Finite Group Databasebeta

← C2×#120×D31 #16858925

Subgroups — C2×#120×D31 #16858925

Order 3968.

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×D31C211yesmaximal, characteristic, complemented
2C22×D4×D31C213yesmaximal, complemented
3C2×#11758459C213yesmaximal, complemented
4C2×#11758530C211yesmaximal, characteristic, complemented
5C2×#11758514C213yesmaximal, complemented
6#120×D31C2116yesmaximal, direct product factor
7C2×C22⋊C4×D31C213yesmaximal, complemented
8C2×C31×#120C211yesmaximal, characteristic, complemented
9C25×C31C2211yescharacteristic, complemented
10C24×D31—224no—
11C24×D31C2213yes—
12C24×D31C2218yescomplemented
13C24×D31C2213yescomplemented
14C24×D31C2211yescharacteristic, complemented
15C22×D4×C31C2213yescomplemented
16C22×C22⋊D31C2213yescomplemented
17C22×C22⋊D31C2213yescomplemented
18C2×D4×D31—224no—
19C2×D4×D31C22124yescomplemented
20C22×D124C2213yescomplemented
21C22×C4×D31C2213yescomplemented
22C31×#120C2218yescomplemented
23C2×C22⋊C4×C31C2213yescomplemented
24#11758530C2218yescomplemented
25C2×C22⋊Dic31C2213yescomplemented
26#11758514C22124yescomplemented
27C2×(C2×D31)⋊C4C2216yescomplemented
28#11758459C22124yescomplemented
29C22⋊C4×D31C22124yescomplemented
30C24×C31—212no—
31C24×C31C2313yes—
32C24×C31C2314yescomplemented
33C24×C31C2311yescharacteristic, complemented
34C23×D31—2128no—
35C23×D31—212no—
36C23×D31—224no—
37C23×D31—224no—
38C23×D31—296no—
39C23×D31—212no—
40C23×D31D4124yescomplemented
41C23×D31C23112yes—
42C23×D31C2313yes—
43C23×D31C2311yescharacteristic
44C23×D31C23112yescomplemented
45C23×D31C2314yescomplemented
46C23×D31C2311yescharacteristic, complemented
47C2×D4×C31—212no—
48C2×D4×C31C23112yescomplemented
49C22×C4×C31C2313yescomplemented
50C2×C22⋊D31—212no—
51C2×C22⋊D31—212no—
52C2×C22⋊D31C23112yescomplemented
53C2×C22⋊D31C23112yescomplemented
54C22×Dic31C2313yescomplemented
55D4×D31—296no—
56C2×D124—212no—
57C2×D124C23112yescomplemented
58C2×C4×D31—212no—
59C2×C4×D31C23112yescomplemented
60C22⋊C4×C31C23112yescomplemented
61C22⋊Dic31C23112yescomplemented
62(C2×D31)⋊C4C23124yescomplemented
63C23×C31—232no—
64C23×C31—212no—
65C23×C31—224no—
66C23×C31—26no—
67C23×C31C2×D4112yescomplemented
68C23×C31C2416yes—
69C23×C31C2411yescharacteristic
70C23×C31C2412yescomplemented
71C22×D31—296no—
72C22×D31—2256no—
73C22×D31—212no—
74C22×D31—224no—
75C22×D31—296no—
76C22×D31—248no—
77C22×D31—248no—
78C22×D31—26no—
79C22×D31—224no—
80C22×D31—416no—
81C22×D31C2×D4112yes—
82C22×D31C2×D4148yescomplemented
83C22×D31C2×D4112yescomplemented
84C22×D31C2416yes—
85C22×D31C2412yes—
86C22×D31C2412yescomplemented
87D4×C31—224no—
88C2×C4×C31—26no—
89C2×C4×C31C2416yescomplemented
