Finite Group Databasebeta

← C23×#41×D71 #32911333

Subgroups — C23×#41×D71 #32911333

Order 18176.

65470 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765866C211yesmaximal, characteristic, complemented
2C23×#11765865C213yesmaximal, complemented
3C23×#11765864C213yesmaximal, complemented
4C22×#41×D71C21112yesmaximal, direct product factor
5C23×Q8×D71C211yesmaximal, characteristic, complemented
6C23×D4×D71C213yesmaximal, complemented
7C24×C4×D71C213yesmaximal, complemented
8C23×#41×C71C211yesmaximal, characteristic, complemented
9C25×D71C2213yes—
10C22×#11765866C22156yescomplemented
11C22×#11765865C221168yescomplemented
12C22×#11765864C221168yescomplemented
13C2×#41×D71C2211792yesdirect product factor
14C23×Dic142C2213yescomplemented
15C22×Q8×D71C22156yescomplemented
16C23×D284C2213yescomplemented
17C23×C22⋊D71C2216yescomplemented
18C22×D4×D71C221168yescomplemented
19C24×Dic71C2213yescomplemented
20C23×C4×D71C2211yescharacteristic
21C23×C4×D71C2213yes—
22C23×C4×D71C221168yescomplemented
23C23×C4×D71C2213yescomplemented
24C23×C4×D71C2213yescomplemented
25C22×#41×C71C22156yescomplemented
26C23×Q8×C71C2211yescharacteristic, complemented
27C23×D4×C71C2213yescomplemented
28C24×C4×C71C2213yescomplemented
29C24×D71—248no—
30C24×D71C23184yes—
31C24×D71C2313yes—
32C24×D71C2311yescharacteristic
33C2×#11765866C231448yescomplemented
34C2×#11765865C2311344yescomplemented
35C2×#11765864C2311344yescomplemented
36#41×D71C2314096yesdirect product factor
37C22×Dic142C23184yescomplemented
38C2×Q8×D71C231448yescomplemented
39C22×D284C23184yescomplemented
40C22×C22⋊D71C231168yescomplemented
41C2×D4×D71C2311344yescomplemented
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2313yes—
44C23×Dic71C23184yescomplemented
45C22×C4×D71C23128yes—
46C22×C4×D71C23184yes—
47C22×C4×D71C23184yescomplemented
48C22×C4×D71C23184yescomplemented
49C22×C4×D71C2311344yescomplemented
50C25×C71C2313yes—
51C2×#41×C71C231448yescomplemented
52C22×Q8×C71C23128yescomplemented
53C22×D4×C71C23184yescomplemented
54C23×C4×C71C2313yes—
55C23×C4×C71C2311yescharacteristic
56C23×C4×C71C23184yescomplemented
57C23×D71—224no—
58C23×D71—2672no—
59C23×D71#41116yesdirect product factor
60C23×D71C241336yes—
61C23×D71C24142yes—
62C23×D71C24114yes—
63#11765866C241512yescomplemented
64#11765865C2411536yescomplemented
65#11765864C2411536yescomplemented
66C2×Dic142C241336yescomplemented
67Q8×D71C241512yescomplemented
68C2×D284C241336yescomplemented
69C2×C22⋊D71C241672yescomplemented
70D4×D71C2411536yescomplemented
71C22×Dic71C24114yes—
72C22×Dic71C24142yes—
73C22×Dic71C241336yescomplemented
74C2×C4×D71C241112yes—
75C2×C4×D71C241336yes—
76C2×C4×D71C2411536yescomplemented
77C2×C4×D71C241336yescomplemented
78C2×C4×D71C241336yescomplemented
79C24×C71—224no—
80C24×C71C24142yes—
81C24×C71C2411yescharacteristic
82#41×C71C241512yescomplemented
83C2×Q8×C71C241112yescomplemented
84C2×D4×C71C241336yescomplemented
85C22×C4×C71C24114yes—
86C22×C4×C71C24142yes—
87C22×C4×C71C241336yescomplemented
88C23×C71—2168no—
89C23×C71C2×#4118yescomplemented
90C23×C71C25184yes—
91C23×C71C2517yes—
92C22×D71—2168no—
93C22×D71—21344no—
94C22×D71C2×#411112yesdirect product factor
95C22×D71C251192yes—
96C22×D71C25184yes—
97C22×D71C25128yes—
98Q8×C71C25164yescomplemented
99D4×C71C251192yescomplemented
100C2×C4×C71C25128yes—
101C2×C4×C71C25184yes—
