Finite Group Databasebeta

← C2×#120×D167 #36337352

Subgroups — C2×#120×D167 #36337352

Order 21376.

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×D167C211yesmaximal, characteristic, complemented
2C22×D4×D167C213yesmaximal, complemented
3C2×#19128802C213yesmaximal, complemented
4C2×#19128801C211yesmaximal, characteristic, complemented
5C2×#19128800C213yesmaximal, complemented
6#120×D167C2116yesmaximal, direct product factor
7C2×C22⋊C4×D167C213yesmaximal, complemented
8C2×#120×C167C211yesmaximal, characteristic, complemented
9C24×D167—224no—
10C24×D167C2213yes—
11C24×D167C2213yescomplemented
12C24×D167C2218yescomplemented
13C24×D167C2211yescharacteristic, complemented
14C22×D668C2213yescomplemented
15C22×C22⋊D167C2213yescomplemented
16C22×C22⋊D167C2213yescomplemented
17C2×D4×D167—224no—
18C2×D4×D167C22124yescomplemented
19C22×C4×D167C2213yescomplemented
20#19128802C22124yescomplemented
21#19128801C2218yescomplemented
22#19128800C22124yescomplemented
23C2×C22⋊Dic167C2213yescomplemented
24C2×(C2×D167)⋊C4C2216yescomplemented
25C22⋊C4×D167C22124yescomplemented
26C25×C167C2211yescharacteristic, complemented
27C22×D4×C167C2213yescomplemented
28#120×C167C2218yescomplemented
29C2×C22⋊C4×C167C2213yescomplemented
30C23×D167—2128no—
31C23×D167—224no—
32C23×D167—224no—
33C23×D167—296no—
34C23×D167—212no—
35C23×D167—212no—
36C23×D167D4124yescomplemented
37C23×D167C23112yes—
38C23×D167C2313yes—
39C23×D167C2311yescharacteristic
40C23×D167C23112yescomplemented
41C23×D167C2311yescharacteristic, complemented
42C23×D167C2314yescomplemented
43C2×D668—212no—
44C2×D668C23112yescomplemented
45C2×C22⋊D167—212no—
46C2×C22⋊D167—212no—
47C2×C22⋊D167C23112yescomplemented
48C2×C22⋊D167C23112yescomplemented
49D4×D167—296no—
50C22×Dic167C2313yescomplemented
51C2×C4×D167—212no—
52C2×C4×D167C23112yescomplemented
53C22⋊Dic167C23112yescomplemented
54(C2×D167)⋊C4C23124yescomplemented
55C24×C167—212no—
56C24×C167C2313yes—
57C24×C167C2314yescomplemented
58C24×C167C2311yescharacteristic, complemented
59C2×D4×C167—212no—
60C2×D4×C167C23112yescomplemented
61C22×C4×C167C2313yescomplemented
62C22⋊C4×C167C23112yescomplemented
63C22×D167—296no—
64C22×D167—2256no—
65C22×D167—212no—
66C22×D167—224no—
67C22×D167—296no—
68C22×D167—248no—
69C22×D167—248no—
70C22×D167—26no—
71C22×D167—224no—
72C22×D167—416no—
73C22×D167C2×D4112yes—
74C22×D167C2×D4148yescomplemented
75C22×D167C2×D4112yescomplemented
76C22×D167C2416yes—
77C22×D167C2412yes—
78C22×D167C2412yescomplemented
79D668—224no—
80C22⋊D167—224no—
81C22⋊D167—224no—
82C2×Dic167—26no—
83C2×Dic167C2416yescomplemented
84C4×D167—224no—
85C23×C167—232no—
86C23×C167—212no—
87C23×C167—224no—
88C23×C167—26no—
89C23×C167C2×D4112yescomplemented
90C23×C167C2416yes—
91C23×C167C2411yescharacteristic