90C22⋊D31—224no—
91C22⋊D31—224no—
92C2×Dic31—26no—
93C2×Dic31C2416yescomplemented
94D124—224no—
95C4×D31—224no—
96C22×#120—311nomaximal, Sylow
97C22×C31—232no—
98C22×C31—224no—
99C22×C31—212no—
100C22×C31—26no—
101C22×C31—44no—
102C22×C31C22×D416yes—
103C22×C31C22×D4112yescomplemented
104C22×C31C2511yescharacteristic
105C2×D31—296no—
106C2×D31—224no—
107C2×D31—212no—
108C2×D31—296no—
109C2×D31—26no—
110C2×D31—44no—
111C2×D31—416no—
112C2×D31#120116yesdirect product factor
113C2×D31C22×D4112yes—
114C2×D31C22×D4112yescomplemented
115C4×C31—26no—
116Dic31—26no—
117C26—311no—
118C23×D4—313no—
119C2×#120—311no—
120C2×#120—313no—
121C2×#120—3116no—
122C2×#120—313no—
123C2×#120D3111yescharacteristic, direct product factor
124C22×C22⋊C4—313no—
125C2×C31—212no—
126C2×C31—42no—
127C2×C31C2×#12014yescomplemented
128C2×C31C23×D413yes—
129D31—212no—
130D31—42no—
131D31C2×#12018yesdirect product factor
132C25—313no—
133C25—313no—
134C25—318no—
135C25—311no—
136C25—6224no—
137C25C2×D3111yescharacteristic, complemented
138C22×D4—313no—
139C22×D4—313no—
140C22×D4—3124no—
141C22×D4—313no—
142C22×D4—6224no—
143C22×D4C2×D3113yescomplemented
144C23×C4—313no—
145#120—318no—
146#120—3124no—
147#120—3124no—
148#120C2×D3118yesdirect product factor
149C2×C22⋊C4—313no—
150C2×C22⋊C4—316no—
151C2×C22⋊C4—3124no—
152C2×C22⋊C4C2×D3113yescomplemented
153C31C22×#12011yescharacteristic, complemented, Sylow
154C24—212no—
155C24—311no—
156C24—3124no—
157C24—3112no—
158C24—313no—
159C24—3112no—
160C24—311no—
161C24—314no—
162C24—6224no—
163C24—6224no—
164C24—6296no—
165C24—6212no—
166C24—62128no—
167C24—6212no—
168C24C22×D3113yes—
169C24C22×D3114yescomplemented
170C24C22×D3111yescharacteristic, complemented
171C2×D4—212no—
172C2×D4—3112no—
173C2×D4—3112no—
174C2×D4—3112no—
175C2×D4—6212no—
176C2×D4—6296no—
177C2×D4—6212no—
178C2×D4—6212no—
179C2×D4C22×D31112yescomplemented
180C22×C4—313no—
181C22×C4—3112no—
182C22×C4—6212no—
183C22×C4C22×D3113yescomplemented
184C22⋊C4—3112no—
185C22⋊C4—3124no—
186C22⋊C4C22×D31112yescomplemented
187C23—232no—
188C23—212no—
189C23—224no—
190C23—26no—
191C23—3112no—
192C23—312no—
193C23—3112no—
194C23—3148no—
195C23—316no—
196C23—312no—
197C23—6248no—
198C23—6248no—
199C23—6296no—
200C23—6224no—
201C23—62256no—
202C23—6296no—
203C23—6212no—
204C23—6224no—
205C23—626no—
206C23—12416no—
207C23D4×D31112yescomplemented
208C23C23×D3116yes—
209C23C23×D3111yescharacteristic
210C23C23×D3112yescomplemented
211D4—224no—
212D4—6224no—
213D4—6224no—
214D4—6224no—
215C2×C4—26no—
216C2×C4—316no—
217C2×C4—6224no—
218C2×C4—626no—
219C2×C4C23×D3116yescomplemented
220C22—232no—
221C22—212no—
222C22—224no—
223C22—26no—
224C22—44no—
