Finite Group Databasebeta

← C2×#120×D131 #30679175

Subgroups — C2×#120×D131 #30679175

Order 16768.

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×D131C211yesmaximal, characteristic, complemented
2C22×D4×D131C213yesmaximal, complemented
3C2×#16862216C213yesmaximal, complemented
4C2×#16862215C211yesmaximal, characteristic, complemented
5C2×#16862214C213yesmaximal, complemented
6#120×D131C2116yesmaximal, direct product factor
7C2×C22⋊C4×D131C213yesmaximal, complemented
8C2×#120×C131C211yesmaximal, characteristic, complemented
9C24×D131—224no—
10C24×D131C2213yes—
11C24×D131C2213yescomplemented
12C24×D131C2218yescomplemented
13C24×D131C2211yescharacteristic, complemented
14C22×D524C2213yescomplemented
15C22×C22⋊D131C2213yescomplemented
16C22×C22⋊D131C2213yescomplemented
17C2×D4×D131—224no—
18C2×D4×D131C22124yescomplemented
19C22×C4×D131C2213yescomplemented
20#16862216C22124yescomplemented
21#16862215C2218yescomplemented
22#16862214C22124yescomplemented
23C2×C22⋊Dic131C2213yescomplemented
24C2×(C2×D131)⋊C4C2216yescomplemented
25C22⋊C4×D131C22124yescomplemented
26C25×C131C2211yescharacteristic, complemented
27C22×D4×C131C2213yescomplemented
28#120×C131C2218yescomplemented
29C2×C22⋊C4×C131C2213yescomplemented
30C23×D131—2128no—
31C23×D131—224no—
32C23×D131—224no—
33C23×D131—296no—
34C23×D131—212no—
35C23×D131—212no—
36C23×D131D4124yescomplemented
37C23×D131C23112yes—
38C23×D131C2313yes—
39C23×D131C2311yescharacteristic
40C23×D131C23112yescomplemented
41C23×D131C2311yescharacteristic, complemented
42C23×D131C2314yescomplemented
43C2×D524—212no—
44C2×D524C23112yescomplemented
45C2×C22⋊D131—212no—
46C2×C22⋊D131—212no—
47C2×C22⋊D131C23112yescomplemented
48C2×C22⋊D131C23112yescomplemented
49D4×D131—296no—
50C22×Dic131C2313yescomplemented
51C2×C4×D131—212no—
52C2×C4×D131C23112yescomplemented
53C22⋊Dic131C23112yescomplemented
54(C2×D131)⋊C4C23124yescomplemented
55C24×C131—212no—
56C24×C131C2313yes—
57C24×C131C2314yescomplemented
58C24×C131C2311yescharacteristic, complemented
59C2×D4×C131—212no—
60C2×D4×C131C23112yescomplemented
61C22×C4×C131C2313yescomplemented
62C22⋊C4×C131C23112yescomplemented
63C22×D131—296no—
64C22×D131—2256no—
65C22×D131—212no—
66C22×D131—224no—
67C22×D131—296no—
68C22×D131—248no—
69C22×D131—248no—
70C22×D131—26no—
71C22×D131—224no—
72C22×D131—416no—
73C22×D131C2×D4112yes—
74C22×D131C2×D4148yescomplemented
75C22×D131C2×D4112yescomplemented
76C22×D131C2416yes—
77C22×D131C2412yes—
78C22×D131C2412yescomplemented
79D524—224no—
80C22⋊D131—224no—
81C22⋊D131—224no—
82C2×Dic131—26no—
83C2×Dic131C2416yescomplemented
84C4×D131—224no—
85C23×C131—232no—
86C23×C131—212no—
87C23×C131—224no—
88C23×C131—26no—
89C23×C131C2×D4112yescomplemented
90C23×C131C2416yes—
91C23×C131C2411yescharacteristic
