Finite Group Databasebeta

← C23×#41×D67 #31769712

Subgroups — C23×#41×D67 #31769712

Order 17152.

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×#11761629C211yesmaximal, characteristic, complemented
2C23×#11761628C213yesmaximal, complemented
3C23×#11761627C213yesmaximal, complemented
4C22×#41×D67C21112yesmaximal, direct product factor
5C23×Q8×D67C211yesmaximal, characteristic, complemented
6C23×D4×D67C213yesmaximal, complemented
7C24×C4×D67C213yesmaximal, complemented
8C23×#41×C67C211yesmaximal, characteristic, complemented
9C25×D67C2213yes—
10C22×#11761629C22156yescomplemented
11C22×#11761628C221168yescomplemented
12C22×#11761627C221168yescomplemented
13C2×#41×D67C2211792yesdirect product factor
14C23×Dic134C2213yescomplemented
15C22×Q8×D67C22156yescomplemented
16C23×D268C2213yescomplemented
17C23×C22⋊D67C2216yescomplemented
18C22×D4×D67C221168yescomplemented
19C24×Dic67C2213yescomplemented
20C23×C4×D67C2211yescharacteristic
21C23×C4×D67C2213yes—
22C23×C4×D67C2213yescomplemented
23C23×C4×D67C2213yescomplemented
24C23×C4×D67C221168yescomplemented
25C22×#41×C67C22156yescomplemented
26C23×Q8×C67C2211yescharacteristic, complemented
27C23×D4×C67C2213yescomplemented
28C24×C4×C67C2213yescomplemented
29C24×D67—248no—
30C24×D67C23184yes—
31C24×D67C2313yes—
32C24×D67C2311yescharacteristic
33C2×#11761629C231448yescomplemented
34C2×#11761628C2311344yescomplemented
35C2×#11761627C2311344yescomplemented
36#41×D67C2314096yesdirect product factor
37C22×Dic134C23184yescomplemented
38C2×Q8×D67C231448yescomplemented
39C22×D268C23184yescomplemented
40C22×C22⋊D67C231168yescomplemented
41C2×D4×D67C2311344yescomplemented
42C23×Dic67C2311yescharacteristic
43C23×Dic67C2313yes—
44C23×Dic67C23184yescomplemented
45C22×C4×D67C23128yes—
46C22×C4×D67C23184yes—
47C22×C4×D67C23184yescomplemented
48C22×C4×D67C23184yescomplemented
49C22×C4×D67C2311344yescomplemented
50C25×C67C2313yes—
51C2×#41×C67C231448yescomplemented
52C22×Q8×C67C23128yescomplemented
53C22×D4×C67C23184yescomplemented
54C23×C4×C67C2313yes—
55C23×C4×C67C2311yescharacteristic
56C23×C4×C67C23184yescomplemented
57C23×D67—224no—
58C23×D67—2672no—
59C23×D67#41116yesdirect product factor
60C23×D67C241336yes—
61C23×D67C24142yes—
62C23×D67C24114yes—
63#11761629C241512yescomplemented
64#11761628C2411536yescomplemented
65#11761627C2411536yescomplemented
66C2×Dic134C241336yescomplemented
67Q8×D67C241512yescomplemented
68C2×D268C241336yescomplemented
69C2×C22⋊D67C241672yescomplemented
70D4×D67C2411536yescomplemented
71C22×Dic67C24114yes—
72C22×Dic67C24142yes—
73C22×Dic67C241336yescomplemented
74C2×C4×D67C241112yes—
75C2×C4×D67C241336yes—
76C2×C4×D67C2411536yescomplemented
77C2×C4×D67C241336yescomplemented
78C2×C4×D67C241336yescomplemented
79C24×C67—224no—
80C24×C67C24142yes—
81C24×C67C2411yescharacteristic
82#41×C67C241512yescomplemented
83C2×Q8×C67C241112yescomplemented
84C2×D4×C67C241336yescomplemented
85C22×C4×C67C24114yes—
86C22×C4×C67C24142yes—
87C22×C4×C67C241336yescomplemented
88C23×C67—2168no—
89C23×C67C2×#4118yescomplemented
90C23×C67C25184yes—
91C23×C67C2517yes—
92C22×D67—2168no—
93C22×D67—21344no—
94C22×D67C2×#411112yesdirect product factor
95C22×D67C251192yes—
96C22×D67C25184yes—
97C22×D67C25128yes—
98Q8×C67C25164yescomplemented
99D4×C67C251192yescomplemented
100C2×C4×C67C25128yes—
101C2×C4×C67C25184yes—