102C2×C4×C71C251192yescomplemented
103C22⋊D71C251384yescomplemented
104C2×Dic71C25128yes—
105C2×Dic71C25184yes—
106C2×Dic71C251192yescomplemented
107D284C251192yescomplemented
108C4×D71C25164yes—
109C4×D71C251192yes—
110C4×D71C251192yescomplemented
111C4×D71C251192yescomplemented
112Dic142C251192yescomplemented
113C22×C71—2168no—
114C22×C71C22×#41128yescomplemented
115C22×C71C26124yes—
116C22×C71C2617yes—
117C2×D71—2168no—
118C2×D71—2384no—
119C2×D71C22×#411112yesdirect product factor
120C2×D71C26124yes—
121C2×D71C2618yes—
122C4×C71C26124yes—
123C4×C71C2618yes—
124Dic71C26124yes—
125Dic71C2618yes—
126C24×#41—711nomaximal, Sylow
127C2×C71—224no—
128C2×C71C23×#41114yescomplemented
129C2×C71C2711yescharacteristic
130D71—224no—
131D71C23×#41116yesdirect product factor
132C23×#41—71112no—
133C23×#41—711no—
134C23×#41—713no—
135C23×#41—713no—
136C23×#41D7111yescharacteristic, direct product factor
137C24×Q8—711no—
138C24×D4—713no—
139C25×C4—713no—
140C71C24×#4111yescharacteristic, complemented, Sylow
141C26—713no—
142C22×#41—711792no—
143C22×#41—7156no—
144C22×#41—71168no—
145C22×#41—71168no—
146C22×#41C2×D71156yesdirect product factor
147C23×Q8—7156no—
148C23×Q8—713no—
149C23×Q8C2×D7111yescharacteristic, complemented
150C23×D4—716no—
151C23×D4—71168no—
152C23×D4—713no—
153C23×D4C2×D7113yescomplemented
154C24×C4—713no—
155C24×C4—713no—
156C24×C4—713no—
157C24×C4—711no—
158C24×C4—713no—
159C24×C4—71168no—
160C24×C4C2×D7113yescomplemented
161C25—711no—
162C25—7184no—
163C25—713no—
164C25—14248no—
165C25C22×D7113yes—
166C2×#41—714096no—
167C2×#41—71448no—
168C2×#41—711344no—
169C2×#41—711344no—
170C2×#41C22×D711448yesdirect product factor
171C22×Q8—71448no—
172C22×Q8—7184no—
173C22×Q8C22×D71128yescomplemented
174C22×D4—71168no—
175C22×D4—711344no—
176C22×D4—7184no—
177C22×D4C22×D71184yescomplemented
178C23×C4—711no—
179C23×C4—713no—
180C23×C4—7184no—
181C23×C4—7128no—
182C23×C4—7184no—
183C23×C4—7184no—
184C23×C4—7184no—
185C23×C4—711344no—
186C23×C4C22×D7113yes—
187C23×C4C22×D7111yescharacteristic
188C23×C4C22×D71184yescomplemented
189C24—224no—
190C24—7116no—
191C24—7114no—
192C24—71336no—
193C24—7142no—
194C24—142672no—
195C24—14224no—
196C24C23×D71142yes—
197C24C23×D7111yescharacteristic
198#41—71512no—
199#41—711536no—
200#41—711536no—
201#41C23×D711512yesdirect product factor
202C2×Q8—71512no—
203C2×Q8—71336no—
204C2×Q8C23×D711112yescomplemented
205C2×D4—71672no—
206C2×D4—711536no—
207C2×D4—71336no—
208C2×D4C23×D711336yescomplemented
209C22×C4—7114no—
210C22×C4—7142no—
211C22×C4—71336no—
212C22×C4—71112no—
213C22×C4—71336no—
214C22×C4—71336no—
215C22×C4—71336no—
216C22×C4—711536no—
217C22×C4C23×D71114yes—
218C22×C4C23×D71142yes—
219C22×C4C23×D711336yescomplemented
220C23—2168no—
221C23—71112no—
222C23—7128no—
223C23—71192no—
224C23—7184no—
225C23—1421344no—
226C23—142168no—
227C23#41×D7118yesdirect product factor
228C23C24×D71184yes—
229C23C24×D7117yes—
230Q8—71192no—
231Q8C24×D71164yescomplemented
232D4—71384no—
233D4—71192no—
234D4C24×D711192yescomplemented
235C2×C4—7128no—
236C2×C4—7184no—
237C2×C4—71192no—
238C2×C4—7164no—
239C2×C4—71192no—
240C2×C4—71192no—
241C2×C4—71192no—
242C2×C4C24×D71128yes—
243C2×C4C24×D71184yes—
244C2×C4C24×D711192yescomplemented
245C22—2168no—
246C22—71112no—
247C22—718no—
248C22—7124no—
249C22—142384no—
250C22—142168no—
