Finite Group Databasebeta

← C2×#120×D101 #27184346

Subgroups — C2×#120×D101 #27184346

Order 12928.

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×D101C211yesmaximal, characteristic, complemented
2C22×D4×D101C213yesmaximal, complemented
3C2×#15692225C213yesmaximal, complemented
4C2×#15692224C211yesmaximal, characteristic, complemented
5C2×#15692223C213yesmaximal, complemented
6#120×D101C2116yesmaximal, direct product factor
7C2×C22⋊C4×D101C213yesmaximal, complemented
8C2×#120×C101C211yesmaximal, characteristic, complemented
9C24×D101—224no—
10C24×D101C2213yes—
11C24×D101C2213yescomplemented
12C24×D101C2218yescomplemented
13C24×D101C2211yescharacteristic, complemented
14C22×D404C2213yescomplemented
15C22×C22⋊D101C2213yescomplemented
16C22×C22⋊D101C2213yescomplemented
17C2×D4×D101—224no—
18C2×D4×D101C22124yescomplemented
19C22×C4×D101C2213yescomplemented
20#15692225C22124yescomplemented
21#15692224C2218yescomplemented
22#15692223C22124yescomplemented
23C2×C22⋊Dic101C2213yescomplemented
24C2×#13108051C2216yescomplemented
25C22⋊C4×D101C22124yescomplemented
26C25×C101C2211yescharacteristic, complemented
27C22×D4×C101C2213yescomplemented
28#120×C101C2218yescomplemented
29C2×C22⋊C4×C101C2213yescomplemented
30C23×D101—2128no—
31C23×D101—212no—
32C23×D101—224no—
33C23×D101—224no—
34C23×D101—296no—
35C23×D101—212no—
36C23×D101D4124yescomplemented
37C23×D101C23112yes—
38C23×D101C2313yes—
39C23×D101C2311yescharacteristic
40C23×D101C23112yescomplemented
41C23×D101C2314yescomplemented
42C23×D101C2311yescharacteristic, complemented
43C2×D404—212no—
44C2×D404C23112yescomplemented
45C2×C22⋊D101—212no—
46C2×C22⋊D101—212no—
47C2×C22⋊D101C23112yescomplemented
48C2×C22⋊D101C23112yescomplemented
49D4×D101—296no—
50C22×Dic101C2313yescomplemented
51C2×C4×D101—212no—
52C2×C4×D101C23112yescomplemented
53C22⋊Dic101C23112yescomplemented
54#13108051C23124yescomplemented
55C24×C101—212no—
56C24×C101C2313yes—
57C24×C101C2314yescomplemented
58C24×C101C2311yescharacteristic, complemented
59C2×D4×C101—212no—
60C2×D4×C101C23112yescomplemented
61C22×C4×C101C2313yescomplemented
62C22⋊C4×C101C23112yescomplemented
63C23×C101—232no—
64C23×C101—212no—
65C23×C101—224no—
66C23×C101—26no—
67C23×C101C2×D4112yescomplemented
68C23×C101C2416yes—
69C23×C101C2411yescharacteristic
70C23×C101C2412yescomplemented
71C22×D101—296no—
72C22×D101—2256no—
73C22×D101—212no—
74C22×D101—224no—
75C22×D101—296no—
76C22×D101—248no—
77C22×D101—248no—
78C22×D101—26no—
79C22×D101—224no—
80C22×D101—416no—
81C22×D101C2×D4112yes—
82C22×D101C2×D4148yescomplemented
83C22×D101C2×D4112yescomplemented
84C22×D101C2416yes—
85C22×D101C2412yes—
86C22×D101C2412yescomplemented
87D4×C101—224no—
88C2×C4×C101—26no—
89C2×C4×C101C2416yescomplemented
90C22⋊D101—224no—
91C22⋊D101—224no—
92C2×Dic101—26no—
93C2×Dic101C2416yescomplemented
94D404—224no—
95C4×D101—224no—
96C22×C101—232no—
97C22×C101—224no—
98C22×C101—212no—
99C22×C101—26no—
100C22×C101—44no—
101C22×C101C22×D416yes—
102C22×C101C22×D4112yescomplemented
103C22×C101C2511yescharacteristic
104C2×D101—296no—
105C2×D101—224no—
106C2×D101—212no—
107C2×D101—296no—
108C2×D101—26no—
109C2×D101—44no—
110C2×D101—416no—
111C2×D101#120116yesdirect product factor
112C2×D101C22×D4112yes—
113C2×D101C22×D4112yescomplemented
114C4×C101—26no—
115Dic101—26no—
116C2×C101—212no—
117C2×C101—42no—
118C2×C101C2×#12014yescomplemented
119C2×C101C23×D413yes—
120D101—212no—
121D101—42no—
122D101C2×#12018yesdirect product factor
123C22×#120—1011nomaximal, Sylow
124C101C22×#12011yescharacteristic, complemented, Sylow
125C26—1011no—
126C23×D4—1013no—
127C2×#120—1011no—
128C2×#120—1013no—
129C2×#120—10116no—
130C2×#120—1013no—
131C2×#120D10111yescharacteristic, direct product factor
132C22×C22⋊C4—1013no—
133C25—1013no—
134C25—1013no—
135C25—1018no—
136C25—1011no—
137C25—20224no—
138C25C2×D10111yescharacteristic, complemented
139C22×D4—1013no—
140C22×D4—1013no—
141C22×D4—10124no—
142C22×D4—1013no—
143C22×D4—20224no—
144C22×D4C2×D10113yescomplemented
145C23×C4—1013no—
146#120—1018no—
147#120—10124no—
148#120—10124no—
149#120C2×D10118yesdirect product factor
150C2×C22⋊C4—1013no—
