Finite Group Databasebeta

← #470×D97 #27151334

Subgroups — #470×D97 #27151334

Order 12416.

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×D97C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic
3C22⋊Q8×D97C212yesmaximal
4#20260994C211yesmaximal, characteristic, complemented
5#20260993C211yesmaximal, characteristic, complemented
6#20260992C212yesmaximal, complemented
7#20260991C212yesmaximal, complemented
8#20260990C211yesmaximal, characteristic, complemented
9#106×D97C211yesmaximal, characteristic, complemented
10#103×D97C212yesmaximal, complemented
11#470×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2212yes—
13C22×C4×D97C2211yescharacteristic
14C2×Dic388C2211yescharacteristic, complemented
15C2×#13095501C2212yescomplemented
16Q16×D97—24no—
17C2×#13095495C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20M16×D97—24no—
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690790C2212yes—
24#15690789C2212yes—
25#15690788—22no—
26#15690787—22no—
27#15690786C2212yes—
28C4⋊C4×D97—22no—
29C4⋊C4×D97C2212yes—
30C22⋊C4×D97—22no—
31#15690742C2211yescharacteristic, complemented
32#15690741C2212yescomplemented
33#15690735C2212yescomplemented
34#15690734C2212yescomplemented
35#15690733C2212yescomplemented
36C2×Q16×C97C2211yescharacteristic, complemented
37C2×M16×C97C2211yescharacteristic
38C22⋊Q8×C97C2212yes—
39#106×C97C2211yescharacteristic, complemented
40#103×C97C2212yescomplemented
41C23×D97D411yescharacteristic
42C2×Dic194—22no—
43C2×Dic194C2312yes—
44Q8×D97—24no—
45Q8×D97—24no—
46C22×Dic97D411yescharacteristic
47C2×C4×D97—22no—
48C2×C4×D97—22no—
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97D411yescharacteristic
52C2×C4×D97D411yescharacteristic
53C2×C4×D97C2311yescharacteristic
54C2×C4×D97C2311yescharacteristic
55Dic388—22no—
56#13095501—24no—
57#13095495—22no—
58#13095494—22no—
59#13095494—22no—
60C2×#11715895C2311yescharacteristic
61C2×#11715895C2311yescharacteristic
62C8×D97—22no—
63C8×D97D414yes—
64C4⋊Dic97—22no—
65C4⋊Dic97C2312yes—
66#13095490—22no—
67#13095490—22no—
68#13095490—22no—
69C22⋊Dic97—22no—
70#13095488—22no—
71#13095488—22no—
72C2×Q8×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74Q16×C97—22no—
75M16×C97—22no—
76C2×C8×C97C2311yescharacteristic
77C2×C8×C97C2311yescharacteristic
78C4⋊C4×C97—22no—
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—22no—
81C23×C97C2×D411yescharacteristic
82C22×D97—22no—
83C22×D97—24no—
84C22×D97C2×D411yescharacteristic
85C22×D97C2×D411yescharacteristic
86Q8×C97—22no—
87Q8×C97—22no—
88C2×C4×C97—21no—
89C2×C4×C97—21no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2411yescharacteristic
94C2×Dic97—22no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97C2×D411yescharacteristic
98C2×Dic97C2×D411yescharacteristic
99C4×D97—22no—
100C4×D97—21no—
101C4×D97—21no—
102C4×D97—22no—
103C4×D97—44no—
104C4×D97—44no—
105C4×D97C2×D412yes—
106C4×D97#4112yes—
107Dic194—22no—
108Dic194—22no—
109Dic194—44no—
110C8×C97—21no—
111C8×C97C2×D412yes—
112#11715895—21no—
113#11715895C2×D412yes—
114C22×C97—21no—
115C22×C97—22no—
116C22×C97C22×D411yescharacteristic
117C2×D97—21no—
118C2×D97—22no—
119C2×D97—44no—
120C2×D97#12112yes—
121C2×D97#13714yes—
122C4×C97—21no—
123C4×C97—42no—
124C4×C97—42no—
125C4×C97C22×D411yescharacteristic
126C4×C97C2×#4111yescharacteristic
127Dic97—21no—
128Dic97—42no—
129Dic97—42no—
130Dic97#12112yes—
131C2×C97—41no—
132C2×C97C2×#12111yescharacteristic
133C2×C97C2×#13712yes—
134D97—41no—
135D97#47014yesdirect product factor
136C2×#470—971nomaximal, Sylow
