Finite Group Databasebeta

← C23×#13133282 #28329280

Subgroups — C23×#13133282 #28329280

Order 13568.

5758 conjugacy classes of subgroups; only the 1208 normal subgroups are shown.

normal subgroups only

This group has 5758 conjugacy classes of subgroups — too many to draw. Only its 1208 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
1C23×#11719936C211yesmaximal, characteristic, complemented
2C23×C4×#6157C211yesmaximal, characteristic, complemented
3C22×#13133282C2128yesmaximal, direct product factor
4C23×#11719949C211yesmaximal, characteristic
5C24×#6157—21no—
6C22×#11719936—27no—
7C22×#11719936C22114yescomplemented
8C23×#88324—21no—
9C22×C4×#6157—27no—
10C22×C4×#6157C22114yescomplemented
11C2×#13133282C221112yesdirect product factor
12C22×#11719949—27no—
13C22×#11719949—27no—
14C22×#11719949C4114yescomplemented
15C22×#11719949C22114yes—
16C23×D212—21no—
17C23×D212C411yescharacteristic, complemented
18C23×C4×D53—21no—
19C23×C4×D53C2211yescharacteristic
20C23×Q8×C53C411yescharacteristic, complemented
21C23×#6157—216no—
22C23×#6157—214no—
23C2×#11719936—242no—
24C2×#11719936C23128yescomplemented
25C22×#88324—214no—
26C2×C4×#6157—242no—
27C2×C4×#6157C23128yescomplemented
28#13133282C23164yesdirect product factor
29C24×D53—21no—
30C24×D53D411yescharacteristic
31C2×#11719949—284no—
32C2×#11719949—242no—
33C2×#11719949—242no—
34C2×#11719949C2×C4184yescomplemented
35C2×#11719949C23128yes—
36C22×D212—214no—
37C22×D212—214no—
38C22×D212—27no—
39C22×D212C2×C4114yescomplemented
40C23×Dic53D411yescharacteristic
41C22×C4×D53—214no—
42C22×C4×D53—214no—
43C22×C4×D53C2×C417yes—
44C22×C4×D53D417yes—
45C22×C4×D53D417yes—
46C22×C4×D53C2317yes—
47C22×Q8×C53—27no—
48C22×Q8×C53C2×C4114yescomplemented
49C23×C4×C53—21no—
50C23×C4×C53C2×C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C23×D53—28no—
53C23×D53—214no—
54C23×D53—48no—
55C23×D53C22⋊C417yes—
56C23×D53C2×D417yes—
57C22×#6157—2112no—
58C22×#6157—228no—
59C2×Q8×C53—242no—
60C2×Q8×C53C22×C4128yescomplemented
61C22×C4×C53—214no—
62C22×C4×C53C22⋊C417yes—
63C22×C4×C53C22×C417yes—
64C22×Dic53C22⋊C417yes—
65C22×Dic53C2×D417yes—
66#11719949—2168no—
67#11719949—228no—
68#11719949—228no—
69#11719949C22×C4156yescomplemented
70#11719949C2418yes—
71C2×D212—284no—
72C2×D212—228no—
73C2×D212—242no—
74C2×D212C22×C4128yescomplemented
75C2×C4×D53—284no—
76C2×C4×D53—228no—
77C2×C4×D53C22⋊C4142yes—
78C2×C4×D53C22×C4121yes—
79C2×C4×D53C2×D4121yes—
80C2×C4×D53C2×D4121yes—
81C2×C4×D53C2417yes—
82C2×#88324—228no—
83C4×#6157—228no—
84C4×#6157C2418yescomplemented
85#11719936—228no—
86#11719936C2418yescomplemented
87C23×C53#10418yescomplemented
88C23×C53C2×C22⋊C417yes—
89C22×D53—228no—
90C22×D53—228no—
91C22×D53—228no—
92C22×D53—456no—
93C22×D53C2×C22⋊C4121yes—
94C22×D53C22×D417yes—
95C2×#6157—2112no—
96C2×#6157—28no—
97Q8×C53—228no—
98Q8×C53C23×C418yescomplemented
99C2×C4×C53—228no—
100C2×C4×C53C2×C22⋊C4121yes—
101C2×C4×C53C23×C417yes—
102C2×Dic53C2×C22⋊C4121yes—
103C2×Dic53C22×D417yes—
104D212—256no—
105D212—28no—
106D212—228no—
107D212C23×C418yescomplemented
108C4×D53—256no—
109C4×D53—28no—
110C4×D53C2×C22⋊C4142yes—
111C4×D53C23×C417yes—
112C4×D53C22×D417yes—
113C4×D53C22×D417yes—
114C4×D53C2511yescharacteristic
115#88324—28no—
116C23×#104—531nomaximal, Sylow
117C22×C53C2×#104128yescomplemented
118C22×C53C22×C22⋊C417yes—
119C2×D53—214no—
120C2×D53—242no—
121C2×D53—28no—
122C2×D53—456no—
123C2×D53C22×C22⋊C417yes—
124C2×D53C23×D411yescharacteristic
125#6157—216no—
126C4×C53—28no—
127C4×C53C22×C22⋊C417yes—
128C4×C53C24×C411yescharacteristic
129Dic53C22×C22⋊C417yes—
130Dic53C23×D411yescharacteristic
131C23×#41—531no—
132C23×M16—531no—
133C23×C42—531no—
134C22×#104—5328no—
135C2×C53C22×#104114yescomplemented
136C2×C53C23×C22⋊C411yescharacteristic
137D53—21no—
138D53—27no—
139D53—48no—
140C22×#41—5314no—
141C22×#41—5314no—
