Finite Group Databasebeta

← #496×D197 #38753142

Subgroups — #496×D197 #38753142

Order 25216.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D197C212yesmaximal, complemented
2C2×D8×D197C212yesmaximal, complemented
3#127×D197C211yesmaximal, characteristic, complemented
4#124×D197C211yesmaximal, characteristic, complemented
5#27161469C211yesmaximal, characteristic, complemented
6#27161468C211yesmaximal, characteristic, complemented
7#27161467C212yesmaximal, complemented
8#27161466C211yesmaximal, characteristic, complemented
9#27161465C212yesmaximal, complemented
10#97×D197C211yesmaximal, characteristic, complemented
11#496×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2211yescharacteristic
13C2×D4×D197—22no—
14C2×D4×D197C2212yes—
15C2×D4×D197C2211yescharacteristic
16C2×#15691347C2212yescomplemented
17C2×#15691346C2212yescomplemented
18C2×#15691345C2212yescomplemented
19SD16×D197—28no—
20C2×D1576C2212yescomplemented
21C2×#15691343C2212yescomplemented
22C2×#15691343C2212yescomplemented
23D8×D197—28no—
24C2×C8×D197C2212yes—
25#20263323C2211yescharacteristic, complemented
26#20263322—22no—
27#20263322C2212yescomplemented
28#20263315C2211yescharacteristic, complemented
29#20263314C2211yescharacteristic, complemented
30#20263313C2211yescharacteristic, complemented
31#20263312—22no—
32C22⋊C4×D197—22no—
33C42×D197C2211yescharacteristic
34#20263237C2211yescharacteristic, complemented
35#20263236C2212yescomplemented
36C2×SD16×C197C2212yescomplemented
37C2×D8×C197C2212yescomplemented
38#127×C197C2211yescharacteristic, complemented
39#124×C197C2211yescharacteristic, complemented
40#97×C197C2211yescharacteristic, complemented
41C23×D197—21no—
42C23×D197—22no—
43C2×Dic394—21no—
44C2×Dic394C2311yescharacteristic
45Q8×D197—24no—
46C2×D788—22no—
47C2×D788C2311yescharacteristic
48C2×D788C2312yes—
49C2×C22⋊D197—21no—
50C2×C22⋊D197—22no—
51C2×C22⋊D197—24no—
52D4×D197—24no—
53D4×D197—24no—
54D4×D197—24no—
55D4×D197—48no—
56C2×C4×D197—21no—
57C2×C4×D197D412yes—
58C2×C4×D197C2311yescharacteristic
59#15691347—24no—
60#15691346—24no—
61#15691345—24no—
62D1576—24no—
63#15691343—24no—
64#15691343—24no—
65C2×#13096802C2312yes—
66C8×D197D418yescomplemented
67C22⋊Dic197—22no—
68#15691335—22no—
69#15691335—22no—
70C4×Dic197D412yes—
71C4×Dic197C2311yescharacteristic
72C2×Q8×C197C2311yescharacteristic
73C2×D4×C197—22no—
74C2×D4×C197C2311yescharacteristic
75C2×D4×C197C2312yes—
76SD16×C197—24no—
77D8×C197—24no—
78C2×C8×C197C2312yes—
79C22⋊C4×C197—22no—
80C42×C197C2311yescharacteristic
81C22×D197—21no—
82C22×D197—22no—
83C22×D197—42no—
84C22×D197—44no—
85C22×D197—48no—
86C22×D197—44no—
87C22×D197C2×D411yescharacteristic
88Dic394—22no—
89Dic394—42no—
90D788—22no—
91D788—22no—
92D788—22no—
93D788—44no—
94C22⋊D197—48no—
95C22⋊D197—42no—
96C22⋊D197—44no—
97C2×Dic197—21no—
98C2×Dic197C2×D412yes—
99C2×Dic197C2×D411yescharacteristic
100C4×D197—24no—
101C4×D197—42no—
102C4×D197C2×D414yes—
103#13096802C2×D414yescomplemented
104C23×C197—21no—
105C23×C197—22no—
106Q8×C197—22no—
107D4×C197—22no—
108D4×C197—22no—
109D4×C197—22no—
110D4×C197—44no—
111C2×C4×C197—21no—
112C2×C4×C197C2×D412yes—
113C2×C4×C197C2411yescharacteristic
114C8×C197C2×D414yescomplemented
115C22×C197—41no—
116C22×C197—42no—
117C22×C197—44no—
118C22×C197—42no—
119C22×C197C22×D411yescharacteristic
120C2×D197—41no—
121C2×D197—42no—
122C2×D197—44no—
123C2×D197—42no—
124C2×D197—88no—
125C2×D197—84no—
126C2×D197#12712yes—
127C2×D197AGL(1,ℤ/8)14yes—
128C4×C197—22no—
129C4×C197—41no—
130C4×C197C22×D412yes—
131Dic197—22no—
132Dic197—41no—
133Dic197#12712yes—
134C2×C197—82no—
135C2×C197—81no—
136C2×C197C2×#12711yescharacteristic
137C2×C197C2×AGL(1,ℤ/8)12yes—
138D197—82no—
139D197—81no—
140D197#49614yesdirect product factor
141C197C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1971nomaximal, Sylow
143C22×SD16—1972no—
144C22×D8—1972no—
145C2×#127—1971no—
146C2×#124—1971no—
