Finite Group Databasebeta

← C23×#4008 #13044486

Subgroups — C23×#4008 #13044486

Order 1280.

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×#711C211yesmaximal, characteristic
2C23×C4×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C23×#700C211yesmaximal, characteristic, complemented
4C22×#4008C2128yesmaximal, direct product factor
5C24×AGL(1,𝔽5)—21no—
6C23×C5×D4C411yescharacteristic, complemented
7C23×C22⋊D5—21no—
8C24×Dic5—21no—
9C22×#711—27no—
10C22×#711—27no—
11C22×#711C4114yescomplemented
12C22×#711C22114yes—
13C23×C4×D5C2211yescharacteristic
14C23×Dic10C411yescharacteristic, complemented
15C23×#170—21no—
16C22×C4×AGL(1,𝔽5)—27no—
17C22×C4×AGL(1,𝔽5)C22114yescomplemented
18C22×#700—27no—
19C22×#700C22114yescomplemented
20C2×#4008C221112yesdirect product factor
21C23×#104—51nomaximal, Sylow
22C25×C5—21no—
23C24×D5D411yescharacteristic
24C23×AGL(1,𝔽5)—216no—
25C23×AGL(1,𝔽5)—214no—
26C22×C5×D4—27no—
27C22×C5×D4C2×C4114yescomplemented
28C23×C4×C5C2×C411yescharacteristic
29C22×C22⋊D5—214no—
30C22×C22⋊D5—214no—
31C23×Dic5—21no—
32C23×Dic5—214no—
33C23×Dic5—214no—
34C23×Dic5D411yescharacteristic
35C2×#711—284no—
36C2×#711—242no—
37C2×#711—242no—
38C2×#711C2×C4184yescomplemented
39C2×#711C23128yes—
40C22×C4×D5C2×C417yes—
41C22×C4×D5D417yes—
42C22×C4×D5D417yes—
43C22×C4×D5C2317yes—
44C22×Dic10—27no—
45C22×Dic10C2×C4114yescomplemented
46C22×#170—214no—
47C2×C4×AGL(1,𝔽5)—242no—
48C2×C4×AGL(1,𝔽5)C23128yescomplemented
49C2×#700—242no—
50C2×#700C23128yescomplemented
51#4008C23164yesdirect product factor
52C23×#41—51no—
53C23×M16—51no—
54C23×C42—51no—
55C22×#104—528no—
56C24×C5—214no—
57C24×C5—48no—
58C24×C5C22⋊C411yescharacteristic
59C23×D5—28no—
60C23×D5C22⋊C417yes—
61C23×D5C2×D417yes—
62C22×AGL(1,𝔽5)—2112no—
63C22×AGL(1,𝔽5)—228no—
64C2×C5×D4—242no—
65C2×C5×D4C22×C4128yescomplemented
66C22×C4×C5C22⋊C417yes—
67C22×C4×C5C22×C417yes—
68C2×C22⋊D5—284no—
69C2×C22⋊D5—228no—
70C22×Dic5—214no—
71C22×Dic5—284no—
72C22×Dic5—228no—
73C22×Dic5C22⋊C417yes—
74C22×Dic5C2×D417yes—
75#711—2168no—
76#711—228no—
77#711—228no—
78#711C22×C4156yescomplemented
79#711C2418yes—
80C2×C4×D5C22⋊C4142yes—
81C2×C4×D5C22×C4121yes—
82C2×C4×D5C2×D4121yes—
83C2×C4×D5C2×D4121yes—
84C2×C4×D5C2417yes—
85C2×Dic10—242no—
86C2×Dic10C22×C4128yescomplemented
87C2×#170—228no—
88C4×AGL(1,𝔽5)—228no—
89C4×AGL(1,𝔽5)C2418yescomplemented
90#700—228no—
91#700C2418yescomplemented
92C22×#41—514no—
93C22×#41—514no—