92C23×C131C2412yescomplemented
93D4×C131—224no—
94C2×C4×C131—26no—
95C2×C4×C131C2416yescomplemented
96C22×C131—232no—
97C22×C131—224no—
98C22×C131—212no—
99C22×C131—26no—
100C22×C131—44no—
101C22×C131C22×D416yes—
102C22×C131C22×D4112yescomplemented
103C22×C131C2511yescharacteristic
104C2×D131—296no—
105C2×D131—224no—
106C2×D131—212no—
107C2×D131—296no—
108C2×D131—26no—
109C2×D131—44no—
110C2×D131—416no—
111C2×D131#120116yesdirect product factor
112C2×D131C22×D4112yes—
113C2×D131C22×D4112yescomplemented
114C4×C131—26no—
115Dic131—26no—
116C2×C131—212no—
117C2×C131—42no—
118C2×C131C2×#12014yescomplemented
119C2×C131C23×D413yes—
120D131—212no—
121D131—42no—
122D131C2×#12018yesdirect product factor
123C131C22×#12011yescharacteristic, complemented, Sylow
124C22×#120—1311nomaximal, Sylow
125C26—1311no—
126C23×D4—1313no—
127C2×#120—1311no—
128C2×#120—1313no—
129C2×#120—13116no—
130C2×#120—1313no—
131C2×#120D13111yescharacteristic, direct product factor
132C22×C22⋊C4—1313no—
133C25—26224no—
134C25—1313no—
135C25—1318no—
136C25—1311no—
137C25—1313no—
138C25C2×D13111yescharacteristic, complemented
139C22×D4—26224no—
140C22×D4—1313no—
141C22×D4—1313no—
142C22×D4—13124no—
143C22×D4—1313no—
144C22×D4C2×D13113yescomplemented
145C23×C4—1313no—
146#120—1318no—
147#120—13124no—
148#120—13124no—
149#120C2×D13118yesdirect product factor
150C2×C22⋊C4—1313no—
151C2×C22⋊C4—1316no—
152C2×C22⋊C4—13124no—
153C2×C22⋊C4C2×D13113yescomplemented
154C24—212no—
155C24—262128no—
156C24—26224no—
157C24—26224no—
158C24—26296no—
159C24—26212no—
160C24—26212no—
161C24—13112no—
162C24—1311no—
163C24—1314no—
164C24—13124no—
165C24—13112no—
166C24—1313no—
167C24—1311no—
168C24C22×D13113yes—
169C24C22×D13114yescomplemented
170C24C22×D13111yescharacteristic, complemented
171C2×D4—212no—
172C2×D4—26212no—
173C2×D4—26296no—
174C2×D4—26212no—
175C2×D4—26212no—
176C2×D4—13112no—
177C2×D4—13112no—
178C2×D4—13112no—
179C2×D4C22×D131112yescomplemented
180C22×C4—26212no—
181C22×C4—1313no—
182C22×C4—13112no—
183C22×C4C22×D13113yescomplemented
184C22⋊C4—13112no—
185C22⋊C4—13124no—
186C22⋊C4C22×D131112yescomplemented
187C23—232no—
188C23—212no—
189C23—224no—
190C23—26no—
191C23—262256no—
192C23—26296no—
193C23—26248no—
194C23—26248no—
195C23—26296no—
196C23—26224no—
197C23—26212no—
198C23—26224no—
199C23—2626no—
200C23—52416no—
201C23—1312no—
202C23—13112no—
203C23—13148no—
204C23—1316no—
205C23—13112no—
206C23—1312no—
207C23D4×D131112yescomplemented
208C23C23×D13116yes—
209C23C23×D13111yescharacteristic
210C23C23×D13112yescomplemented
211D4—224no—
212D4—26224no—
213D4—26224no—
214D4—26224no—
215C2×C4—26no—
216C2×C4—26224no—
217C2×C4—2626no—
218C2×C4—1316no—
219C2×C4C23×D13116yescomplemented
220C22—232no—
221C22—212no—
222C22—224no—
223C22—26no—
224C22—44no—
