Finite Group Databasebeta

← #469×D97 #27151328

Subgroups — #469×D97 #27151328

Order 12416.

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

normal subgroups only

This group has 596 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×D8×D97C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#121×D97C212yesmaximal, complemented
4#20260989C211yesmaximal, characteristic, complemented
5#20260988C211yesmaximal, characteristic, complemented
6#20260987C212yesmaximal, complemented
7#20260986C212yesmaximal, complemented
8#20260985C211yesmaximal, characteristic, complemented
9#106×D97C211yesmaximal, characteristic, complemented
10#102×D97C212yesmaximal, complemented
11#469×C97C211yesmaximal, characteristic, complemented
12C2×D4×D97—22no—
13C2×D4×D97C2212yes—
14C22×C4×D97C2211yescharacteristic
15C2×D776C2211yescharacteristic, complemented
16C2×#13095496C2212yescomplemented
17D8×D97—24no—
18C2×#13095495C2211yescharacteristic, complemented
19C2×#13095494C2211yescharacteristic, complemented
20C2×#13095494C2211yescharacteristic, complemented
21M16×D97—24no—
22C2×C8×D97C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
24#15690785C2212yescomplemented
25#15690784C2212yescomplemented
26#15690783—22no—
27#15690782—22no—
28#15690781C2212yescomplemented
29C4⋊C4×D97C2212yes—
30C22⋊C4×D97—22no—
31#15690742C2211yescharacteristic, complemented
32#15690741C2212yescomplemented
33#15690732C2212yescomplemented
34#15690731C2212yescomplemented
35#15690730C2212yescomplemented
36C2×D8×C97C2211yescharacteristic, complemented
37C2×M16×C97C2211yescharacteristic, complemented
38#121×C97C2212yescomplemented
39#106×C97C2211yescharacteristic, complemented
40#102×C97C2212yescomplemented
41C23×D97—22no—
42C23×D97D411yescharacteristic
43C2×D388—22no—
44C2×D388C2312yes—
45C2×C22⋊D97—22no—
46C2×C22⋊D97—22no—
47C2×C22⋊D97—22no—
48D4×D97—24no—
49D4×D97—24no—
50D4×D97—48no—
51C22×Dic97D411yescharacteristic
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55C2×C4×D97D411yescharacteristic
56C2×C4×D97D411yescharacteristic
57C2×C4×D97C2311yescharacteristic
58C2×C4×D97C2311yescharacteristic
59D776—22no—
60#13095496—24no—
61#13095495—22no—
62#13095494—22no—
63#13095494—22no—
64C2×#11715895C2311yescharacteristic
65C2×#11715895C2311yescharacteristic
66C8×D97—22no—
67C8×D97D414yescomplemented
68C4⋊Dic97C2312yes—
69#13095490—22no—
70C22⋊Dic97—22no—
71#13095488—22no—
72#13095488—22no—
73C2×D4×C97—22no—
74C2×D4×C97C2312yes—
75C22×C4×C97C2311yescharacteristic
76D8×C97—22no—
77M16×C97—22no—
78C2×C8×C97C2311yescharacteristic
79C2×C8×C97C2311yescharacteristic
80C4⋊C4×C97C2312yes—
81C22⋊C4×C97—22no—
82C23×C97—22no—
83C23×C97C2×D411yescharacteristic
84C22×D97—22no—
85C22×D97—24no—
86C22×D97—22no—
87C22×D97—44no—
88C22×D97—44no—
89C22×D97—44no—
90C22×D97C2×D411yescharacteristic
91C22×D97C2×D411yescharacteristic
92D4×C97—22no—
93D4×C97—22no—
94D4×C97—44no—
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—22no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2411yescharacteristic
100C22⋊D97—44no—
101C22⋊D97—44no—
102C22⋊D97—44no—
103C2×Dic97—22no—
104C2×Dic97—22no—
105C2×Dic97C2×D411yescharacteristic
106C2×Dic97C2×D411yescharacteristic
107D388—22no—
108D388—22no—
109D388—44no—
110C4×D97—22no—
111C4×D97—21no—
112C4×D97—21no—
113C4×D97—22no—
114C4×D97—44no—
115C4×D97C2×D412yes—
116C4×D97#4112yes—
117C8×C97—21no—
118C8×C97C2×D412yescomplemented
119#11715895—21no—
120#11715895C2×D412yescomplemented
121C22×C97—21no—
122C22×C97—22no—
123C22×C97—42no—
124C22×C97—42no—
125C22×C97—42no—
126C22×C97C22×D411yescharacteristic
127C2×D97—21no—
128C2×D97—22no—
129C2×D97—44no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—88no—
134C2×D97#12112yes—
135C2×D97AGL(1,ℤ/8)14yes—
136C4×C97—21no—
137C4×C97—42no—
138C4×C97C22×D411yescharacteristic
139C4×C97C2×#4111yescharacteristic
140Dic97—21no—
141Dic97—42no—
142Dic97#12112yes—
143C2×C97—41no—
144C2×C97—82no—
145C2×C97C2×#12111yescharacteristic
146C2×C97C2×AGL(1,ℤ/8)12yes—
147D97—41no—
148D97—82no—
149D97#46914yesdirect product factor
150C2×#469—971nomaximal, Sylow