137C97C2×#47011yescharacteristic, complemented, Sylow
138C22×Q16—971no—
139C22×M16—971no—
140C2×C22⋊Q8—972no—
141#470—971no—
142#470—971no—
143#470—972no—
144#470—971no—
145#470—972no—
146#470D9711yescharacteristic, direct product factor
147C2×#106—971no—
148C2×#103—972no—
149C22×Q8—972no—
150C23×C4—971no—
151C2×Q16—972no—
152C2×Q16—971no—
153C2×Q16—1944no—
154C2×Q16C2×D9711yescharacteristic, complemented
155C2×M16—971no—
156C2×M16—971no—
157C2×M16—971no—
158C2×M16—1944no—
159C2×M16C2×D9711yescharacteristic
160C22×C8—971no—
161C22×C8—971no—
162C22⋊Q8—972no—
163C22⋊Q8—972no—
164C22⋊Q8—972no—
165C22⋊Q8—1942no—
166C22⋊Q8—1942no—
167C22⋊Q8C2×D9712yes—
168C2×C4⋊C4—972no—
169C2×C4⋊C4—1942no—
170C2×C22⋊C4—1942no—
171#106—971no—
172#106—972no—
173#106C2×D9711yescharacteristic, complemented
174#103—972no—
175#103—972no—
176#103—972no—
177#103C2×D9712yescomplemented
178C24—971no—
179C2×Q8—972no—
180C2×Q8—1942no—
181C2×Q8—1944no—
182C2×Q8—1944no—
183C2×Q8C22×D9712yes—
184C22×C4—971no—
185C22×C4—971no—
186C22×C4—971no—
187C22×C4—971no—
188C22×C4—971no—
189C22×C4—1942no—
190C22×C4—1942no—
191C22×C4—1942no—
192C22×C4—1942no—
193C22×C4C22×D9711yescharacteristic
194Q16—22no—
195Q16—1944no—
196Q16—1942no—
197M16—22no—
198M16—1942no—
199M16—1942no—
200M16—1942no—
201C2×C8—971no—
202C2×C8—974no—
203C2×C8—971no—
204C2×C8—1942no—
205C2×C8C22×D9711yescharacteristic
206C2×C8C22×D9711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—972no—
209C4⋊C4—1942no—
210C4⋊C4—1942no—
211C4⋊C4—1942no—
212C4⋊C4—1942no—
213C4⋊C4C22×D9712yes—
214C22⋊C4—22no—
215C22⋊C4—1942no—
216C22⋊C4—1942no—
217C22⋊C4—1942no—
218C23—971no—
219C23—971no—
220C23—1942no—
221C23—1944no—
222C23D4×D9711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—3884no—
226Q8—1942no—
227Q8—1942no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—971no—
233C2×C4—972no—
234C2×C4—972no—
235C2×C4—971no—
236C2×C4—3884no—
237C2×C4—3884no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4—1942no—
243C2×C4—1941no—
244C2×C4—1941no—
245C2×C4D4×D9711yescharacteristic
246C2×C4C23×D9711yescharacteristic
247C8—21no—
248C8—972no—
249C8—1941no—
250C8D4×D9712yes—
251C22—21no—
252C22—22no—
253C22—972no—
254C22—974no—
255C22—3884no—
256C22—1941no—
257C22—1942no—
258C22C2×D4×D9711yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—972no—
263C4—3882no—
264C4—3882no—
265C4—1941no—
266C4C2×D4×D9711yescharacteristic
267C4#41×D9711yescharacteristic
268C2—41no—
269C2—974no—
270C2—3881no—
271C2#121×D9711yescharacteristic
272C2#137×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#470×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#470D97
hypocenterC97C2×#470
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D97C2semidirect
4. #20260994C2semidirect
5. #20260993C2semidirect
6. #20260992C2semidirect
7. #20260991C2semidirect
8. #20260990C2semidirect
9. #106×D97C2semidirect
10. #103×D97C2semidirect
11. #470×C97C2semidirect
14. C2×Dic388C22semidirect
15. C2×#13095501C22semidirect
31. #15690742C22semidirect
32. #15690741C22semidirect
33. #15690735C22semidirect
34. #15690734C22semidirect
35. #15690733C22semidirect
36. C2×Q16×C97C22semidirect
39. #106×C97C22semidirect
40. #103×C97C22semidirect
135. D97#470direct (N × H)
137. C97C2×#470semidirect
146. #470D97direct (N × H)
154. C2×Q16C2×D97semidirect
173. #106C2×D97semidirect
177. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D972
C2×M16×D972
C22⋊Q8×D972
#202609942
#202609932
#202609922
#202609912
#202609902
#106×D972
#103×D972
#470×C972
C2×#47097