92C23×C167C2412yescomplemented
93D4×C167—224no—
94C2×C4×C167—26no—
95C2×C4×C167C2416yescomplemented
96C22×C167—232no—
97C22×C167—224no—
98C22×C167—212no—
99C22×C167—26no—
100C22×C167—44no—
101C22×C167C22×D416yes—
102C22×C167C22×D4112yescomplemented
103C22×C167C2511yescharacteristic
104C2×D167—296no—
105C2×D167—224no—
106C2×D167—212no—
107C2×D167—296no—
108C2×D167—26no—
109C2×D167—44no—
110C2×D167—416no—
111C2×D167#120116yesdirect product factor
112C2×D167C22×D4112yes—
113C2×D167C22×D4112yescomplemented
114C4×C167—26no—
115Dic167—26no—
116C2×C167—212no—
117C2×C167—42no—
118C2×C167C2×#12014yescomplemented
119C2×C167C23×D413yes—
120D167—212no—
121D167—42no—
122D167C2×#12018yesdirect product factor
123C167C22×#12011yescharacteristic, complemented, Sylow
124C22×#120—1671nomaximal, Sylow
125C26—1671no—
126C23×D4—1673no—
127C2×#120—1671no—
128C2×#120—1673no—
129C2×#120—16716no—
130C2×#120—1673no—
131C2×#120D16711yescharacteristic, direct product factor
132C22×C22⋊C4—1673no—
133C25—33424no—
134C25—1673no—
135C25—1678no—
136C25—1671no—
137C25—1673no—
138C25C2×D16711yescharacteristic, complemented
139C22×D4—33424no—
140C22×D4—1673no—
141C22×D4—1673no—
142C22×D4—16724no—
143C22×D4—1673no—
144C22×D4C2×D16713yescomplemented
145C23×C4—1673no—
146#120—1678no—
147#120—16724no—
148#120—16724no—
149#120C2×D16718yesdirect product factor
150C2×C22⋊C4—1673no—
151C2×C22⋊C4—1676no—
152C2×C22⋊C4—16724no—
153C2×C22⋊C4C2×D16713yescomplemented
154C24—212no—
155C24—334128no—
156C24—33424no—
157C24—33424no—
158C24—33496no—
159C24—33412no—
160C24—33412no—
161C24—16712no—
162C24—1671no—
163C24—1674no—
164C24—16724no—
165C24—16712no—
166C24—1673no—
167C24—1671no—
168C24C22×D16713yes—
169C24C22×D16714yescomplemented
170C24C22×D16711yescharacteristic, complemented
171C2×D4—212no—
172C2×D4—33412no—
173C2×D4—33496no—
174C2×D4—33412no—
175C2×D4—33412no—
176C2×D4—16712no—
177C2×D4—16712no—
178C2×D4—16712no—
179C2×D4C22×D167112yescomplemented
180C22×C4—33412no—
181C22×C4—1673no—
182C22×C4—16712no—
183C22×C4C22×D16713yescomplemented
184C22⋊C4—16712no—
185C22⋊C4—16724no—
186C22⋊C4C22×D167112yescomplemented
187C23—232no—
188C23—212no—
189C23—224no—
190C23—26no—
191C23—334256no—
192C23—33496no—
193C23—33496no—
194C23—33448no—
195C23—33448no—
196C23—33424no—
197C23—33412no—
198C23—33424no—
199C23—3346no—
200C23—66816no—
201C23—1672no—
202C23—16712no—
203C23—16748no—
204C23—1676no—
205C23—16712no—
206C23—1672no—
207C23D4×D167112yescomplemented
208C23C23×D16716yes—
209C23C23×D16711yescharacteristic
210C23C23×D16712yescomplemented
211D4—224no—
212D4—33424no—
213D4—33424no—
214D4—33424no—
215C2×C4—26no—
216C2×C4—33424no—
217C2×C4—3346no—
218C2×C4—1676no—
219C2×C4C23×D16716yescomplemented
220C22—232no—
221C22—224no—
222C22—212no—
223C22—26no—
224C22—44no—