151C97C2×#46911yescharacteristic, complemented, Sylow
152C22×D8—971no—
153C22×M16—971no—
154C2×#121—972no—
155#469—971no—
156#469—971no—
157#469—972no—
158#469—971no—
159#469—972no—
160#469D9711yescharacteristic, direct product factor
161C2×#106—971no—
162C2×#102—972no—
163C22×D4—972no—
164C22×D4—1942no—
165C23×C4—971no—
166C2×D8—972no—
167C2×D8—971no—
168C2×D8—1944no—
169C2×D8C2×D9711yescharacteristic, complemented
170C2×M16—971no—
171C2×M16—971no—
172C2×M16—971no—
173C2×M16—1944no—
174C2×M16C2×D9711yescharacteristic, complemented
175C22×C8—971no—
176C22×C8—971no—
177#121—972no—
178#121—972no—
179#121—972no—
180#121—1942no—
181#121—1942no—
182#121C2×D9712yescomplemented
183C2×C4⋊C4—972no—
184C2×C22⋊C4—1942no—
185#106—971no—
186#106—972no—
187#106C2×D9711yescharacteristic, complemented
188#102—972no—
189#102—972no—
190#102—972no—
191#102C2×D9712yescomplemented
192C24—971no—
193C24—1942no—
194C2×D4—22no—
195C2×D4—972no—
196C2×D4—3888no—
197C2×D4—1942no—
198C2×D4—1944no—
199C2×D4—1944no—
200C2×D4—1942no—
201C2×D4—1942no—
202C2×D4—1942no—
203C2×D4C22×D9712yes—
204C22×C4—971no—
205C22×C4—971no—
206C22×C4—971no—
207C22×C4—971no—
208C22×C4—971no—
209C22×C4—1942no—
210C22×C4—1942no—
211C22×C4—1942no—
212C22×C4C22×D9711yescharacteristic
213D8—22no—
214D8—1944no—
215D8—1942no—
216M16—22no—
217M16—1942no—
218M16—1942no—
219M16—1942no—
220C2×C8—971no—
221C2×C8—974no—
222C2×C8—971no—
223C2×C8—1942no—
224C2×C8C22×D9711yescharacteristic
225C2×C8C22×D9711yescharacteristic
226C4⋊C4—972no—
227C4⋊C4—1942no—
228C4⋊C4C22×D9712yes—
229C22⋊C4—22no—
230C22⋊C4—1942no—
231C22⋊C4—1942no—
232C22⋊C4—1942no—
233C23—22no—
234C23—971no—
235C23—971no—
236C23—3884no—
237C23—3884no—
238C23—3884no—
239C23—1942no—
240C23—1944no—
241C23—1942no—
242C23D4×D9711yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—3884no—
247D4—3884no—
248D4—3884no—
249D4—3884no—
250D4—1942no—
251D4—1942no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—971no—
256C2×C4—972no—
257C2×C4—972no—
258C2×C4—971no—
259C2×C4—3884no—
260C2×C4—1942no—
261C2×C4—1942no—
262C2×C4—1942no—
263C2×C4—1942no—
264C2×C4—1941no—
265C2×C4—1941no—
266C2×C4D4×D9711yescharacteristic
267C2×C4C23×D9711yescharacteristic
268C8—21no—
269C8—972no—
270C8—1941no—
271C8D4×D9712yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—7768no—
278C22—972no—
279C22—974no—
280C22—3884no—
281C22—3882no—
282C22—3882no—
283C22—3882no—
284C22—1941no—
285C22—1942no—
286C22C2×D4×D9711yescharacteristic
287C4—21no—
288C4—42no—
289C4—972no—
290C4—3882no—
291C4—1941no—
292C4C2×D4×D9711yescharacteristic
293C4#41×D9711yescharacteristic
294C2—41no—
295C2—82no—
296C2—7762no—
297C2—974no—
298C2—3881no—
299C2#121×D9711yescharacteristic
300C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#469×C97C2
socleC22×C97C22×D4
hypercenter#469D97
hypocenterC97C2×#469
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D97C2semidirect
2. C2×M16×D97C2semidirect
3. #121×D97C2semidirect
4. #20260989C2semidirect
5. #20260988C2semidirect
6. #20260987C2semidirect
7. #20260986C2semidirect
8. #20260985C2semidirect
9. #106×D97C2semidirect
10. #102×D97C2semidirect
11. #469×C97C2semidirect
15. C2×D776C22semidirect
16. C2×#13095496C22semidirect
18. C2×#13095495C22semidirect
19. C2×#13095494C22semidirect
20. C2×#13095494C22semidirect
24. #15690785C22semidirect
25. #15690784C22semidirect
28. #15690781C22semidirect
31. #15690742C22semidirect
32. #15690741C22semidirect
33. #15690732C22semidirect
34. #15690731C22semidirect
35. #15690730C22semidirect
36. C2×D8×C97C22semidirect
37. C2×M16×C97C22semidirect
38. #121×C97C22semidirect
39. #106×C97C22semidirect
40. #102×C97C22semidirect
67. C8×D97D4semidirect
118. C8×C97C2×D4semidirect
120. #11715895C2×D4semidirect
149. D97#469direct (N × H)
151. C97C2×#469semidirect
160. #469D97direct (N × H)
169. C2×D8C2×D97semidirect
174. C2×M16C2×D97semidirect
182. #121C2×D97semidirect
187. #106C2×D97semidirect
191. #102C2×D97semidirect
271. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D972
C2×M16×D972
#121×D972
#202609892
#202609882
#202609872
#202609862
#202609852
#106×D972
#102×D972
#469×C972
C2×#46997