Finite Group Databasebeta

← C23×#41×D53 #28297973

Subgroups — C23×#41×D53 #28297973

Order 13568.

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×#11719949C211yesmaximal, characteristic, complemented
2C23×#11719945C213yesmaximal, complemented
3C23×#11719947C213yesmaximal, complemented
4C22×#41×D53C21112yesmaximal, direct product factor
5C23×Q8×D53C211yesmaximal, characteristic, complemented
6C23×D4×D53C213yesmaximal, complemented
7C24×C4×D53C213yesmaximal, complemented
8C23×#41×C53C211yesmaximal, characteristic, complemented
9C25×D53C2213yes—
10C22×#11719949C22156yescomplemented
11C22×#11719945C221168yescomplemented
12C22×#11719947C221168yescomplemented
13C2×#41×D53C2211792yesdirect product factor
14C23×Dic106C2213yescomplemented
15C22×Q8×D53C22156yescomplemented
16C23×D212C2213yescomplemented
17C23×C22⋊D53C2216yescomplemented
18C22×D4×D53C221168yescomplemented
19C24×Dic53C2213yescomplemented
20C23×C4×D53C2211yescharacteristic
21C23×C4×D53C2213yes—
22C23×C4×D53C2213yescomplemented
23C23×C4×D53C2213yescomplemented
24C23×C4×D53C221168yescomplemented
25C22×#41×C53C22156yescomplemented
26C23×Q8×C53C2211yescharacteristic, complemented
27C23×D4×C53C2213yescomplemented
28C24×C4×C53C2213yescomplemented
29C24×D53—248no—
30C24×D53C23184yes—
31C24×D53C2313yes—
32C24×D53C2311yescharacteristic
33C2×#11719949C231448yescomplemented
34C2×#11719945C2311344yescomplemented
35C2×#11719947C2311344yescomplemented
36#41×D53C2314096yesdirect product factor
37C22×Dic106C23184yescomplemented
38C2×Q8×D53C231448yescomplemented
39C22×D212C23184yescomplemented
40C22×C22⋊D53C231168yescomplemented
41C2×D4×D53C2311344yescomplemented
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2313yes—
44C23×Dic53C23184yescomplemented
45C22×C4×D53C23128yes—
46C22×C4×D53C23184yes—
47C22×C4×D53C2311344yescomplemented
48C22×C4×D53C23184yescomplemented
49C22×C4×D53C23184yescomplemented
50C25×C53C2313yes—
51C2×#41×C53C231448yescomplemented
52C22×Q8×C53C23128yescomplemented
53C22×D4×C53C23184yescomplemented
54C23×C4×C53C2313yes—
55C23×C4×C53C2311yescharacteristic
56C23×C4×C53C23184yescomplemented
57C24×C53—224no—
58C24×C53C24142yes—
59C24×C53C2411yescharacteristic
60C23×D53—224no—
61C23×D53—2672no—
62C23×D53#41116yesdirect product factor
63C23×D53C241336yes—
64C23×D53C24142yes—
65C23×D53C24114yes—
66#41×C53C241512yescomplemented
67C2×Q8×C53C241112yescomplemented
68C2×D4×C53C241336yescomplemented
69C22×C4×C53C24114yes—
70C22×C4×C53C24142yes—
71C22×C4×C53C241336yescomplemented
72C2×C22⋊D53C241672yescomplemented
73C22×Dic53C24114yes—
74C22×Dic53C24142yes—
75C22×Dic53C241336yescomplemented
76#11719949C241512yescomplemented
77Q8×D53C241512yescomplemented
78#11719947C2411536yescomplemented
79D4×D53C2411536yescomplemented
80#11719945C2411536yescomplemented
81C2×D212C241336yescomplemented
82C2×C4×D53C241112yes—
83C2×C4×D53C241336yes—
84C2×C4×D53C2411536yescomplemented
85C2×C4×D53C241336yescomplemented
86C2×C4×D53C241336yescomplemented
87C2×Dic106C241336yescomplemented
88C23×C53—2168no—
89C23×C53C2×#4118yescomplemented
90C23×C53C25184yes—
91C23×C53C2517yes—
92C22×D53—2168no—
93C22×D53—21344no—
94C22×D53C2×#411112yesdirect product factor
95C22×D53C25184yes—
96C22×D53C251192yes—
97C22×D53C25128yes—
98Q8×C53C25164yescomplemented
99D4×C53C251192yescomplemented
100C2×C4×C53C25184yes—