225C22—26296no—
226C22—26296no—
227C22—26212no—
228C22—26224no—
229C22—2626no—
230C22—52416no—
231C22—5244no—
232C22—13112no—
233C22—13112no—
234C22—13116no—
235C22C2×D4×D13116yes—
236C22C2×D4×D131112yescomplemented
237C22C24×D13111yescharacteristic
238C4—26no—
239C4—2626no—
240C2—212no—
241C2—42no—
242C2—26212no—
243C2—5242no—
244C2—1318no—
245C2#120×D13114yesdirect product factor
246C2C22×D4×D13113yes—
subgroupG/H
centerC23C23×D131
commutator, cosocleC22×C131C25
FrattiniC22C24×D131
FittingC2×#120×C131C2
socleC23×C131C24
hypercenterC2×#120D131
hypocenterC131C22×#120
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C25×D131C2semidirect
2. C22×D4×D131C2semidirect
3. C2×#16862216C2semidirect
4. C2×#16862215C2semidirect
5. C2×#16862214C2semidirect
6. #120×D131C2direct (N × H)
7. C2×C22⋊C4×D131C2semidirect
8. C2×#120×C131C2semidirect
11. C24×D131C22semidirect
12. C24×D131C22semidirect
13. C24×D131C22semidirect
14. C22×D524C22semidirect
15. C22×C22⋊D131C22semidirect
16. C22×C22⋊D131C22semidirect
18. C2×D4×D131C22semidirect
19. C22×C4×D131C22semidirect
20. #16862216C22semidirect
21. #16862215C22semidirect
22. #16862214C22semidirect
23. C2×C22⋊Dic131C22semidirect
24. C2×(C2×D131)⋊C4C22semidirect
25. C22⋊C4×D131C22semidirect
26. C25×C131C22semidirect
27. C22×D4×C131C22semidirect
28. #120×C131C22semidirect
29. C2×C22⋊C4×C131C22semidirect
36. C23×D131D4semidirect
40. C23×D131C23semidirect
41. C23×D131C23semidirect
42. C23×D131C23semidirect
44. C2×D524C23semidirect
47. C2×C22⋊D131C23semidirect
48. C2×C22⋊D131C23semidirect
50. C22×Dic131C23semidirect
52. C2×C4×D131C23semidirect
53. C22⋊Dic131C23semidirect
54. (C2×D131)⋊C4C23semidirect
57. C24×C131C23semidirect
58. C24×C131C23semidirect
60. C2×D4×C131C23semidirect
61. C22×C4×C131C23semidirect
62. C22⋊C4×C131C23semidirect
74. C22×D131C2×D4semidirect
75. C22×D131C2×D4semidirect
78. C22×D131C24semidirect
83. C2×Dic131C24semidirect
89. C23×C131C2×D4semidirect
92. C23×C131C24semidirect
95. C2×C4×C131C24semidirect
102. C22×C131C22×D4semidirect
111. C2×D131#120direct (N × H)
113. C2×D131C22×D4semidirect
118. C2×C131C2×#120semidirect
122. D131C2×#120direct (N × H)
123. C131C22×#120semidirect
131. C2×#120D131direct (N × H)
138. C25C2×D131semidirect
144. C22×D4C2×D131semidirect
149. #120C2×D131direct (N × H)
153. C2×C22⋊C4C2×D131semidirect
169. C24C22×D131semidirect
170. C24C22×D131semidirect
179. C2×D4C22×D131semidirect
183. C22×C4C22×D131semidirect
186. C22⋊C4C22×D131semidirect
207. C23D4×D131semidirect
210. C23C23×D131semidirect
219. C2×C4C23×D131semidirect
236. C22C2×D4×D131semidirect
245. C2#120×D131direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D1312
C22×D4×D1312
C2×#168622162
C2×#168622152
C2×#168622142
#120×D1312
C2×C22⋊C4×D1312
C2×#120×C1312
C22×#120131