94C22×#41—107no—
95C22×#41—107no—
96C23×Q8—51no—
97C23×D4—101no—
98C23×D4AGL(1,𝔽5)11yescharacteristic, complemented
99C24×C4—51no—
100C24×C4—101no—
101C24×C4—101no—
102C22×M16—514no—
103C22×M16—107no—
104C23×C8—101no—
105C22×C42—514no—
106C22×C42—107no—
107C2×#104—5112no—
108C23×C5—228no—
109C23×C5—456no—
110C23×C5#10418yescomplemented
111C23×C5C2×C22⋊C417yes—
112C22×D5—228no—
113C22×D5—228no—
114C22×D5C2×C22⋊C4121yes—
115C22×D5C22×D417yes—
116C2×AGL(1,𝔽5)—2112no—
117C2×AGL(1,𝔽5)—28no—
118C5×D4—228no—
119C5×D4C23×C418yescomplemented
120C2×C4×C5C2×C22⋊C4121yes—
121C2×C4×C5C23×C417yes—
122C22⋊D5—256no—
123C22⋊D5—28no—
124C2×Dic5—228no—
125C2×Dic5—256no—
126C2×Dic5—28no—
127C2×Dic5C2×C22⋊C4121yes—
128C2×Dic5C22×D417yes—
129C4×D5C2×C22⋊C4142yes—
130C4×D5C23×C417yes—
131C4×D5C22×D417yes—
132C4×D5C22×D417yes—
133C4×D5C2511yescharacteristic
134Dic10—228no—
135Dic10C23×C418yescomplemented
136#170—28no—
137C25—21no—
138C25—51no—
139C2×#41—584no—
140C2×#41—528no—
141C2×#41—1042no—
142C2×#41—1042no—
143C2×#41—1084no—
144C22×Q8—514no—
145C22×Q8—107no—
146C22×D4—27no—
147C22×D4—1014no—
148C22×D4—1014no—
149C22×D4C2×AGL(1,𝔽5)114yescomplemented
150C23×C4—51no—
151C23×C4—57no—
152C23×C4—57no—
153C23×C4—57no—
154C23×C4—57no—
155C23×C4—1014no—
156C23×C4—1014no—
157C23×C4—1014no—
158C23×C4—1016no—
159C23×C4—101no—
160C23×C4C2×AGL(1,𝔽5)11yescharacteristic
161C2×M16—528no—
162C2×M16—1042no—
163C22×C8—1014no—
164C2×C42—528no—
165C2×C42—1042no—
166#104—564no—
167C22×C5—28no—
168C22×C5—456no—
169C22×C5C2×#104128yescomplemented
170C22×C5C22×C22⋊C417yes—
171C2×D5—214no—
172C2×D5—242no—
173C2×D5C22×C22⋊C417yes—
174C2×D5C23×D411yescharacteristic
175AGL(1,𝔽5)—216no—
176C4×C5C22×C22⋊C417yes—
177C4×C5C24×C411yescharacteristic
178Dic5—28no—
179Dic5C22×C22⋊C417yes—
180Dic5C23×D411yescharacteristic
181C24—214no—
182C24—48no—
183C24—57no—
184C24—57no—
185C24—108no—
186C24C22⋊AGL(1,𝔽5)11yescharacteristic
187#41—58no—
188#41—556no—
189#41—1028no—
190#41—1028no—
191#41—10168no—
192C2×Q8—528no—
193C2×Q8—1042no—
194C2×D4—242no—
195C2×D4—1028no—
196C2×D4—1084no—
197C2×D4C22×AGL(1,𝔽5)128yescomplemented
198C22×C4—57no—
199C22×C4—57no—
200C22×C4—521no—
201C22×C4—521no—
202C22×C4—542no—
203C22×C4—57no—
204C22×C4—521no—
205C22×C4—1028no—
206C22×C4—1084no—
207C22×C4—1028no—
208C22×C4—10112no—
209C22×C4—1014no—
210C22×C4C22⋊AGL(1,𝔽5)17yes—
211C22×C4C22×AGL(1,𝔽5)17yes—
212M16—58no—
213M16—1028no—
214C2×C8—1028no—
