Finite Group Databasebeta

← #470×D197 #38752310

Subgroups — #470×D197 #38752310

Order 25216.

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×D197C211yesmaximal, characteristic, complemented
2C2×M16×D197C211yesmaximal, characteristic
3C22⋊Q8×D197C212yesmaximal
4#27161338C211yesmaximal, characteristic, complemented
5#27161337C211yesmaximal, characteristic, complemented
6#27161336C212yesmaximal, complemented
7#27161335C212yesmaximal, complemented
8#27161334C211yesmaximal, characteristic, complemented
9#106×D197C211yesmaximal, characteristic, complemented
10#103×D197C212yesmaximal, complemented
11#470×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2212yes—
13C22×C4×D197C2211yescharacteristic
14C2×Dic788C2211yescharacteristic, complemented
15C2×#15691348C2212yescomplemented
16Q16×D197—24no—
17C2×#15691342C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19C2×#15691341C2211yescharacteristic
20M16×D197—24no—
21C2×C8×D197C2211yescharacteristic
22C2×C8×D197C2211yescharacteristic
23#20263306C2212yes—
24#20263305C2212yes—
25#20263304—22no—
26#20263303—22no—
27#20263302C2212yes—
28C4⋊C4×D197—22no—
29C4⋊C4×D197C2212yes—
30C22⋊C4×D197—22no—
31#20263258C2211yescharacteristic, complemented
32#20263257C2212yescomplemented
33#20263251C2212yescomplemented
34#20263250C2212yescomplemented
35#20263249C2212yescomplemented
36C2×Q16×C197C2211yescharacteristic, complemented
37C2×M16×C197C2211yescharacteristic
38C22⋊Q8×C197C2212yes—
39#106×C197C2211yescharacteristic, complemented
40#103×C197C2212yescomplemented
41C23×D197D411yescharacteristic
42C2×Dic394—22no—
43C2×Dic394C2312yes—
44Q8×D197—24no—
45Q8×D197—24no—
46C22×Dic197D411yescharacteristic
47C2×C4×D197—22no—
48C2×C4×D197—22no—
49C2×C4×D197—22no—
50C2×C4×D197—22no—
51C2×C4×D197D411yescharacteristic
52C2×C4×D197D411yescharacteristic
53C2×C4×D197C2311yescharacteristic
54C2×C4×D197C2311yescharacteristic
55Dic788—22no—
56#15691348—24no—
57#15691342—22no—
58#15691341—22no—
59#15691341—22no—
60C2×#13096802C2311yescharacteristic
61C2×#13096802C2311yescharacteristic
62C8×D197—22no—
63C8×D197D414yes—
64C4⋊Dic197—22no—
65C4⋊Dic197C2312yes—
66#15691337—22no—
67#15691337—22no—
68#15691337—22no—
69C22⋊Dic197—22no—
70#15691335—22no—
71#15691335—22no—
72C2×Q8×C197C2312yes—
73C22×C4×C197C2311yescharacteristic
74Q16×C197—22no—
75M16×C197—22no—
76C2×C8×C197C2311yescharacteristic
77C2×C8×C197C2311yescharacteristic
78C4⋊C4×C197—22no—
79C4⋊C4×C197C2312yes—
80C22⋊C4×C197—22no—
81C22×D197—22no—
82C22×D197—24no—
83C22×D197C2×D411yescharacteristic
84C22×D197C2×D411yescharacteristic
85Dic394—22no—
86Dic394—22no—
87Dic394—44no—
88C2×Dic197—22no—
89C2×Dic197—22no—
90C2×Dic197—22no—
91C2×Dic197C2×D411yescharacteristic
92C2×Dic197C2×D411yescharacteristic
93C4×D197—22no—
94C4×D197—21no—
95C4×D197—21no—
96C4×D197—22no—
97C4×D197—44no—
98C4×D197—44no—
99C4×D197C2×D412yes—
100C4×D197#4112yes—
101#13096802—21no—
102#13096802C2×D412yes—
103C23×C197C2×D411yescharacteristic
104Q8×C197—22no—
105Q8×C197—22no—
106C2×C4×C197—21no—
107C2×C4×C197—21no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2411yescharacteristic
112C8×C197—21no—
113C8×C197C2×D412yes—
114C22×C197—21no—
115C22×C197—22no—
116C22×C197C22×D411yescharacteristic
117C2×D197—21no—
118C2×D197—22no—
119C2×D197—44no—
120C2×D197#12112yes—
121C2×D197#13714yes—
122C4×C197—21no—
123C4×C197—42no—
124C4×C197—42no—
125C4×C197C22×D411yescharacteristic
126C4×C197C2×#4111yescharacteristic
127Dic197—21no—
128Dic197—42no—
129Dic197—42no—
130Dic197#12112yes—
131C2×C197—41no—
132C2×C197C2×#12111yescharacteristic
133C2×C197C2×#13712yes—
134D197—41no—
135D197#47014yesdirect product factor
