10C24×D97C2213yes—
11C24×D97C2213yescomplemented
12C24×D97C2218yescomplemented
13C24×D97C2211yescharacteristic, complemented
14C22×D388C2213yescomplemented
15C22×C22⋊D97C2213yescomplemented
16C22×C22⋊D97C2213yescomplemented
17C2×D4×D97—224no—
18C2×D4×D97C22124yescomplemented
19C22×C4×D97C2213yescomplemented
20#15690780C22124yescomplemented
21#15690779C2218yescomplemented
22#15690778C22124yescomplemented
23C2×C22⋊Dic97C2213yescomplemented
24C2×#13095488C2216yescomplemented
25C22⋊C4×D97C22124yescomplemented
26C25×C97C2211yescharacteristic, complemented
27C22×D4×C97C2213yescomplemented
28#120×C97C2218yescomplemented
29C2×C22⋊C4×C97C2213yescomplemented
30C23×D97—2128no—
31C23×D97—212no—
32C23×D97—224no—
33C23×D97—224no—
34C23×D97—296no—
35C23×D97—212no—
36C23×D97D4124yescomplemented
37C23×D97C23112yes—
38C23×D97C2313yes—
39C23×D97C2311yescharacteristic
40C23×D97C23112yescomplemented
41C23×D97C2314yescomplemented
42C23×D97C2311yescharacteristic, complemented
43C2×D388—212no—
44C2×D388C23112yescomplemented
45C2×C22⋊D97—212no—
46C2×C22⋊D97—212no—
47C2×C22⋊D97C23112yescomplemented
48C2×C22⋊D97C23112yescomplemented
49D4×D97—296no—
50C22×Dic97C2313yescomplemented
51C2×C4×D97—212no—
52C2×C4×D97C23112yescomplemented
53C22⋊Dic97C23112yescomplemented
54#13095488C23124yescomplemented
55C24×C97—212no—
56C24×C97C2313yes—
57C24×C97C2314yescomplemented
58C24×C97C2311yescharacteristic, complemented
59C2×D4×C97—212no—
60C2×D4×C97C23112yescomplemented
61C22×C4×C97C2313yescomplemented
62C22⋊C4×C97C23112yescomplemented
63C23×C97—232no—
64C23×C97—212no—
65C23×C97—224no—
66C23×C97—26no—
67C23×C97C2×D4112yescomplemented
68C23×C97C2416yes—
69C23×C97C2411yescharacteristic
70C23×C97C2412yescomplemented
71C22×D97—296no—
72C22×D97—2256no—
73C22×D97—212no—
74C22×D97—224no—
75C22×D97—296no—
76C22×D97—248no—
77C22×D97—248no—
78C22×D97—26no—
79C22×D97—224no—
80C22×D97—416no—
81C22×D97C2×D4112yes—
82C22×D97C2×D4148yescomplemented
83C22×D97C2×D4112yescomplemented
84C22×D97C2416yes—
85C22×D97C2412yes—
86C22×D97C2412yescomplemented
87D4×C97—224no—
88C2×C4×C97—26no—
89C2×C4×C97C2416yescomplemented
90C22⋊D97—224no—
91C22⋊D97—224no—
92C2×Dic97—26no—
93C2×Dic97C2416yescomplemented
94D388—224no—
95C4×D97—224no—
96C22×C97—232no—
97C22×C97—224no—
98C22×C97—212no—
99C22×C97—26no—
100C22×C97—44no—
101C22×C97C22×D416yes—
102C22×C97C22×D4112yescomplemented
103C22×C97C2511yescharacteristic
104C2×D97—296no—
105C2×D97—224no—
106C2×D97—212no—
107C2×D97—296no—
108C2×D97—26no—
109C2×D97—44no—
110C2×D97—416no—
111C2×D97#120116yesdirect product factor
112C2×D97C22×D4112yes—
113C2×D97C22×D4112yescomplemented
114C4×C97—26no—
115Dic97—26no—
116C2×C97—212no—
117C2×C97—42no—
118C2×C97C2×#12014yescomplemented
119C2×C97C23×D413yes—
120D97—212no—
121D97—42no—
122D97C2×#12018yesdirect product factor
123C22×#120—971nomaximal, Sylow
124C97C22×#12011yescharacteristic, complemented, Sylow
125C26—971no—
126C23×D4—973no—
127C2×#120—971no—
128C2×#120—973no—
129C2×#120—9716no—
130C2×#120—973no—
131C2×#120D9711yescharacteristic, direct product factor
132C22×C22⋊C4—973no—
133C25—973no—
134C25—973no—
135C25—978no—
136C25—971no—
137C25—19424no—
138C25C2×D9711yescharacteristic, complemented
139C22×D4—973no—
140C22×D4—973no—
141C22×D4—9724no—
142C22×D4—973no—
143C22×D4—19424no—
144C22×D4C2×D9713yescomplemented
145C23×C4—973no—
146#120—978no—
147#120—9724no—
148#120—9724no—
149#120C2×D9718yesdirect product factor
150C2×C22⋊C4—973no—
151C2×C22⋊C4—976no—
152C2×C22⋊C4—9724no—
153C2×C22⋊C4C2×D9713yescomplemented
154C24—212no—
155C24—971no—
156C24—9724no—
157C24—9712no—
158C24—973no—
159C24—9712no—
160C24—971no—
161C24—974no—
162C24—19412no—
163C24—19424no—
164C24—19424no—
165C24—19496no—
166C24—194128no—
167C24—19412no—