215C42—58no—
216C42—1028no—
217C2×C5—48no—
218C2×C5C22×#104114yescomplemented
219C2×C5C23×C22⋊C411yescharacteristic
220D5—21no—
221D5—27no—
222C23—228no—
223C23—456no—
224C23—57no—
225C23—521no—
226C23—1028no—
227C23—1028no—
228C23#400818yesdirect product factor
229C23C2×C22⋊AGL(1,𝔽5)17yes—
230Q8—58no—
231Q8—1028no—
232D4—228no—
233D4—108no—
234D4—1056no—
235D4C23×AGL(1,𝔽5)18yescomplemented
236C2×C4—57no—
237C2×C4—521no—
238C2×C4—57no—
239C2×C4—57no—
240C2×C4—542no—
241C2×C4—51no—
242C2×C4—57no—
243C2×C4—108no—
244C2×C4—1056no—
245C2×C4—108no—
246C2×C4—10112no—
247C2×C4—1028no—
248C2×C4C2×C22⋊AGL(1,𝔽5)121yes—
249C2×C4C23×AGL(1,𝔽5)17yes—
250C8—108no—
251C5C23×#10411yescharacteristic, complemented, Sylow
252C22—28no—
253C22—456no—
254C22—51no—
255C22—57no—
256C22—1014no—
257C22—1042no—
258C22C2×#4008128yesdirect product factor
259C22C22×C22⋊AGL(1,𝔽5)17yes—
260C4—51no—
261C4—57no—
262C4—1016no—
263C4—108no—
264C4C22×C22⋊AGL(1,𝔽5)17yes—
265C4C24×AGL(1,𝔽5)11yescharacteristic
266C2—48no—
267C2—101no—
268C2—107no—
269C2C22×#4008114yesdirect product factor
270C2C23×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC24C22⋊AGL(1,𝔽5)
commutatorC4×C5C24×C4
FrattiniC4C24×AGL(1,𝔽5)
FittingC23×C5×D4C4
socleC24×C5C22⋊C4
prime radicalC23×#711C2
cosocleC4×D5C25
hypercenterC23×D4AGL(1,𝔽5)
hypocenterC5C23×#104
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
2. C23×C4×AGL(1,𝔽5)C2semidirect
3. C23×#700C2semidirect
4. C22×#4008C2direct (N × H)
6. C23×C5×D4C4semidirect
11. C22×#711C4semidirect
14. C23×Dic10C4semidirect
17. C22×C4×AGL(1,𝔽5)C22semidirect
19. C22×#700C22semidirect
20. C2×#4008C22direct (N × H)
27. C22×C5×D4C2×C4semidirect
38. C2×#711C2×C4semidirect
45. C22×Dic10C2×C4semidirect
48. C2×C4×AGL(1,𝔽5)C23semidirect
50. C2×#700C23semidirect
51. #4008C23direct (N × H)
65. C2×C5×D4C22×C4semidirect
78. #711C22×C4semidirect
86. C2×Dic10C22×C4semidirect
89. C4×AGL(1,𝔽5)C24semidirect
91. #700C24semidirect
98. C23×D4AGL(1,𝔽5)semidirect
110. C23×C5#104semidirect
119. C5×D4C23×C4semidirect
135. Dic10C23×C4semidirect
149. C22×D4C2×AGL(1,𝔽5)semidirect
169. C22×C5C2×#104semidirect
197. C2×D4C22×AGL(1,𝔽5)semidirect
218. C2×C5C22×#104semidirect
228. C23#4008direct (N × H)
235. D4C23×AGL(1,𝔽5)semidirect
251. C5C23×#104semidirect
258. C22C2×#4008direct (N × H)
269. C2C22×#4008direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#7112
C23×C4×AGL(1,𝔽5)2
C23×#7002
C22×#40082
C23×#1045