136C197C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—1971nomaximal, Sylow
138C22×Q16—1971no—
139C22×M16—1971no—
140C2×C22⋊Q8—1972no—
141#470—1971no—
142#470—1971no—
143#470—1972no—
144#470—1971no—
145#470—1972no—
146#470D19711yescharacteristic, direct product factor
147C2×#106—1971no—
148C2×#103—1972no—
149C22×Q8—1972no—
150C23×C4—1971no—
151C2×Q16—3944no—
152C2×Q16—1972no—
153C2×Q16—1971no—
154C2×Q16C2×D19711yescharacteristic, complemented
155C2×M16—3944no—
156C2×M16—1971no—
157C2×M16—1971no—
158C2×M16—1971no—
159C2×M16C2×D19711yescharacteristic
160C22×C8—1971no—
161C22×C8—1971no—
162C22⋊Q8—3942no—
163C22⋊Q8—3942no—
164C22⋊Q8—1972no—
165C22⋊Q8—1972no—
166C22⋊Q8—1972no—
167C22⋊Q8C2×D19712yes—
168C2×C4⋊C4—3942no—
169C2×C4⋊C4—1972no—
170C2×C22⋊C4—3942no—
171#106—1971no—
172#106—1972no—
173#106C2×D19711yescharacteristic, complemented
174#103—1972no—
175#103—1972no—
176#103—1972no—
177#103C2×D19712yescomplemented
178C24—1971no—
179C2×Q8—3942no—
180C2×Q8—3944no—
181C2×Q8—3944no—
182C2×Q8—1972no—
183C2×Q8C22×D19712yes—
184C22×C4—3942no—
185C22×C4—3942no—
186C22×C4—3942no—
187C22×C4—3942no—
188C22×C4—1971no—
189C22×C4—1971no—
190C22×C4—1971no—
191C22×C4—1971no—
192C22×C4—1971no—
193C22×C4C22×D19711yescharacteristic
194Q16—22no—
195Q16—3944no—
196Q16—3942no—
197M16—22no—
198M16—3942no—
199M16—3942no—
200M16—3942no—
201C2×C8—3942no—
202C2×C8—1971no—
203C2×C8—1971no—
204C2×C8—1974no—
205C2×C8C22×D19711yescharacteristic
206C2×C8C22×D19711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—3942no—
209C4⋊C4—3942no—
210C4⋊C4—3942no—
211C4⋊C4—3942no—
212C4⋊C4—1972no—
213C4⋊C4C22×D19712yes—
214C22⋊C4—22no—
215C22⋊C4—3942no—
216C22⋊C4—3942no—
217C22⋊C4—3942no—
218C23—3942no—
219C23—3944no—
220C23—1971no—
221C23—1971no—
222C23D4×D19711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—7884no—
226Q8—3942no—
227Q8—3942no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—7884no—
233C2×C4—7884no—
234C2×C4—3942no—
235C2×C4—3942no—
236C2×C4—3942no—
237C2×C4—3942no—
238C2×C4—3942no—
239C2×C4—3941no—
240C2×C4—3941no—
241C2×C4—1971no—
242C2×C4—1971no—
243C2×C4—1972no—
244C2×C4—1972no—
245C2×C4D4×D19711yescharacteristic
246C2×C4C23×D19711yescharacteristic
247C8—21no—
248C8—3941no—
249C8—1972no—
250C8D4×D19712yes—
251C22—22no—
252C22—21no—
253C22—7884no—
254C22—3942no—
255C22—3941no—
256C22—1974no—
257C22—1972no—
258C22C2×D4×D19711yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—7882no—
263C4—7882no—
264C4—3941no—
265C4—1972no—
266C4C2×D4×D19711yescharacteristic
267C4#41×D19711yescharacteristic
268C2—41no—
269C2—7881no—
270C2—1974no—
271C2#121×D19711yescharacteristic
272C2#137×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#470×C197C2
socleC22×C197C22×D4
prime radicalC23×D197D4
hypercenter#470D197
hypocenterC197C2×#470
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D197C2semidirect
4. #27161338C2semidirect
5. #27161337C2semidirect
6. #27161336C2semidirect
7. #27161335C2semidirect
8. #27161334C2semidirect
9. #106×D197C2semidirect
10. #103×D197C2semidirect
11. #470×C197C2semidirect
14. C2×Dic788C22semidirect
15. C2×#15691348C22semidirect
31. #20263258C22semidirect
32. #20263257C22semidirect
33. #20263251C22semidirect
34. #20263250C22semidirect
35. #20263249C22semidirect
36. C2×Q16×C197C22semidirect
39. #106×C197C22semidirect
40. #103×C197C22semidirect
135. D197#470direct (N × H)
136. C197C2×#470semidirect
146. #470D197direct (N × H)
154. C2×Q16C2×D197semidirect
173. #106C2×D197semidirect
177. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1972
C2×M16×D1972
C22⋊Q8×D1972
#271613382
#271613372
#271613362
#271613352
#271613342
#106×D1972
#103×D1972
#470×C1972
C2×#470197