Finite Group Databasebeta

← D23×#470 #15682995

Subgroups — D23×#470 #15682995

Order 2944.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×Q16×D23C211yesmaximal, characteristic, complemented
2C2×M16×D23C211yesmaximal, characteristic
3C22⋊Q8×D23C212yesmaximal
4#13092676C211yesmaximal, characteristic, complemented
5#13092675C211yesmaximal, characteristic, complemented
6#13092674C212yesmaximal, complemented
7#13092673C212yesmaximal, complemented
8#13092672C211yesmaximal, characteristic, complemented
9#106×D23C211yesmaximal, characteristic, complemented
10#103×D23C212yesmaximal, complemented
11C23×#470C211yesmaximal, characteristic, complemented
12C2×Q8×D23C2212yes—
13C22×C4×D23C2211yescharacteristic
14C2×Q16×C23C2211yescharacteristic, complemented
15C2×M16×C23C2211yescharacteristic
16C23×C22⋊Q8C2212yes—
17#10625117C2212yes—
18C2×#67275C2212yescomplemented
19#10625095C2212yes—
20C2×#67267C2211yescharacteristic
21Q16×D23—24no—
22M16×D23—24no—
23C2×Dic92C2211yescharacteristic, complemented
24C2×#67262C2211yescharacteristic
25C2×#67262C2211yescharacteristic
26C2×C8×D23C2211yescharacteristic
27C2×C8×D23C2211yescharacteristic
28#10625068C2212yes—
29#10625067—22no—
30C4⋊C4×D23—22no—
31C4⋊C4×D23C2212yes—
32C22⋊C4×D23—22no—
33#10625049—22no—
34C23×#106C2211yescharacteristic, complemented
35C23×#103C2212yescomplemented
36#10625017C2212yescomplemented
37#10624999C2211yescharacteristic, complemented
38#10624998C2212yescomplemented
39#10624992C2212yescomplemented
40#10624990C2212yescomplemented
41C23×D23D411yescharacteristic
42C2×Q8×C23C2312yes—
43C22×C4×C23C2311yescharacteristic
44C22×Dic23D411yescharacteristic
45Q8×D23—24no—
46Q8×D23—24no—
47C2×C4×D23—22no—
48C2×C4×D23—22no—
49C2×C4×D23—22no—
50C2×C4×D23—22no—
51C2×C4×D23D411yescharacteristic
52C2×C4×D23D411yescharacteristic
53C2×C4×D23C2311yescharacteristic
54C2×C4×D23C2311yescharacteristic
55C2×Dic46—22no—
56C2×Dic46C2312yes—
57Q16×C23—22no—
58M16×C23—22no—
59C2×C8×C23C2311yescharacteristic
60C2×C8×C23C2311yescharacteristic
61C4⋊C4×C23—22no—
62C4⋊C4×C23C2312yes—
63C22⋊C4×C23—22no—
64C22⋊Dic23—22no—
65#67275—24no—
66(C2×D23)⋊C4—22no—
67(C2×D23)⋊C4—22no—
68C4⋊Dic23—22no—
69C4⋊Dic23C2312yes—
70Dic23⋊C4—22no—
71Dic23⋊C4—22no—
72Dic23⋊C4—22no—
73#67267—22no—
74C2×C23⋊C8C2311yescharacteristic
75C2×C23⋊C8C2311yescharacteristic
76Dic92—22no—
77#67262—22no—
78#67262—22no—
79C8×D23—22no—
80C8×D23D414yes—
81C23×C23C2×D411yescharacteristic
82C22×D23—22no—
83C22×D23—24no—
84C22×D23C2×D411yescharacteristic
85C22×D23C2×D411yescharacteristic
86Q8×C23—22no—
87Q8×C23—22no—
88C2×C4×C23—21no—
89C2×C4×C23—21no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23C2×D411yescharacteristic
93C2×C4×C23C2411yescharacteristic
94C2×Dic23—22no—
95C2×Dic23—22no—
96C2×Dic23—22no—
97C2×Dic23C2×D411yescharacteristic
98C2×Dic23C2×D411yescharacteristic
99C4×D23—22no—
100C4×D23—21no—
101C4×D23—21no—
102C4×D23—22no—
103C4×D23—44no—
104C4×D23—44no—
105C4×D23C2×D412yes—
106C4×D23#4112yes—
107Dic46—22no—
108Dic46—22no—
109Dic46—44no—
110C8×C23—21no—
111C8×C23C2×D412yes—
112C23⋊C8—21no—
113C23⋊C8C2×D412yes—
114C2×#470—231nomaximal, Sylow
115C22×C23—21no—
116C22×C23—22no—
117C22×C23C22×D411yescharacteristic
118C2×D23—21no—
119C2×D23—22no—
120C2×D23—44no—
121C2×D23#12112yes—
122C2×D23#13714yes—
123C4×C23—21no—
124C4×C23—42no—
125C4×C23—42no—
126C4×C23C22×D411yescharacteristic
127C4×C23C2×#4111yescharacteristic
128Dic23—21no—
129Dic23—42no—
130Dic23—42no—
131Dic23#12112yes—
132C22×Q16—231no—
133C22×M16—231no—
134C2×C22⋊Q8—232no—
135#470—231no—
136#470—231no—