225C22—33496no—
226C22—33496no—
227C22—33424no—
228C22—33412no—
229C22—3346no—
230C22—66816no—
231C22—6684no—
232C22—16712no—
233C22—16716no—
234C22—16712no—
235C22C2×D4×D16716yes—
236C22C2×D4×D167112yescomplemented
237C22C24×D16711yescharacteristic
238C4—26no—
239C4—3346no—
240C2—212no—
241C2—42no—
242C2—33412no—
243C2—6682no—
244C2—1678no—
245C2#120×D16714yesdirect product factor
246C2C22×D4×D16713yes—
subgroupG/H
centerC23C23×D167
commutator, cosocleC22×C167C25
FrattiniC22C24×D167
FittingC2×#120×C167C2
socleC23×C167C24
hypercenterC2×#120D167
hypocenterC167C22×#120
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C25×D167C2semidirect
2. C22×D4×D167C2semidirect
3. C2×#19128802C2semidirect
4. C2×#19128801C2semidirect
5. C2×#19128800C2semidirect
6. #120×D167C2direct (N × H)
7. C2×C22⋊C4×D167C2semidirect
8. C2×#120×C167C2semidirect
11. C24×D167C22semidirect
12. C24×D167C22semidirect
13. C24×D167C22semidirect
14. C22×D668C22semidirect
15. C22×C22⋊D167C22semidirect
16. C22×C22⋊D167C22semidirect
18. C2×D4×D167C22semidirect
19. C22×C4×D167C22semidirect
20. #19128802C22semidirect
21. #19128801C22semidirect
22. #19128800C22semidirect
23. C2×C22⋊Dic167C22semidirect
24. C2×(C2×D167)⋊C4C22semidirect
25. C22⋊C4×D167C22semidirect
26. C25×C167C22semidirect
27. C22×D4×C167C22semidirect
28. #120×C167C22semidirect
29. C2×C22⋊C4×C167C22semidirect
36. C23×D167D4semidirect
40. C23×D167C23semidirect
41. C23×D167C23semidirect
42. C23×D167C23semidirect
44. C2×D668C23semidirect
47. C2×C22⋊D167C23semidirect
48. C2×C22⋊D167C23semidirect
50. C22×Dic167C23semidirect
52. C2×C4×D167C23semidirect
53. C22⋊Dic167C23semidirect
54. (C2×D167)⋊C4C23semidirect
57. C24×C167C23semidirect
58. C24×C167C23semidirect
60. C2×D4×C167C23semidirect
61. C22×C4×C167C23semidirect
62. C22⋊C4×C167C23semidirect
74. C22×D167C2×D4semidirect
75. C22×D167C2×D4semidirect
78. C22×D167C24semidirect
83. C2×Dic167C24semidirect
89. C23×C167C2×D4semidirect
92. C23×C167C24semidirect
95. C2×C4×C167C24semidirect
102. C22×C167C22×D4semidirect
111. C2×D167#120direct (N × H)
113. C2×D167C22×D4semidirect
118. C2×C167C2×#120semidirect
122. D167C2×#120direct (N × H)
123. C167C22×#120semidirect
131. C2×#120D167direct (N × H)
138. C25C2×D167semidirect
144. C22×D4C2×D167semidirect
149. #120C2×D167direct (N × H)
153. C2×C22⋊C4C2×D167semidirect
169. C24C22×D167semidirect
170. C24C22×D167semidirect
179. C2×D4C22×D167semidirect
183. C22×C4C22×D167semidirect
186. C22⋊C4C22×D167semidirect
207. C23D4×D167semidirect
210. C23C23×D167semidirect
219. C2×C4C23×D167semidirect
236. C22C2×D4×D167semidirect
245. C2#120×D167direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D1672
C22×D4×D1672
C2×#191288022
C2×#191288012
C2×#191288002
#120×D1672
C2×C22⋊C4×D1672
C2×#120×C1672
C22×#120167