168C24C22×D9713yes—
169C24C22×D9714yescomplemented
170C24C22×D9711yescharacteristic, complemented
171C2×D4—212no—
172C2×D4—9712no—
173C2×D4—9712no—
174C2×D4—9712no—
175C2×D4—19412no—
176C2×D4—19496no—
177C2×D4—19412no—
178C2×D4—19412no—
179C2×D4C22×D97112yescomplemented
180C22×C4—973no—
181C22×C4—9712no—
182C22×C4—19412no—
183C22×C4C22×D9713yescomplemented
184C22⋊C4—9712no—
185C22⋊C4—9724no—
186C22⋊C4C22×D97112yescomplemented
187C23—232no—
188C23—212no—
189C23—224no—
190C23—26no—
191C23—9712no—
192C23—972no—
193C23—9712no—
194C23—9748no—
195C23—976no—
196C23—972no—
197C23—38816no—
198C23—19424no—
199C23—19496no—
200C23—19448no—
201C23—19448no—
202C23—194256no—
203C23—19496no—
204C23—1946no—
205C23—19412no—
206C23—19424no—
207C23D4×D97112yescomplemented
208C23C23×D9716yes—
209C23C23×D9711yescharacteristic
210C23C23×D9712yescomplemented
211D4—224no—
212D4—19424no—
213D4—19424no—
214D4—19424no—
215C2×C4—26no—
216C2×C4—976no—
217C2×C4—19424no—
218C2×C4—1946no—
219C2×C4C23×D9716yescomplemented
220C22—232no—
221C22—224no—
222C22—212no—
223C22—26no—
224C22—44no—
225C22—9716no—
226C22—9712no—
227C22—9712no—
228C22—38816no—
229C22—3884no—
230C22—19496no—
231C22—19496no—
232C22—1946no—
233C22—19424no—
234C22—19412no—
235C22C2×D4×D9716yes—
236C22C2×D4×D97112yescomplemented
237C22C24×D9711yescharacteristic
238C4—26no—
239C4—1946no—
240C2—212no—
241C2—42no—
242C2—978no—
243C2—3882no—
244C2—19412no—
245C2#120×D9714yesdirect product factor
246C2C22×D4×D9713yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#120×C97C2
socleC23×C97C24
hypercenterC2×#120D97
hypocenterC97C22×#120
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C25×D97C2semidirect
2. C22×D4×D97C2semidirect
3. C2×#15690780C2semidirect
4. C2×#15690779C2semidirect
5. C2×#15690778C2semidirect
6. #120×D97C2direct (N × H)
7. C2×C22⋊C4×D97C2semidirect
8. C2×#120×C97C2semidirect
11. C24×D97C22semidirect
12. C24×D97C22semidirect
13. C24×D97C22semidirect
14. C22×D388C22semidirect
15. C22×C22⋊D97C22semidirect
16. C22×C22⋊D97C22semidirect
18. C2×D4×D97C22semidirect
19. C22×C4×D97C22semidirect
20. #15690780C22semidirect
21. #15690779C22semidirect
22. #15690778C22semidirect
23. C2×C22⋊Dic97C22semidirect
24. C2×#13095488C22semidirect
25. C22⋊C4×D97C22semidirect
26. C25×C97C22semidirect
27. C22×D4×C97C22semidirect
28. #120×C97C22semidirect
29. C2×C22⋊C4×C97C22semidirect
36. C23×D97D4semidirect
40. C23×D97C23semidirect
41. C23×D97C23semidirect
42. C23×D97C23semidirect
44. C2×D388C23semidirect
47. C2×C22⋊D97C23semidirect
48. C2×C22⋊D97C23semidirect
50. C22×Dic97C23semidirect
52. C2×C4×D97C23semidirect
53. C22⋊Dic97C23semidirect
54. #13095488C23semidirect
57. C24×C97C23semidirect
58. C24×C97C23semidirect
60. C2×D4×C97C23semidirect
61. C22×C4×C97C23semidirect
62. C22⋊C4×C97C23semidirect
67. C23×C97C2×D4semidirect
70. C23×C97C24semidirect
82. C22×D97C2×D4semidirect
83. C22×D97C2×D4semidirect
86. C22×D97C24semidirect
89. C2×C4×C97C24semidirect
93. C2×Dic97C24semidirect
102. C22×C97C22×D4semidirect
111. C2×D97#120direct (N × H)
113. C2×D97C22×D4semidirect
118. C2×C97C2×#120semidirect
122. D97C2×#120direct (N × H)
124. C97C22×#120semidirect
131. C2×#120D97direct (N × H)
138. C25C2×D97semidirect
144. C22×D4C2×D97semidirect
149. #120C2×D97direct (N × H)
153. C2×C22⋊C4C2×D97semidirect
169. C24C22×D97semidirect
170. C24C22×D97semidirect
179. C2×D4C22×D97semidirect
183. C22×C4C22×D97semidirect
186. C22⋊C4C22×D97semidirect
207. C23D4×D97semidirect
210. C23C23×D97semidirect
219. C2×C4C23×D97semidirect
236. C22C2×D4×D97semidirect
245. C2#120×D97direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D972
C22×D4×D972
C2×#156907802
C2×#156907792
C2×#156907782
#120×D972
C2×C22⋊C4×D972
C2×#120×C972
C22×#12097