101C2×C4×C53C25128yes—
102C2×C4×C53C251192yescomplemented
103C22⋊D53C251384yescomplemented
104C2×Dic53C25184yes—
105C2×Dic53C25128yes—
106C2×Dic53C251192yescomplemented
107D212C251192yescomplemented
108C4×D53C25164yes—
109C4×D53C251192yes—
110C4×D53C251192yescomplemented
111C4×D53C251192yescomplemented
112Dic106C251192yescomplemented
113C24×#41—531nomaximal, Sylow
114C22×C53—2168no—
115C22×C53C22×#41128yescomplemented
116C22×C53C26124yes—
117C22×C53C2617yes—
118C2×D53—2168no—
119C2×D53—2384no—
120C2×D53C22×#411112yesdirect product factor
121C2×D53C26124yes—
122C2×D53C2618yes—
123C4×C53C26124yes—
124C4×C53C2618yes—
125Dic53C26124yes—
126Dic53C2618yes—
127C23×#41—53112no—
128C23×#41—531no—
129C23×#41—533no—
130C23×#41—533no—
131C23×#41D5311yescharacteristic, direct product factor
132C24×Q8—531no—
133C24×D4—533no—
134C25×C4—533no—
135C2×C53—224no—
136C2×C53C23×#41114yescomplemented
137C2×C53C2711yescharacteristic
138D53—224no—
139D53C23×#41116yesdirect product factor
140C26—533no—
141C22×#41—531792no—
142C22×#41—5356no—
143C22×#41—53168no—
144C22×#41—53168no—
145C22×#41C2×D53156yesdirect product factor
146C23×Q8—5356no—
147C23×Q8—533no—
148C23×Q8C2×D5311yescharacteristic, complemented
149C23×D4—536no—
150C23×D4—53168no—
151C23×D4—533no—
152C23×D4C2×D5313yescomplemented
153C24×C4—533no—
154C24×C4—533no—
155C24×C4—533no—
156C24×C4—531no—
157C24×C4—533no—
158C24×C4—53168no—
159C24×C4C2×D5313yescomplemented
160C53C24×#4111yescharacteristic, complemented, Sylow
161C25—531no—
162C25—5384no—
163C25—533no—
164C25—10648no—
165C25C22×D5313yes—
166C2×#41—534096no—
167C2×#41—53448no—
168C2×#41—531344no—
169C2×#41—531344no—
170C2×#41C22×D531448yesdirect product factor
171C22×Q8—53448no—
172C22×Q8—5384no—
173C22×Q8C22×D53128yescomplemented
174C22×D4—53168no—
175C22×D4—531344no—
176C22×D4—5384no—
177C22×D4C22×D53184yescomplemented
178C23×C4—531no—
179C23×C4—533no—
180C23×C4—5384no—
181C23×C4—5328no—
182C23×C4—5384no—
183C23×C4—5384no—
184C23×C4—5384no—
185C23×C4—531344no—
186C23×C4C22×D5313yes—
187C23×C4C22×D5311yescharacteristic
188C23×C4C22×D53184yescomplemented
189C24—224no—
190C24—5316no—
191C24—5314no—
192C24—53336no—
193C24—5342no—
194C24—106672no—
195C24—10624no—
196C24C23×D53142yes—
197C24C23×D5311yescharacteristic
198#41—53512no—
199#41—531536no—
200#41—531536no—
201#41C23×D531512yesdirect product factor
202C2×Q8—53512no—
203C2×Q8—53336no—
204C2×Q8C23×D531112yescomplemented
205C2×D4—53672no—
206C2×D4—531536no—
207C2×D4—53336no—
208C2×D4C23×D531336yescomplemented
209C22×C4—5314no—
210C22×C4—5342no—
211C22×C4—53336no—
212C22×C4—53112no—
213C22×C4—53336no—
214C22×C4—53336no—
215C22×C4—53336no—
216C22×C4—531536no—
217C22×C4C23×D53114yes—
218C22×C4C23×D53142yes—
219C22×C4C23×D531336yescomplemented
220C23—2168no—
221C23—53112no—
222C23—5328no—
223C23—53192no—
224C23—5384no—
225C23—1061344no—
226C23—106168no—
227C23#41×D5318yesdirect product factor
228C23C24×D53184yes—
229C23C24×D5317yes—
230Q8—53192no—
231Q8C24×D53164yescomplemented
232D4—53384no—
233D4—53192no—
234D4C24×D531192yescomplemented
235C2×C4—5328no—
236C2×C4—5384no—
237C2×C4—53192no—
238C2×C4—5364no—
239C2×C4—53192no—
240C2×C4—53192no—
241C2×C4—53192no—
242C2×C4C24×D53128yes—
243C2×C4C24×D53184yes—
244C2×C4C24×D531192yescomplemented
245C22—2168no—
246C22—53112no—
247C22—538no—
248C22—5324no—
