Finite Group Databasebeta

← C23×#14556656 #34031172

Subgroups — C23×#14556656 #34031172

Order 18688.

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×#11924038C211yesmaximal, characteristic, complemented
2C23×C4×#64408C211yesmaximal, characteristic, complemented
3C22×#14556656C2128yesmaximal, direct product factor
4C23×#11924029C211yesmaximal, characteristic
5C24×#64408—21no—
6C22×#11924038—27no—
7C22×#11924038C22114yescomplemented
8C23×#10596833—21no—
9C22×C4×#64408—27no—
10C22×C4×#64408C22114yescomplemented
11C2×#14556656C221112yesdirect product factor
12C22×#11924029—27no—
13C22×#11924029—27no—
14C22×#11924029C4114yescomplemented
15C22×#11924029C22114yes—
16C23×D292—21no—
17C23×D292C411yescharacteristic, complemented
18C23×C4×D73—21no—
19C23×C4×D73C2211yescharacteristic
20C23×Q8×C73C411yescharacteristic, complemented
21C23×#64408—216no—
22C23×#64408—214no—
23C2×#11924038—242no—
24C2×#11924038C23128yescomplemented
25C22×#10596833—214no—
26C2×C4×#64408—242no—
27C2×C4×#64408C23128yescomplemented
28#14556656C23164yesdirect product factor
29C24×D73—21no—
30C24×D73D411yescharacteristic
31C2×#11924029—284no—
32C2×#11924029—242no—
33C2×#11924029—242no—
34C2×#11924029C2×C4184yescomplemented
35C2×#11924029C23128yes—
36C22×D292—214no—
37C22×D292—214no—
38C22×D292—27no—
39C22×D292C2×C4114yescomplemented
40C23×Dic73D411yescharacteristic
41C22×C4×D73—214no—
42C22×C4×D73—214no—
43C22×C4×D73C2×C417yes—
44C22×C4×D73D417yes—
45C22×C4×D73D417yes—
46C22×C4×D73C2317yes—
47C22×Q8×C73—27no—
48C22×Q8×C73C2×C4114yescomplemented
49C23×C4×C73—21no—
50C23×C4×C73C2×C411yescharacteristic
51C22×#64408—2112no—
52C22×#64408—228no—
53#11924038—228no—
54#11924038C2418yescomplemented
55C2×#10596833—228no—
56C4×#64408—228no—
57C4×#64408C2418yescomplemented
58C23×D73—28no—
59C23×D73—214no—
60C23×D73—48no—
61C23×D73C22⋊C417yes—
62C23×D73C2×D417yes—
63#11924029—2168no—
64#11924029—228no—
65#11924029—228no—
66#11924029C22×C4156yescomplemented
67#11924029C2418yes—
68C2×D292—284no—
69C2×D292—228no—
70C2×D292—242no—
71C2×D292C22×C4128yescomplemented
72C22×Dic73C22⋊C417yes—
73C22×Dic73C2×D417yes—
74C2×C4×D73—284no—
75C2×C4×D73—228no—
76C2×C4×D73C22⋊C4142yes—
77C2×C4×D73C22×C4121yes—
78C2×C4×D73C2×D4121yes—
79C2×C4×D73C2×D4121yes—
80C2×C4×D73C2417yes—
81C24×C73C22⋊C411yescharacteristic
82C2×Q8×C73—242no—
83C2×Q8×C73C22×C4128yescomplemented
84C22×C4×C73—214no—
85C22×C4×C73C22⋊C417yes—
86C22×C4×C73C22×C417yes—
87C23×C73#10418yescomplemented
88C23×C73C2×C22⋊C417yes—
89C22×D73—228no—
90C22×D73—228no—
91C22×D73—228no—
92C22×D73—456no—
93C22×D73C2×C22⋊C4121yes—
94C22×D73C22×D417yes—
95C2×#64408—2112no—
96C2×#64408—28no—
97Q8×C73—228no—
98Q8×C73C23×C418yescomplemented
99C2×C4×C73—228no—
100C2×C4×C73C2×C22⋊C4121yes—
101C2×C4×C73C23×C417yes—
102C2×Dic73C2×C22⋊C4121yes—
103C2×Dic73C22×D417yes—
104D292—256no—
105D292—28no—
106D292—228no—
107D292C23×C418yescomplemented
108C4×D73—256no—
109C4×D73—28no—
110C4×D73C2×C22⋊C4142yes—
111C4×D73C23×C417yes—
112C4×D73C22×D417yes—
113C4×D73C22×D417yes—
114C4×D73C2511yescharacteristic
115#10596833—28no—
116C22×C73C2×#104128yescomplemented
117C22×C73C22×C22⋊C417yes—
118C2×D73—214no—
119C2×D73—242no—
120C2×D73—28no—
121C2×D73—456no—
122C2×D73C22×C22⋊C417yes—
123C2×D73C23×D411yescharacteristic
124#64408—216no—
125C4×C73—28no—
126C4×C73C22×C22⋊C417yes—
127C4×C73C24×C411yescharacteristic
128Dic73C22×C22⋊C417yes—
129Dic73C23×D411yescharacteristic
130C23×#104—731nomaximal, Sylow
131C2×C73C22×#104114yescomplemented
132C2×C73C23×C22⋊C411yescharacteristic
133D73—21no—
134D73—27no—
135D73—48no—
136C23×#41—731no—
137C23×M16—731no—
138C23×C42—731no—
139C22×#104—7328no—
140C73C23×#10411yescharacteristic, complemented, Sylow
141C22×#41—7314no—
142C22×#41—7314no—
143C22×#41—1467no—
144C22×#41—1467no—
