Finite Group Databasebeta

← C23×#41×D13 #16776800

Subgroups — C23×#41×D13 #16776800

Order 3328.

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×#6131C211yesmaximal, characteristic, complemented
2C23×#6127C213yesmaximal, complemented
3C23×#6129C213yesmaximal, complemented
4C22×#41×D13C21112yesmaximal, direct product factor
5C23×Q8×D13C211yesmaximal, characteristic, complemented
6C23×D4×D13C213yesmaximal, complemented
7C24×C4×D13C213yesmaximal, complemented
8C23×C13×#41C211yesmaximal, characteristic, complemented
9C25×D13C2213yes—
10C22×C13×#41C22156yescomplemented
11C23×Q8×C13C2211yescharacteristic, complemented
12C23×D4×C13C2213yescomplemented
13C24×C4×C13C2213yescomplemented
14C23×C22⋊D13C2216yescomplemented
15C24×Dic13C2213yescomplemented
16C2×#41×D13C2211792yesdirect product factor
17C22×#6131C22156yescomplemented
18C22×Q8×D13C22156yescomplemented
19C22×#6129C221168yescomplemented
20C22×D4×D13C221168yescomplemented
21C22×#6127C221168yescomplemented
22C23×D52C2213yescomplemented
23C23×C4×D13C2211yescharacteristic
24C23×C4×D13C2213yes—
25C23×C4×D13C221168yescomplemented
26C23×C4×D13C2213yescomplemented
27C23×C4×D13C2213yescomplemented
28C23×Dic26C2213yescomplemented
29C25×C13C2313yes—
30C24×D13—248no—
31C24×D13C2313yes—
32C24×D13C23184yes—
33C24×D13C2311yescharacteristic
34C2×C13×#41C231448yescomplemented
35C22×Q8×C13C23128yescomplemented
36C22×D4×C13C23184yescomplemented
37C23×C4×C13C2313yes—
38C23×C4×C13C2311yescharacteristic
39C23×C4×C13C23184yescomplemented
40C22×C22⋊D13C231168yescomplemented
41C23×Dic13C2313yes—
42C23×Dic13C2311yescharacteristic
43C23×Dic13C23184yescomplemented
44#41×D13C2314096yesdirect product factor
45C2×#6131C231448yescomplemented
46C2×Q8×D13C231448yescomplemented
47C2×#6129C2311344yescomplemented
48C2×D4×D13C2311344yescomplemented
49C2×#6127C2311344yescomplemented
50C22×D52C23184yescomplemented
51C22×C4×D13C23128yes—
52C22×C4×D13C23184yes—
53C22×C4×D13C2311344yescomplemented
54C22×C4×D13C23184yescomplemented
55C22×C4×D13C23184yescomplemented
56C22×Dic26C23184yescomplemented
57C24×#41—131nomaximal, Sylow
58C24×C13—224no—
59C24×C13C24142yes—
60C24×C13C2411yescharacteristic
61C23×D13—224no—
62C23×D13—2672no—
63C23×D13#41116yesdirect product factor
64C23×D13C24142yes—
65C23×D13C241336yes—
66C23×D13C24114yes—
67C13×#41C241512yescomplemented
68C2×Q8×C13C241112yescomplemented
69C2×D4×C13C241336yescomplemented
70C22×C4×C13C24142yes—
71C22×C4×C13C24114yes—
72C22×C4×C13C241336yescomplemented
73C2×C22⋊D13C241672yescomplemented
74C22×Dic13C24142yes—
75C22×Dic13C24114yes—
76C22×Dic13C241336yescomplemented
77#6131C241512yescomplemented
78Q8×D13C241512yescomplemented
79#6129C2411536yescomplemented
80D4×D13C2411536yescomplemented
81#6127C2411536yescomplemented
82C2×D52C241336yescomplemented
83C2×C4×D13C241112yes—
84C2×C4×D13C241336yes—
85C2×C4×D13C2411536yescomplemented
86C2×C4×D13C241336yescomplemented
87C2×C4×D13C241336yescomplemented
88C2×Dic26C241336yescomplemented
89C23×#41—13112no—
90C23×#41—131no—
91C23×#41—133no—
92C23×#41—133no—
93C23×#41D1311yescharacteristic, direct product factor
94C24×Q8—131no—
95C24×D4—133no—
96C25×C4—133no—
97C23×C13—2168no—
98C23×C13C2×#4118yescomplemented
99C23×C13C25184yes—
100C23×C13C2517yes—
101C22×D13—2168no—
102C22×D13—21344no—
103C22×D13C2×#411112yesdirect product factor