225C22—3112no—
226C22—3116no—
227C22—3112no—
228C22—6296no—
229C22—6296no—
230C22—6212no—
231C22—6224no—
232C22—626no—
233C22—12416no—
234C22—1244no—
235C22C2×D4×D3116yes—
236C22C2×D4×D31112yescomplemented
237C22C24×D3111yescharacteristic
238C4—26no—
239C4—626no—
240C2—212no—
241C2—42no—
242C2—318no—
243C2—6212no—
244C2—1242no—
245C2#120×D3114yesdirect product factor
246C2C22×D4×D3113yes—
subgroupG/H
centerC23C23×D31
commutator, cosocleC22×C31C25
FrattiniC22C24×D31
FittingC2×C31×#120C2
socleC23×C31C24
hypercenterC2×#120D31
hypocenterC31C22×#120
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C25×D31C2semidirect
2. C22×D4×D31C2semidirect
3. C2×#11758459C2semidirect
4. C2×#11758530C2semidirect
5. C2×#11758514C2semidirect
6. #120×D31C2direct (N × H)
7. C2×C22⋊C4×D31C2semidirect
8. C2×C31×#120C2semidirect
9. C25×C31C22semidirect
12. C24×D31C22semidirect
13. C24×D31C22semidirect
14. C24×D31C22semidirect
15. C22×D4×C31C22semidirect
16. C22×C22⋊D31C22semidirect
17. C22×C22⋊D31C22semidirect
19. C2×D4×D31C22semidirect
20. C22×D124C22semidirect
21. C22×C4×D31C22semidirect
22. C31×#120C22semidirect
23. C2×C22⋊C4×C31C22semidirect
24. #11758530C22semidirect
25. C2×C22⋊Dic31C22semidirect
26. #11758514C22semidirect
27. C2×(C2×D31)⋊C4C22semidirect
28. #11758459C22semidirect
29. C22⋊C4×D31C22semidirect
32. C24×C31C23semidirect
33. C24×C31C23semidirect
40. C23×D31D4semidirect
44. C23×D31C23semidirect
45. C23×D31C23semidirect
46. C23×D31C23semidirect
48. C2×D4×C31C23semidirect
49. C22×C4×C31C23semidirect
52. C2×C22⋊D31C23semidirect
53. C2×C22⋊D31C23semidirect
54. C22×Dic31C23semidirect
57. C2×D124C23semidirect
59. C2×C4×D31C23semidirect
60. C22⋊C4×C31C23semidirect
61. C22⋊Dic31C23semidirect
62. (C2×D31)⋊C4C23semidirect
67. C23×C31C2×D4semidirect
70. C23×C31C24semidirect
82. C22×D31C2×D4semidirect
83. C22×D31C2×D4semidirect
86. C22×D31C24semidirect
89. C2×C4×C31C24semidirect
93. C2×Dic31C24semidirect
103. C22×C31C22×D4semidirect
112. C2×D31#120direct (N × H)
114. C2×D31C22×D4semidirect
123. C2×#120D31direct (N × H)
127. C2×C31C2×#120semidirect
131. D31C2×#120direct (N × H)
137. C25C2×D31semidirect
143. C22×D4C2×D31semidirect
148. #120C2×D31direct (N × H)
152. C2×C22⋊C4C2×D31semidirect
153. C31C22×#120semidirect
169. C24C22×D31semidirect
170. C24C22×D31semidirect
179. C2×D4C22×D31semidirect
183. C22×C4C22×D31semidirect
186. C22⋊C4C22×D31semidirect
207. C23D4×D31semidirect
210. C23C23×D31semidirect
219. C2×C4C23×D31semidirect
236. C22C2×D4×D31semidirect
245. C2#120×D31direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D312
C22×D4×D312
C2×#117584592
C2×#117585302
C2×#117585142
#120×D312
C2×C22⋊C4×D312
C2×C31×#1202
C22×#12031