145C23×Q8#6440811yescharacteristic, complemented
146C23×D4—731no—
147C23×D4—1461no—
148C24×C4—731no—
149C24×C4—1461no—
150C24×C4—1461no—
151C22×M16—7314no—
152C22×M16—1467no—
153C23×C8—1461no—
154C22×C42—7314no—
155C22×C42—1467no—
156C2×#104—73112no—
157C25—731no—
158C25—1461no—
159C2×#41—7328no—
160C2×#41—7384no—
161C2×#41—14684no—
162C2×#41—14642no—
163C2×#41—14642no—
164C22×Q8—27no—
165C22×Q8C2×#64408114yescomplemented
166C22×D4—7314no—
167C22×D4—14614no—
168C22×D4—14614no—
169C22×D4—1467no—
170C23×C4—21no—
171C23×C4—731no—
172C23×C4—737no—
173C23×C4—737no—
174C23×C4—737no—
175C23×C4—737no—
176C23×C4—14614no—
177C23×C4—14614no—
178C23×C4—14614no—
179C23×C4—14616no—
180C23×C4C2×#6440811yescharacteristic
181C2×M16—7328no—
182C2×M16—14642no—
183C22×C8—14614no—
184C2×C42—7328no—
185C2×C42—14642no—
186#104—7364no—
187C24—2928no—
188C24—737no—
189C24—737no—
190C24—14614no—
191C24—1468no—
192C24C22⋊#6440811yescharacteristic
193#41—7356no—
194#41—738no—
195#41—14628no—
196#41—14628no—
197#41—146168no—
198C2×Q8—242no—
199C2×Q8C22×#64408128yescomplemented
200C2×D4—7328no—
201C2×D4—14628no—
202C2×D4—14684no—
203C2×D4—14642no—
204C22×C4—214no—
205C22×C4—737no—
206C22×C4—737no—
207C22×C4—7321no—
208C22×C4—737no—
209C22×C4—7342no—
210C22×C4—7321no—
211C22×C4—7321no—
212C22×C4—14628no—
213C22×C4—14684no—
214C22×C4—14628no—
215C22×C4—146112no—
216C22×C4C22⋊#6440817yes—
217C22×C4C22×#6440817yes—
218M16—738no—
219M16—14628no—
220C2×C8—14628no—
221C42—738no—
222C42—14628no—
223C23—29256no—
224C23—737no—
225C23—7321no—
226C23—14628no—
227C23—14628no—
228C23—14628no—
229C23#1455665618yesdirect product factor
230C23C2×C22⋊#6440817yes—
231Q8—228no—
232Q8C23×#6440818yescomplemented
233D4—738no—
234D4—1468no—
235D4—14656no—
236D4—14628no—
237C2×C4—228no—
238C2×C4—737no—
239C2×C4—7321no—
240C2×C4—737no—
241C2×C4—737no—
242C2×C4—7342no—
243C2×C4—731no—
244C2×C4—737no—
245C2×C4—1468no—
246C2×C4—14656no—
247C2×C4—1468no—
248C2×C4—146112no—
249C2×C4C2×C22⋊#64408121yes—
250C2×C4C23×#6440817yes—
251C8—1468no—
252C22—29256no—
253C22—731no—
254C22—737no—
255C22—1468no—
256C22—14614no—
257C22—14642no—
258C22C2×#14556656128yesdirect product factor
259C22C22×C22⋊#6440817yes—
260C4—28no—
261C4—731no—
262C4—737no—
263C4—14616no—
264C4C22×C22⋊#6440817yes—
265C4C24×#6440811yescharacteristic
266C2—2928no—
267C2—1461no—
268C2—1467no—
269C2C22×#14556656114yesdirect product factor
270C2C23×C22⋊#6440811yescharacteristic
subgroupG/H
centerC24C22⋊#64408
commutatorC4×C73C24×C4
FrattiniC4C24×#64408
FittingC23×Q8×C73C4
socleC24×C73C22⋊C4
prime radicalC23×#11924029C2
cosocleC4×D73C25
hypercenterC23×Q8#64408
hypocenterC73C23×#104
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11924038C2semidirect
2. C23×C4×#64408C2semidirect
3. C22×#14556656C2direct (N × H)
7. C22×#11924038C22semidirect
10. C22×C4×#64408C22semidirect
11. C2×#14556656C22direct (N × H)
14. C22×#11924029C4semidirect
17. C23×D292C4semidirect
20. C23×Q8×C73C4semidirect
24. C2×#11924038C23semidirect
27. C2×C4×#64408C23semidirect
28. #14556656C23direct (N × H)
34. C2×#11924029C2×C4semidirect
39. C22×D292C2×C4semidirect
48. C22×Q8×C73C2×C4semidirect
54. #11924038C24semidirect
57. C4×#64408C24semidirect
66. #11924029C22×C4semidirect
71. C2×D292C22×C4semidirect
83. C2×Q8×C73C22×C4semidirect
87. C23×C73#104semidirect
98. Q8×C73C23×C4semidirect
107. D292C23×C4semidirect
116. C22×C73C2×#104semidirect
131. C2×C73C22×#104semidirect
140. C73C23×#104semidirect
145. C23×Q8#64408semidirect
165. C22×Q8C2×#64408semidirect
199. C2×Q8C22×#64408semidirect
229. C23#14556656direct (N × H)
232. Q8C23×#64408semidirect
258. C22C2×#14556656direct (N × H)
269. C2C22×#14556656direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240382
C23×C4×#644082
C22×#145566562
C23×#119240292
C23×#10473