251C22C2×#41×D71128yesdirect product factor
252C22C25×D71124yes—
253C22C25×D7117yes—
254C4—718no—
255C4—7124no—
256C4C25×D7118yes—
257C4C25×D71124yes—
258C2—224no—
259C2—7116no—
260C2—14224no—
261C2C22×#41×D71114yesdirect product factor
262C2C26×D7111yescharacteristic
subgroupG/H
centerC23×C4C22×D71
commutator, cosocleC2×C71C27
FrattiniC2C26×D71
FittingC23×#41×C71C2
socleC24×C71C24
hypercenterC23×#41D71
hypocenterC71C24×#41
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C23×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C23×#11765866C2semidirect
2. C23×#11765865C2semidirect
3. C23×#11765864C2semidirect
4. C22×#41×D71C2direct (N × H)
5. C23×Q8×D71C2semidirect
6. C23×D4×D71C2semidirect
7. C24×C4×D71C2semidirect
8. C23×#41×C71C2semidirect
10. C22×#11765866C22semidirect
11. C22×#11765865C22semidirect
12. C22×#11765864C22semidirect
13. C2×#41×D71C22direct (N × H)
14. C23×Dic142C22semidirect
15. C22×Q8×D71C22semidirect
16. C23×D284C22semidirect
17. C23×C22⋊D71C22semidirect
18. C22×D4×D71C22semidirect
19. C24×Dic71C22semidirect
22. C23×C4×D71C22semidirect
23. C23×C4×D71C22semidirect
24. C23×C4×D71C22semidirect
25. C22×#41×C71C22semidirect
26. C23×Q8×C71C22semidirect
27. C23×D4×C71C22semidirect
28. C24×C4×C71C22semidirect
33. C2×#11765866C23semidirect
34. C2×#11765865C23semidirect
35. C2×#11765864C23semidirect
36. #41×D71C23direct (N × H)
37. C22×Dic142C23semidirect
38. C2×Q8×D71C23semidirect
39. C22×D284C23semidirect
40. C22×C22⋊D71C23semidirect
41. C2×D4×D71C23semidirect
44. C23×Dic71C23semidirect
47. C22×C4×D71C23semidirect
48. C22×C4×D71C23semidirect
49. C22×C4×D71C23semidirect
51. C2×#41×C71C23semidirect
52. C22×Q8×C71C23semidirect
53. C22×D4×C71C23semidirect
56. C23×C4×C71C23semidirect
59. C23×D71#41direct (N × H)
63. #11765866C24semidirect
64. #11765865C24semidirect
65. #11765864C24semidirect
66. C2×Dic142C24semidirect
67. Q8×D71C24semidirect
68. C2×D284C24semidirect
69. C2×C22⋊D71C24semidirect
70. D4×D71C24semidirect
73. C22×Dic71C24semidirect
76. C2×C4×D71C24semidirect
77. C2×C4×D71C24semidirect
78. C2×C4×D71C24semidirect
82. #41×C71C24semidirect
83. C2×Q8×C71C24semidirect
84. C2×D4×C71C24semidirect
87. C22×C4×C71C24semidirect
89. C23×C71C2×#41semidirect
94. C22×D71C2×#41direct (N × H)
98. Q8×C71C25semidirect
99. D4×C71C25semidirect
102. C2×C4×C71C25semidirect
103. C22⋊D71C25semidirect
106. C2×Dic71C25semidirect
107. D284C25semidirect
110. C4×D71C25semidirect
111. C4×D71C25semidirect
112. Dic142C25semidirect
114. C22×C71C22×#41semidirect
119. C2×D71C22×#41direct (N × H)
128. C2×C71C23×#41semidirect
131. D71C23×#41direct (N × H)
136. C23×#41D71direct (N × H)
140. C71C24×#41semidirect
146. C22×#41C2×D71direct (N × H)
149. C23×Q8C2×D71semidirect
153. C23×D4C2×D71semidirect
160. C24×C4C2×D71semidirect
170. C2×#41C22×D71direct (N × H)
173. C22×Q8C22×D71semidirect
177. C22×D4C22×D71semidirect
188. C23×C4C22×D71semidirect
201. #41C23×D71direct (N × H)
204. C2×Q8C23×D71semidirect
208. C2×D4C23×D71semidirect
219. C22×C4C23×D71semidirect
227. C23#41×D71direct (N × H)
231. Q8C24×D71semidirect
234. D4C24×D71semidirect
244. C2×C4C24×D71semidirect
251. C22C2×#41×D71direct (N × H)
261. C2C22×#41×D71direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658662
C23×#117658652
C23×#117658642
C22×#41×D712
C23×Q8×D712
C23×D4×D712
C24×C4×D712
C23×#41×C712
C24×#4171