104C22×D13C25184yes—
105C22×D13C251192yes—
106C22×D13C25128yes—
107Q8×C13C25164yescomplemented
108D4×C13C251192yescomplemented
109C2×C4×C13C25184yes—
110C2×C4×C13C25128yes—
111C2×C4×C13C251192yescomplemented
112C22⋊D13C251384yescomplemented
113C2×Dic13C25184yes—
114C2×Dic13C25128yes—
115C2×Dic13C251192yescomplemented
116D52C251192yescomplemented
117C4×D13C25164yes—
118C4×D13C251192yes—
119C4×D13C251192yescomplemented
120C4×D13C251192yescomplemented
121Dic26C251192yescomplemented
122C26—133no—
123C22×#41—131792no—
124C22×#41—1356no—
125C22×#41—13168no—
126C22×#41—13168no—
127C22×#41C2×D13156yesdirect product factor
128C23×Q8—1356no—
129C23×Q8—133no—
130C23×Q8C2×D1311yescharacteristic, complemented
131C23×D4—136no—
132C23×D4—13168no—
133C23×D4—133no—
134C23×D4C2×D1313yescomplemented
135C24×C4—133no—
136C24×C4—133no—
137C24×C4—133no—
138C24×C4—131no—
139C24×C4—133no—
140C24×C4—13168no—
141C24×C4C2×D1313yescomplemented
142C22×C13—2168no—
143C22×C13C22×#41128yescomplemented
144C22×C13C26124yes—
145C22×C13C2617yes—
146C2×D13—2168no—
147C2×D13—2384no—
148C2×D13C22×#411112yesdirect product factor
149C2×D13C26124yes—
150C2×D13C2618yes—
151C4×C13C26124yes—
152C4×C13C2618yes—
153Dic13C26124yes—
154Dic13C2618yes—
155C25—131no—
156C25—1384no—
157C25—133no—
158C25—2648no—
159C25C22×D1313yes—
160C2×#41—134096no—
161C2×#41—13448no—
162C2×#41—131344no—
163C2×#41—131344no—
164C2×#41C22×D131448yesdirect product factor
165C22×Q8—13448no—
166C22×Q8—1384no—
167C22×Q8C22×D13128yescomplemented
168C22×D4—13168no—
169C22×D4—131344no—
170C22×D4—1384no—
171C22×D4C22×D13184yescomplemented
172C23×C4—131no—
173C23×C4—133no—
174C23×C4—1384no—
175C23×C4—1328no—
176C23×C4—1384no—
177C23×C4—1384no—
178C23×C4—1384no—
179C23×C4—131344no—
180C23×C4C22×D1313yes—
181C23×C4C22×D1311yescharacteristic
182C23×C4C22×D13184yescomplemented
183C2×C13—224no—
184C2×C13C23×#41114yescomplemented
185C2×C13C2711yescharacteristic
186D13—224no—
187D13C23×#41116yesdirect product factor
188C24—224no—
189C24—1316no—
190C24—1314no—
191C24—13336no—
192C24—1342no—
193C24—26672no—
194C24—2624no—
195C24C23×D13142yes—
196C24C23×D1311yescharacteristic
197#41—13512no—
198#41—131536no—
199#41—131536no—
200#41C23×D131512yesdirect product factor
201C2×Q8—13512no—
202C2×Q8—13336no—
203C2×Q8C23×D131112yescomplemented
204C2×D4—13672no—
205C2×D4—131536no—
206C2×D4—13336no—
207C2×D4C23×D131336yescomplemented
208C22×C4—1314no—
209C22×C4—1342no—
210C22×C4—13336no—
211C22×C4—13112no—
212C22×C4—13336no—
213C22×C4—13336no—
214C22×C4—13336no—
215C22×C4—131536no—
216C22×C4C23×D13114yes—
217C22×C4C23×D13142yes—
218C22×C4C23×D131336yescomplemented
219C13C24×#4111yescharacteristic, complemented, Sylow
220C23—2168no—
221C23—13112no—
222C23—1328no—
223C23—13192no—
224C23—1384no—
225C23—261344no—
226C23—26168no—
227C23#41×D1318yesdirect product factor
228C23C24×D13184yes—
229C23C24×D1317yes—
230Q8—13192no—
231Q8C24×D13164yescomplemented
232D4—13384no—
233D4—13192no—
234D4C24×D131192yescomplemented
235C2×C4—1328no—
236C2×C4—1384no—
237C2×C4—13192no—
238C2×C4—1364no—
239C2×C4—13192no—
240C2×C4—13192no—
241C2×C4—13192no—
242C2×C4C24×D13128yes—
243C2×C4C24×D13184yes—
244C2×C4C24×D131192yescomplemented
245C22—2168no—
246C22—13112no—
247C22—138no—
248C22—1324no—
249C22—26384no—