142C22×#41—1067no—
143C22×#41—1067no—
144C23×Q8#615711yescharacteristic, complemented
145C23×D4—531no—
146C23×D4—1061no—
147C24×C4—531no—
148C24×C4—1061no—
149C24×C4—1061no—
150C22×M16—5314no—
151C22×M16—1067no—
152C23×C8—1061no—
153C22×C42—5314no—
154C22×C42—1067no—
155C2×#104—53112no—
156C53C23×#10411yescharacteristic, complemented, Sylow
157C25—531no—
158C25—1061no—
159C2×#41—5328no—
160C2×#41—5384no—
161C2×#41—10684no—
162C2×#41—10642no—
163C2×#41—10642no—
164C22×Q8—27no—
165C22×Q8C2×#6157114yescomplemented
166C22×D4—5314no—
167C22×D4—10614no—
168C22×D4—10614no—
169C22×D4—1067no—
170C23×C4—21no—
171C23×C4—531no—
172C23×C4—537no—
173C23×C4—537no—
174C23×C4—537no—
175C23×C4—537no—
176C23×C4—10614no—
177C23×C4—10614no—
178C23×C4—10614no—
179C23×C4—10616no—
180C23×C4C2×#615711yescharacteristic
181C2×M16—5328no—
182C2×M16—10642no—
183C22×C8—10614no—
184C2×C42—5328no—
185C2×C42—10642no—
186#104—5364no—
187C24—537no—
188C24—537no—
189C24—1068no—
190C24—10614no—
191C24—2128no—
192C24C22⋊#615711yescharacteristic
193#41—5356no—
194#41—538no—
195#41—106168no—
196#41—10628no—
197#41—10628no—
198C2×Q8—242no—
199C2×Q8C22×#6157128yescomplemented
200C2×D4—5328no—
201C2×D4—10684no—
202C2×D4—10628no—
203C2×D4—10642no—
204C22×C4—214no—
205C22×C4—537no—
206C22×C4—537no—
207C22×C4—5342no—
208C22×C4—5321no—
209C22×C4—5321no—
210C22×C4—5321no—
211C22×C4—537no—
212C22×C4—10684no—
213C22×C4—10628no—
214C22×C4—10628no—
215C22×C4—106112no—
216C22×C4C22⋊#615717yes—
217C22×C4C22×#615717yes—
218M16—538no—
219M16—10628no—
220C2×C8—10628no—
221C42—538no—
222C42—10628no—
223C23—5321no—
224C23—537no—
225C23—10628no—
226C23—10628no—
227C23—10628no—
228C23—21256no—
229C23#1313328218yesdirect product factor
230C23C2×C22⋊#615717yes—
231Q8—228no—
232Q8C23×#615718yescomplemented
233D4—538no—
234D4—1068no—
235D4—10656no—
236D4—10628no—
237C2×C4—228no—
238C2×C4—5321no—
239C2×C4—537no—
240C2×C4—5342no—
241C2×C4—537no—
242C2×C4—537no—
243C2×C4—537no—
244C2×C4—531no—
245C2×C4—1068no—
246C2×C4—10656no—
247C2×C4—1068no—
248C2×C4—106112no—
249C2×C4C2×C22⋊#6157121yes—
250C2×C4C23×#615717yes—
251C8—1068no—
252C22—531no—
253C22—537no—
254C22—10614no—
255C22—10642no—
256C22—1068no—
257C22—21256no—
258C22C2×#13133282128yesdirect product factor
259C22C22×C22⋊#615717yes—
260C4—28no—
261C4—531no—
262C4—537no—
263C4—10616no—
264C4C22×C22⋊#615717yes—
265C4C24×#615711yescharacteristic
266C2—1061no—
267C2—1067no—
268C2—2128no—
269C2C22×#13133282114yesdirect product factor
270C2C23×C22⋊#615711yescharacteristic
subgroupG/H
centerC24C22⋊#6157
commutatorC4×C53C24×C4
FrattiniC4C24×#6157
FittingC23×Q8×C53C4
socleC24×C53C22⋊C4
prime radicalC23×#11719949C2
cosocleC4×D53C25
hypercenterC23×Q8#6157
hypocenterC53C23×#104
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11719936C2semidirect
2. C23×C4×#6157C2semidirect
3. C22×#13133282C2direct (N × H)
7. C22×#11719936C22semidirect
10. C22×C4×#6157C22semidirect
11. C2×#13133282C22direct (N × H)
14. C22×#11719949C4semidirect
17. C23×D212C4semidirect
20. C23×Q8×C53C4semidirect
24. C2×#11719936C23semidirect
27. C2×C4×#6157C23semidirect
28. #13133282C23direct (N × H)
34. C2×#11719949C2×C4semidirect
39. C22×D212C2×C4semidirect
48. C22×Q8×C53C2×C4semidirect
60. C2×Q8×C53C22×C4semidirect
69. #11719949C22×C4semidirect
74. C2×D212C22×C4semidirect
84. C4×#6157C24semidirect
86. #11719936C24semidirect
87. C23×C53#104semidirect
98. Q8×C53C23×C4semidirect
107. D212C23×C4semidirect
117. C22×C53C2×#104semidirect
135. C2×C53C22×#104semidirect
144. C23×Q8#6157semidirect
156. C53C23×#104semidirect
165. C22×Q8C2×#6157semidirect
199. C2×Q8C22×#6157semidirect
229. C23#13133282direct (N × H)
232. Q8C23×#6157semidirect
258. C22C2×#13133282direct (N × H)
269. C2C22×#13133282direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199362
C23×C4×#61572
C22×#131332822
C23×#117199492
C23×#10453