102C2×C4×C67C251192yescomplemented
103C22⋊D67C251384yescomplemented
104C2×Dic67C25128yes—
105C2×Dic67C25184yes—
106C2×Dic67C251192yescomplemented
107D268C251192yescomplemented
108C4×D67C25164yes—
109C4×D67C251192yes—
110C4×D67C251192yescomplemented
111C4×D67C251192yescomplemented
112Dic134C251192yescomplemented
113C22×C67—2168no—
114C22×C67C22×#41128yescomplemented
115C22×C67C26124yes—
116C22×C67C2617yes—
117C2×D67—2168no—
118C2×D67—2384no—
119C2×D67C22×#411112yesdirect product factor
120C2×D67C26124yes—
121C2×D67C2618yes—
122C4×C67C26124yes—
123C4×C67C2618yes—
124Dic67C26124yes—
125Dic67C2618yes—
126C24×#41—671nomaximal, Sylow
127C2×C67—224no—
128C2×C67C23×#41114yescomplemented
129C2×C67C2711yescharacteristic
130D67—224no—
131D67C23×#41116yesdirect product factor
132C23×#41—67112no—
133C23×#41—671no—
134C23×#41—673no—
135C23×#41—673no—
136C23×#41D6711yescharacteristic, direct product factor
137C24×Q8—671no—
138C24×D4—673no—
139C25×C4—673no—
140C67C24×#4111yescharacteristic, complemented, Sylow
141C26—673no—
142C22×#41—671792no—
143C22×#41—6756no—
144C22×#41—67168no—
145C22×#41—67168no—
146C22×#41C2×D67156yesdirect product factor
147C23×Q8—6756no—
148C23×Q8—673no—
149C23×Q8C2×D6711yescharacteristic, complemented
150C23×D4—676no—
151C23×D4—67168no—
152C23×D4—673no—
153C23×D4C2×D6713yescomplemented
154C24×C4—673no—
155C24×C4—673no—
156C24×C4—673no—
157C24×C4—671no—
158C24×C4—673no—
159C24×C4—67168no—
160C24×C4C2×D6713yescomplemented
161C25—671no—
162C25—6784no—
163C25—673no—
164C25—13448no—
165C25C22×D6713yes—
166C2×#41—674096no—
167C2×#41—67448no—
168C2×#41—671344no—
169C2×#41—671344no—
170C2×#41C22×D671448yesdirect product factor
171C22×Q8—67448no—
172C22×Q8—6784no—
173C22×Q8C22×D67128yescomplemented
174C22×D4—67168no—
175C22×D4—671344no—
176C22×D4—6784no—
177C22×D4C22×D67184yescomplemented
178C23×C4—671no—
179C23×C4—673no—
180C23×C4—6784no—
181C23×C4—6728no—
182C23×C4—6784no—
183C23×C4—6784no—
184C23×C4—6784no—
185C23×C4—671344no—
186C23×C4C22×D6713yes—
187C23×C4C22×D6711yescharacteristic
188C23×C4C22×D67184yescomplemented
189C24—224no—
190C24—6716no—
191C24—6714no—
192C24—67336no—
193C24—6742no—
194C24—134672no—
195C24—13424no—
196C24C23×D67142yes—
197C24C23×D6711yescharacteristic
198#41—67512no—
199#41—671536no—
200#41—671536no—
201#41C23×D671512yesdirect product factor
202C2×Q8—67512no—
203C2×Q8—67336no—
204C2×Q8C23×D671112yescomplemented
205C2×D4—67672no—
206C2×D4—671536no—
207C2×D4—67336no—
208C2×D4C23×D671336yescomplemented
209C22×C4—6714no—
210C22×C4—6742no—
211C22×C4—67336no—
212C22×C4—67112no—
213C22×C4—67336no—
214C22×C4—67336no—
215C22×C4—67336no—
216C22×C4—671536no—
217C22×C4C23×D67114yes—
218C22×C4C23×D67142yes—
219C22×C4C23×D671336yescomplemented
220C23—2168no—
221C23—67112no—
222C23—6728no—
223C23—67192no—
224C23—6784no—
225C23—1341344no—
226C23—134168no—
227C23#41×D6718yesdirect product factor
228C23C24×D67184yes—
229C23C24×D6717yes—
230Q8—67192no—
231Q8C24×D67164yescomplemented
232D4—67384no—
233D4—67192no—
234D4C24×D671192yescomplemented
235C2×C4—6728no—
236C2×C4—6784no—
237C2×C4—67192no—
238C2×C4—6764no—
239C2×C4—67192no—
240C2×C4—67192no—
241C2×C4—67192no—
242C2×C4C24×D67128yes—
243C2×C4C24×D67184yes—
244C2×C4C24×D671192yescomplemented
245C22—2168no—
246C22—67112no—
247C22—678no—
248C22—6724no—
249C22—134384no—
250C22—134168no—