151C2×C22⋊C4—1016no—
152C2×C22⋊C4—10124no—
153C2×C22⋊C4C2×D10113yescomplemented
154C24—212no—
155C24—1011no—
156C24—10124no—
157C24—10112no—
158C24—1013no—
159C24—10112no—
160C24—1011no—
161C24—1014no—
162C24—20212no—
163C24—20224no—
164C24—20224no—
165C24—20296no—
166C24—202128no—
167C24—20212no—
168C24C22×D10113yes—
169C24C22×D10114yescomplemented
170C24C22×D10111yescharacteristic, complemented
171C2×D4—212no—
172C2×D4—10112no—
173C2×D4—10112no—
174C2×D4—10112no—
175C2×D4—20212no—
176C2×D4—20296no—
177C2×D4—20212no—
178C2×D4—20212no—
179C2×D4C22×D101112yescomplemented
180C22×C4—1013no—
181C22×C4—10112no—
182C22×C4—20212no—
183C22×C4C22×D10113yescomplemented
184C22⋊C4—10112no—
185C22⋊C4—10124no—
186C22⋊C4C22×D101112yescomplemented
187C23—232no—
188C23—212no—
189C23—224no—
190C23—26no—
191C23—10112no—
192C23—1012no—
193C23—10112no—
194C23—10148no—
195C23—1016no—
196C23—1012no—
197C23—40416no—
198C23—20224no—
199C23—20248no—
200C23—20248no—
201C23—20296no—
202C23—202256no—
203C23—20296no—
204C23—2026no—
205C23—20212no—
206C23—20224no—
207C23D4×D101112yescomplemented
208C23C23×D10116yes—
209C23C23×D10111yescharacteristic
210C23C23×D10112yescomplemented
211D4—224no—
212D4—20224no—
213D4—20224no—
214D4—20224no—
215C2×C4—26no—
216C2×C4—1016no—
217C2×C4—20224no—
218C2×C4—2026no—
219C2×C4C23×D10116yescomplemented
220C22—232no—
221C22—212no—
222C22—224no—
223C22—26no—
224C22—44no—
225C22—10112no—
226C22—10116no—
227C22—10112no—
228C22—40416no—
229C22—4044no—
230C22—20296no—
231C22—20296no—
232C22—2026no—
233C22—20212no—
234C22—20224no—
235C22C2×D4×D10116yes—
236C22C2×D4×D101112yescomplemented
237C22C24×D10111yescharacteristic
238C4—26no—
239C4—2026no—
240C2—212no—
241C2—42no—
242C2—1018no—
243C2—4042no—
244C2—20212no—
245C2#120×D10114yesdirect product factor
246C2C22×D4×D10113yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#120×C101C2
socleC23×C101C24
hypercenterC2×#120D101
hypocenterC101C22×#120
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C25×D101C2semidirect
2. C22×D4×D101C2semidirect
3. C2×#15692225C2semidirect
4. C2×#15692224C2semidirect
5. C2×#15692223C2semidirect
6. #120×D101C2direct (N × H)
7. C2×C22⋊C4×D101C2semidirect
8. C2×#120×C101C2semidirect
11. C24×D101C22semidirect
12. C24×D101C22semidirect
13. C24×D101C22semidirect
14. C22×D404C22semidirect
15. C22×C22⋊D101C22semidirect
16. C22×C22⋊D101C22semidirect
18. C2×D4×D101C22semidirect
19. C22×C4×D101C22semidirect
20. #15692225C22semidirect
21. #15692224C22semidirect
22. #15692223C22semidirect
23. C2×C22⋊Dic101C22semidirect
24. C2×#13108051C22semidirect
25. C22⋊C4×D101C22semidirect
26. C25×C101C22semidirect
27. C22×D4×C101C22semidirect
28. #120×C101C22semidirect
29. C2×C22⋊C4×C101C22semidirect
36. C23×D101D4semidirect
40. C23×D101C23semidirect
41. C23×D101C23semidirect
42. C23×D101C23semidirect
44. C2×D404C23semidirect
47. C2×C22⋊D101C23semidirect
48. C2×C22⋊D101C23semidirect
50. C22×Dic101C23semidirect
52. C2×C4×D101C23semidirect
53. C22⋊Dic101C23semidirect
54. #13108051C23semidirect
57. C24×C101C23semidirect
58. C24×C101C23semidirect
60. C2×D4×C101C23semidirect
61. C22×C4×C101C23semidirect
62. C22⋊C4×C101C23semidirect
67. C23×C101C2×D4semidirect
70. C23×C101C24semidirect
82. C22×D101C2×D4semidirect
83. C22×D101C2×D4semidirect
86. C22×D101C24semidirect
89. C2×C4×C101C24semidirect
93. C2×Dic101C24semidirect
102. C22×C101C22×D4semidirect
111. C2×D101#120direct (N × H)
113. C2×D101C22×D4semidirect
118. C2×C101C2×#120semidirect
122. D101C2×#120direct (N × H)
124. C101C22×#120semidirect
131. C2×#120D101direct (N × H)
138. C25C2×D101semidirect
144. C22×D4C2×D101semidirect
149. #120C2×D101direct (N × H)
153. C2×C22⋊C4C2×D101semidirect
169. C24C22×D101semidirect
170. C24C22×D101semidirect
179. C2×D4C22×D101semidirect
183. C22×C4C22×D101semidirect
186. C22⋊C4C22×D101semidirect
207. C23D4×D101semidirect
210. C23C23×D101semidirect
219. C2×C4C23×D101semidirect
236. C22C2×D4×D101semidirect
245. C2#120×D101direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D1012
C22×D4×D1012
C2×#156922252
C2×#156922242
C2×#156922232
#120×D1012
C2×C22⋊C4×D1012
C2×#120×C1012
C22×#120101