147#496—1972no—
148#496—1972no—
149#496—1971no—
150#496—1971no—
151#496—1971no—
152#496D19711yescharacteristic, direct product factor
153C2×#97—1971no—
154C22×Q8—1971no—
155C22×D4—3942no—
156C22×D4—1971no—
157C22×D4—1972no—
158C2×SD16—3948no—
159C2×SD16—1972no—
160C2×SD16—1972no—
161C2×SD16—1972no—
162C2×SD16C2×D19712yescomplemented
163C2×D8—3948no—
164C2×D8—1972no—
165C2×D8—1972no—
166C2×D8—1972no—
167C2×D8C2×D19712yescomplemented
168C22×C8—1972no—
169#127—3942no—
170#127—1972no—
171#127—1971no—
172#127C2×D19711yescharacteristic, complemented
173#124—3942no—
174#124—1971no—
175#124—1971no—
176#124—1971no—
177#124C2×D19711yescharacteristic, complemented
178C2×C22⋊C4—3942no—
179C2×C42—1971no—
180#97—1972no—
181#97—1971no—
182#97C2×D19711yescharacteristic, complemented
183C24—3941no—
184C24—3942no—
185C2×Q8—3941no—
186C2×Q8—3944no—
187C2×Q8—1971no—
188C2×Q8C22×D19711yescharacteristic
189C2×D4—22no—
190C2×D4—7888no—
191C2×D4—3944no—
192C2×D4—3944no—
193C2×D4—3944no—
194C2×D4—3941no—
195C2×D4—3942no—
196C2×D4—3944no—
197C2×D4—3942no—
198C2×D4—1971no—
199C2×D4—1972no—
200C2×D4C22×D19711yescharacteristic
201C2×D4C22×D19712yes—
202C22×C4—3941no—
203C22×C4—1972no—
204C22×C4—1971no—
205SD16—24no—
206SD16—3944no—
207SD16—3944no—
208SD16—3944no—
209D8—24no—
210D8—3944no—
211D8—3944no—
212D8—3944no—
213C2×C8—1972no—
214C2×C8—1978no—
215C2×C8C22×D19712yes—
216C22⋊C4—22no—
217C22⋊C4—3942no—
218C22⋊C4—3942no—
219C22⋊C4—3942no—
220C42—1972no—
221C42—1971no—
222C42C22×D19711yescharacteristic
223C23—21no—
224C23—22no—
225C23—7888no—
226C23—7882no—
227C23—7884no—
228C23—7884no—
229C23—3941no—
230C23—3942no—
231C23—1971no—
232Q8—22no—
233Q8—7882no—
234Q8—3942no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—7884no—
240D4—7888no—
241D4—7882no—
242D4—7884no—
243D4—3942no—
244D4—3942no—
245D4—3942no—
246C2×C4—21no—
247C2×C4—7882no—
248C2×C4—3944no—
249C2×C4—3941no—
250C2×C4—1972no—
251C2×C4—1971no—
252C2×C4—1974no—
253C2×C4D4×D19712yes—
254C2×C4C23×D19711yescharacteristic
255C8—1974no—
256C8D4×D19714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—7884no—
262C22—7882no—
263C22—7881no—
264C22—7882no—
265C22—15768no—
266C22—15764no—
267C22—1974no—
268C22—1972no—
269C22C2×D4×D19711yescharacteristic
270C4—22no—
271C4—41no—
272C4—7881no—
273C4—3942no—
274C4—1972no—
275C4C2×D4×D19712yes—
276C2—82no—
277C2—81no—
278C2—15762no—
279C2—15761no—
280C2—1974no—
281C2#127×D19711yescharacteristic
282C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#496×C197C2
socleC22×C197C22×D4
hypercenter#496D197
hypocenterC197C2×#496
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D197C2semidirect
2. C2×D8×D197C2semidirect
3. #127×D197C2semidirect
4. #124×D197C2semidirect
5. #27161469C2semidirect
6. #27161468C2semidirect
7. #27161467C2semidirect
8. #27161466C2semidirect
9. #27161465C2semidirect
10. #97×D197C2semidirect
11. #496×C197C2semidirect
16. C2×#15691347C22semidirect
17. C2×#15691346C22semidirect
18. C2×#15691345C22semidirect
20. C2×D1576C22semidirect
21. C2×#15691343C22semidirect
22. C2×#15691343C22semidirect
25. #20263323C22semidirect
27. #20263322C22semidirect
28. #20263315C22semidirect
29. #20263314C22semidirect
30. #20263313C22semidirect
34. #20263237C22semidirect
35. #20263236C22semidirect
36. C2×SD16×C197C22semidirect
37. C2×D8×C197C22semidirect
38. #127×C197C22semidirect
39. #124×C197C22semidirect
40. #97×C197C22semidirect
66. C8×D197D4semidirect
103. #13096802C2×D4semidirect
114. C8×C197C2×D4semidirect
140. D197#496direct (N × H)
141. C197C2×#496semidirect
152. #496D197direct (N × H)
162. C2×SD16C2×D197semidirect
167. C2×D8C2×D197semidirect
172. #127C2×D197semidirect
177. #124C2×D197semidirect
182. #97C2×D197semidirect
256. C8D4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1972
C2×D8×D1972
#127×D1972
#124×D1972
#271614692
#271614682
#271614672
#271614662
#271614652
#97×D1972
#496×C1972
C2×#496197