249C22—106384no—
250C22—106168no—
251C22C2×#41×D53128yesdirect product factor
252C22C25×D53124yes—
253C22C25×D5317yes—
254C4—538no—
255C4—5324no—
256C4C25×D5318yes—
257C4C25×D53124yes—
258C2—224no—
259C2—5316no—
260C2—10624no—
261C2C22×#41×D53114yesdirect product factor
262C2C26×D5311yescharacteristic
subgroupG/H
centerC23×C4C22×D53
commutator, cosocleC2×C53C27
FrattiniC2C26×D53
FittingC23×#41×C53C2
socleC24×C53C24
hypercenterC23×#41D53
hypocenterC53C24×#41
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C23×#11719949C2semidirect
2. C23×#11719945C2semidirect
3. C23×#11719947C2semidirect
4. C22×#41×D53C2direct (N × H)
5. C23×Q8×D53C2semidirect
6. C23×D4×D53C2semidirect
7. C24×C4×D53C2semidirect
8. C23×#41×C53C2semidirect
10. C22×#11719949C22semidirect
11. C22×#11719945C22semidirect
12. C22×#11719947C22semidirect
13. C2×#41×D53C22direct (N × H)
14. C23×Dic106C22semidirect
15. C22×Q8×D53C22semidirect
16. C23×D212C22semidirect
17. C23×C22⋊D53C22semidirect
18. C22×D4×D53C22semidirect
19. C24×Dic53C22semidirect
22. C23×C4×D53C22semidirect
23. C23×C4×D53C22semidirect
24. C23×C4×D53C22semidirect
25. C22×#41×C53C22semidirect
26. C23×Q8×C53C22semidirect
27. C23×D4×C53C22semidirect
28. C24×C4×C53C22semidirect
33. C2×#11719949C23semidirect
34. C2×#11719945C23semidirect
35. C2×#11719947C23semidirect
36. #41×D53C23direct (N × H)
37. C22×Dic106C23semidirect
38. C2×Q8×D53C23semidirect
39. C22×D212C23semidirect
40. C22×C22⋊D53C23semidirect
41. C2×D4×D53C23semidirect
44. C23×Dic53C23semidirect
47. C22×C4×D53C23semidirect
48. C22×C4×D53C23semidirect
49. C22×C4×D53C23semidirect
51. C2×#41×C53C23semidirect
52. C22×Q8×C53C23semidirect
53. C22×D4×C53C23semidirect
56. C23×C4×C53C23semidirect
62. C23×D53#41direct (N × H)
66. #41×C53C24semidirect
67. C2×Q8×C53C24semidirect
68. C2×D4×C53C24semidirect
71. C22×C4×C53C24semidirect
72. C2×C22⋊D53C24semidirect
75. C22×Dic53C24semidirect
76. #11719949C24semidirect
77. Q8×D53C24semidirect
78. #11719947C24semidirect
79. D4×D53C24semidirect
80. #11719945C24semidirect
81. C2×D212C24semidirect
84. C2×C4×D53C24semidirect
85. C2×C4×D53C24semidirect
86. C2×C4×D53C24semidirect
87. C2×Dic106C24semidirect
89. C23×C53C2×#41semidirect
94. C22×D53C2×#41direct (N × H)
98. Q8×C53C25semidirect
99. D4×C53C25semidirect
102. C2×C4×C53C25semidirect
103. C22⋊D53C25semidirect
106. C2×Dic53C25semidirect
107. D212C25semidirect
110. C4×D53C25semidirect
111. C4×D53C25semidirect
112. Dic106C25semidirect
115. C22×C53C22×#41semidirect
120. C2×D53C22×#41direct (N × H)
131. C23×#41D53direct (N × H)
136. C2×C53C23×#41semidirect
139. D53C23×#41direct (N × H)
145. C22×#41C2×D53direct (N × H)
148. C23×Q8C2×D53semidirect
152. C23×D4C2×D53semidirect
159. C24×C4C2×D53semidirect
160. C53C24×#41semidirect
170. C2×#41C22×D53direct (N × H)
173. C22×Q8C22×D53semidirect
177. C22×D4C22×D53semidirect
188. C23×C4C22×D53semidirect
201. #41C23×D53direct (N × H)
204. C2×Q8C23×D53semidirect
208. C2×D4C23×D53semidirect
219. C22×C4C23×D53semidirect
227. C23#41×D53direct (N × H)
231. Q8C24×D53semidirect
234. D4C24×D53semidirect
244. C2×C4C24×D53semidirect
251. C22C2×#41×D53direct (N × H)
261. C2C22×#41×D53direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199492
C23×#117199452
C23×#117199472
C22×#41×D532
C23×Q8×D532
C23×D4×D532
C24×C4×D532
C23×#41×C532
C24×#4153