137#470—232no—
138#470—231no—
139#470—232no—
140#470D2311yescharacteristic, direct product factor
141C2×#106—231no—
142C2×#103—232no—
143C2×C23—41no—
144C2×C23C2×#12111yescharacteristic
145C2×C23C2×#13712yes—
146D23—41no—
147D23#47014yesdirect product factor
148C22×Q8—232no—
149C23×C4—231no—
150C2×Q16—232no—
151C2×Q16—231no—
152C2×Q16—464no—
153C2×Q16C2×D2311yescharacteristic, complemented
154C2×M16—231no—
155C2×M16—231no—
156C2×M16—231no—
157C2×M16—464no—
158C2×M16C2×D2311yescharacteristic
159C22×C8—231no—
160C22×C8—231no—
161C22⋊Q8—232no—
162C22⋊Q8—232no—
163C22⋊Q8—232no—
164C22⋊Q8—462no—
165C22⋊Q8—462no—
166C22⋊Q8C2×D2312yes—
167C2×C4⋊C4—232no—
168C2×C4⋊C4—462no—
169C2×C22⋊C4—462no—
170#106—231no—
171#106—232no—
172#106C2×D2311yescharacteristic, complemented
173#103—232no—
174#103—232no—
175#103—232no—
176#103C2×D2312yescomplemented
177C23C2×#47011yescharacteristic, complemented, Sylow
178C24—231no—
179C2×Q8—232no—
180C2×Q8—462no—
181C2×Q8—464no—
182C2×Q8—464no—
183C2×Q8C22×D2312yes—
184C22×C4—231no—
185C22×C4—231no—
186C22×C4—231no—
187C22×C4—231no—
188C22×C4—231no—
189C22×C4—462no—
190C22×C4—462no—
191C22×C4—462no—
192C22×C4—462no—
193C22×C4C22×D2311yescharacteristic
194Q16—22no—
195Q16—464no—
196Q16—462no—
197M16—22no—
198M16—462no—
199M16—462no—
200M16—462no—
201C2×C8—231no—
202C2×C8—234no—
203C2×C8—231no—
204C2×C8—462no—
205C2×C8C22×D2311yescharacteristic
206C2×C8C22×D2311yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—232no—
209C4⋊C4—462no—
210C4⋊C4—462no—
211C4⋊C4—462no—
212C4⋊C4—462no—
213C4⋊C4C22×D2312yes—
214C22⋊C4—22no—
215C22⋊C4—462no—
216C22⋊C4—462no—
217C22⋊C4—462no—
218C23—231no—
219C23—231no—
220C23—462no—
221C23—464no—
222C23D4×D2311yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—462no—
226Q8—462no—
227Q8—924no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—231no—
233C2×C4—232no—
234C2×C4—232no—
235C2×C4—231no—
236C2×C4—462no—
237C2×C4—462no—
238C2×C4—462no—
239C2×C4—461no—
240C2×C4—461no—
241C2×C4—462no—
242C2×C4—462no—
243C2×C4—924no—
244C2×C4—924no—
245C2×C4D4×D2311yescharacteristic
246C2×C4C23×D2311yescharacteristic
247C8—21no—
248C8—232no—
249C8—461no—
250C8D4×D2312yes—
251C22—21no—
252C22—22no—
253C22—232no—
254C22—234no—
255C22—461no—
256C22—462no—
257C22—924no—
258C22C2×D4×D2311yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—232no—
263C4—461no—
264C4—922no—
265C4—922no—
266C4C2×D4×D2311yescharacteristic
267C4#41×D2311yescharacteristic
268C2—41no—
269C2—234no—
270C2—921no—
271C2#121×D2311yescharacteristic
272C2#137×D2312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#470C2
socleC22×C23C22×D4
prime radicalC23×D23D4
hypercenter#470D23
hypocenterC23C2×#470
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D23C2semidirect
4. #13092676C2semidirect
5. #13092675C2semidirect
6. #13092674C2semidirect
7. #13092673C2semidirect
8. #13092672C2semidirect
9. #106×D23C2semidirect
10. #103×D23C2semidirect
11. C23×#470C2semidirect
14. C2×Q16×C23C22semidirect
18. C2×#67275C22semidirect
23. C2×Dic92C22semidirect
34. C23×#106C22semidirect
35. C23×#103C22semidirect
36. #10625017C22semidirect
37. #10624999C22semidirect
38. #10624998C22semidirect
39. #10624992C22semidirect
40. #10624990C22semidirect
140. #470D23direct (N × H)
147. D23#470direct (N × H)
153. C2×Q16C2×D23semidirect
172. #106C2×D23semidirect
176. #103C2×D23semidirect
177. C23C2×#470semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D232
C2×M16×D232
C22⋊Q8×D232
#130926762
#130926752
#130926742
#130926732
#130926722
#106×D232
#103×D232
C23×#4702
C2×#47023