251C22C2×#41×D67128yesdirect product factor
252C22C25×D67124yes—
253C22C25×D6717yes—
254C4—678no—
255C4—6724no—
256C4C25×D6718yes—
257C4C25×D67124yes—
258C2—224no—
259C2—6716no—
260C2—13424no—
261C2C22×#41×D67114yesdirect product factor
262C2C26×D6711yescharacteristic
subgroupG/H
centerC23×C4C22×D67
commutator, cosocleC2×C67C27
FrattiniC2C26×D67
FittingC23×#41×C67C2
socleC24×C67C24
hypercenterC23×#41D67
hypocenterC67C24×#41
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C23×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C23×#11761629C2semidirect
2. C23×#11761628C2semidirect
3. C23×#11761627C2semidirect
4. C22×#41×D67C2direct (N × H)
5. C23×Q8×D67C2semidirect
6. C23×D4×D67C2semidirect
7. C24×C4×D67C2semidirect
8. C23×#41×C67C2semidirect
10. C22×#11761629C22semidirect
11. C22×#11761628C22semidirect
12. C22×#11761627C22semidirect
13. C2×#41×D67C22direct (N × H)
14. C23×Dic134C22semidirect
15. C22×Q8×D67C22semidirect
16. C23×D268C22semidirect
17. C23×C22⋊D67C22semidirect
18. C22×D4×D67C22semidirect
19. C24×Dic67C22semidirect
22. C23×C4×D67C22semidirect
23. C23×C4×D67C22semidirect
24. C23×C4×D67C22semidirect
25. C22×#41×C67C22semidirect
26. C23×Q8×C67C22semidirect
27. C23×D4×C67C22semidirect
28. C24×C4×C67C22semidirect
33. C2×#11761629C23semidirect
34. C2×#11761628C23semidirect
35. C2×#11761627C23semidirect
36. #41×D67C23direct (N × H)
37. C22×Dic134C23semidirect
38. C2×Q8×D67C23semidirect
39. C22×D268C23semidirect
40. C22×C22⋊D67C23semidirect
41. C2×D4×D67C23semidirect
44. C23×Dic67C23semidirect
47. C22×C4×D67C23semidirect
48. C22×C4×D67C23semidirect
49. C22×C4×D67C23semidirect
51. C2×#41×C67C23semidirect
52. C22×Q8×C67C23semidirect
53. C22×D4×C67C23semidirect
56. C23×C4×C67C23semidirect
59. C23×D67#41direct (N × H)
63. #11761629C24semidirect
64. #11761628C24semidirect
65. #11761627C24semidirect
66. C2×Dic134C24semidirect
67. Q8×D67C24semidirect
68. C2×D268C24semidirect
69. C2×C22⋊D67C24semidirect
70. D4×D67C24semidirect
73. C22×Dic67C24semidirect
76. C2×C4×D67C24semidirect
77. C2×C4×D67C24semidirect
78. C2×C4×D67C24semidirect
82. #41×C67C24semidirect
83. C2×Q8×C67C24semidirect
84. C2×D4×C67C24semidirect
87. C22×C4×C67C24semidirect
89. C23×C67C2×#41semidirect
94. C22×D67C2×#41direct (N × H)
98. Q8×C67C25semidirect
99. D4×C67C25semidirect
102. C2×C4×C67C25semidirect
103. C22⋊D67C25semidirect
106. C2×Dic67C25semidirect
107. D268C25semidirect
110. C4×D67C25semidirect
111. C4×D67C25semidirect
112. Dic134C25semidirect
114. C22×C67C22×#41semidirect
119. C2×D67C22×#41direct (N × H)
128. C2×C67C23×#41semidirect
131. D67C23×#41direct (N × H)
136. C23×#41D67direct (N × H)
140. C67C24×#41semidirect
146. C22×#41C2×D67direct (N × H)
149. C23×Q8C2×D67semidirect
153. C23×D4C2×D67semidirect
160. C24×C4C2×D67semidirect
170. C2×#41C22×D67direct (N × H)
173. C22×Q8C22×D67semidirect
177. C22×D4C22×D67semidirect
188. C23×C4C22×D67semidirect
201. #41C23×D67direct (N × H)
204. C2×Q8C23×D67semidirect
208. C2×D4C23×D67semidirect
219. C22×C4C23×D67semidirect
227. C23#41×D67direct (N × H)
231. Q8C24×D67semidirect
234. D4C24×D67semidirect
244. C2×C4C24×D67semidirect
251. C22C2×#41×D67direct (N × H)
261. C2C22×#41×D67direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616292
C23×#117616282
C23×#117616272
C22×#41×D672
C23×Q8×D672
C23×D4×D672
C24×C4×D672
C23×#41×C672
C24×#4167