250C22—26168no—
251C22C2×#41×D13128yesdirect product factor
252C22C25×D13124yes—
253C22C25×D1317yes—
254C4—138no—
255C4—1324no—
256C4C25×D1318yes—
257C4C25×D13124yes—
258C2—224no—
259C2—1316no—
260C2—2624no—
261C2C22×#41×D13114yesdirect product factor
262C2C26×D1311yescharacteristic
subgroupG/H
centerC23×C4C22×D13
commutator, cosocleC2×C13C27
FrattiniC2C26×D13
FittingC23×C13×#41C2
socleC24×C13C24
hypercenterC23×#41D13
hypocenterC13C24×#41
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C23×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C23×#6131C2semidirect
2. C23×#6127C2semidirect
3. C23×#6129C2semidirect
4. C22×#41×D13C2direct (N × H)
5. C23×Q8×D13C2semidirect
6. C23×D4×D13C2semidirect
7. C24×C4×D13C2semidirect
8. C23×C13×#41C2semidirect
10. C22×C13×#41C22semidirect
11. C23×Q8×C13C22semidirect
12. C23×D4×C13C22semidirect
13. C24×C4×C13C22semidirect
14. C23×C22⋊D13C22semidirect
15. C24×Dic13C22semidirect
16. C2×#41×D13C22direct (N × H)
17. C22×#6131C22semidirect
18. C22×Q8×D13C22semidirect
19. C22×#6129C22semidirect
20. C22×D4×D13C22semidirect
21. C22×#6127C22semidirect
22. C23×D52C22semidirect
25. C23×C4×D13C22semidirect
26. C23×C4×D13C22semidirect
27. C23×C4×D13C22semidirect
28. C23×Dic26C22semidirect
34. C2×C13×#41C23semidirect
35. C22×Q8×C13C23semidirect
36. C22×D4×C13C23semidirect
39. C23×C4×C13C23semidirect
40. C22×C22⋊D13C23semidirect
43. C23×Dic13C23semidirect
44. #41×D13C23direct (N × H)
45. C2×#6131C23semidirect
46. C2×Q8×D13C23semidirect
47. C2×#6129C23semidirect
48. C2×D4×D13C23semidirect
49. C2×#6127C23semidirect
50. C22×D52C23semidirect
53. C22×C4×D13C23semidirect
54. C22×C4×D13C23semidirect
55. C22×C4×D13C23semidirect
56. C22×Dic26C23semidirect
63. C23×D13#41direct (N × H)
67. C13×#41C24semidirect
68. C2×Q8×C13C24semidirect
69. C2×D4×C13C24semidirect
72. C22×C4×C13C24semidirect
73. C2×C22⋊D13C24semidirect
76. C22×Dic13C24semidirect
77. #6131C24semidirect
78. Q8×D13C24semidirect
79. #6129C24semidirect
80. D4×D13C24semidirect
81. #6127C24semidirect
82. C2×D52C24semidirect
85. C2×C4×D13C24semidirect
86. C2×C4×D13C24semidirect
87. C2×C4×D13C24semidirect
88. C2×Dic26C24semidirect
93. C23×#41D13direct (N × H)
98. C23×C13C2×#41semidirect
103. C22×D13C2×#41direct (N × H)
107. Q8×C13C25semidirect
108. D4×C13C25semidirect
111. C2×C4×C13C25semidirect
112. C22⋊D13C25semidirect
115. C2×Dic13C25semidirect
116. D52C25semidirect
119. C4×D13C25semidirect
120. C4×D13C25semidirect
121. Dic26C25semidirect
127. C22×#41C2×D13direct (N × H)
130. C23×Q8C2×D13semidirect
134. C23×D4C2×D13semidirect
141. C24×C4C2×D13semidirect
143. C22×C13C22×#41semidirect
148. C2×D13C22×#41direct (N × H)
164. C2×#41C22×D13direct (N × H)
167. C22×Q8C22×D13semidirect
171. C22×D4C22×D13semidirect
182. C23×C4C22×D13semidirect
184. C2×C13C23×#41semidirect
187. D13C23×#41direct (N × H)
200. #41C23×D13direct (N × H)
203. C2×Q8C23×D13semidirect
207. C2×D4C23×D13semidirect
218. C22×C4C23×D13semidirect
219. C13C24×#41semidirect
227. C23#41×D13direct (N × H)
231. Q8C24×D13semidirect
234. D4C24×D13semidirect
244. C2×C4C24×D13semidirect
251. C22C2×#41×D13direct (N × H)
261. C2C22×#41×D13direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#61312
C23×#61272
C23×#61292
C22×#41×D132
C23×Q8×D132
C23×D4×D132
C24×C4×D132
C23×C13×#412
C24×#4113