Finite Group Databasebeta

← #38753192

Subgroups — #38753192

Order 25216.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20263350C212yesmaximal, complemented
2C2×#20263345C211yesmaximal, characteristic, complemented
3C2×D8×D197C211yesmaximal, characteristic, complemented
4#27161642C212yesmaximal, complemented
5#27161467C212yesmaximal, complemented
6#27161466C212yesmaximal, complemented
7#27161463C211yesmaximal, characteristic, complemented
8#27161462C211yesmaximal, characteristic, complemented
9#27161454C211yesmaximal, characteristic, complemented
10#27161033C211yesmaximal, characteristic, complemented
11#493×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2212yes—
13C2×D4×D197C2212yes—
14#20263350—28no—
15#20263345—24no—
16C2×Dic788C2211yescharacteristic, complemented
17C2×#15691347C2212yescomplemented
18C2×#15691346C2212yescomplemented
19C2×#15691346C2212yescomplemented
20C2×D1576C2211yescharacteristic
21C2×#15691343C2212yes—
22C2×#15691343C2212yescomplemented
23D8×D197—24no—
24C2×#15691341C2211yescharacteristic
25C2×C8×D197C2211yescharacteristic
26#20263322C2212yescomplemented
27#20263314C2212yescomplemented
28#20263313C2212yescomplemented
29#20263298—24no—
30#20263294—24no—
31#20263283C2211yescharacteristic
32#20263237C2211yescharacteristic, complemented
33#20263236C2211yescharacteristic, complemented
34C8×Dic197C2211yescharacteristic, complemented
35C2×D8×C197C2211yescharacteristic
36C2×D8×C197C2211yescharacteristic, complemented
37C2×D8×C197C2212yescomplemented
38#127×C197C2212yescomplemented
39C4×C8×C197C2211yescharacteristic, complemented
40C23×D197—22no—
41#15691357—24no—
42#15691357—24no—
43C2×Dic394C2312yes—
44C2×D788C2312yes—
45C2×C22⋊D197—22no—
46C2×C22⋊D197—22no—
47C2×C22⋊D197—24no—
48D4×D197—24no—
49D4×D197—24no—
50C22×Dic197—22no—
51C2×C4×D197C2311yescharacteristic
52Dic788—22no—
53#15691347—24no—
54#15691346—24no—
55#15691346—24no—
56D1576—22no—
57#15691343—24no—
58#15691343—24no—
59#15691341D414yescomplemented
60C2×#13096802C2311yescharacteristic
61C2×#13096802C2311yescharacteristic
62C8×D197D414yescomplemented
63#15691337D412yes—
64C22⋊Dic197—24no—
65#15691335—24no—
66#15691335D412yes—
67C4×Dic197C2311yescharacteristic
68C2×D4×C197—24no—
69C2×D4×C197C2312yes—
70C2×D4×C197C2312yes—
71D8×C197—22no—
72D8×C197—22no—
73D8×C197—24no—
74C2×C8×C197C2311yescharacteristic
75C2×C8×C197C2311yescharacteristic
76C42×C197C2311yescharacteristic
77C22×D197—22no—
78C22×D197—44no—
79C22×D197—84no—
80C22×D197C2×D411yescharacteristic
81Dic394—22no—
82Dic394—22no—
83D788—22no—
84D788—22no—
85C22⋊D197—44no—
86C22⋊D197—44no—
87C22⋊D197—84no—
88C2×Dic197—22no—
89C2×Dic197—44no—
90C2×Dic197C2×D412yes—
91C2×Dic197C2×D411yescharacteristic
92C4×D197C2×D412yes—
93C4×D197C2×D412yes—
94#13096802C2×D412yes—
95#13096802C2×D412yes—
96C23×C197—22no—
97C23×C197—22no—
98D4×C197—22no—
99D4×C197—22no—
100D4×C197—22no—
101D4×C197—22no—
102D4×C197—84no—
103C2×C4×C197C2×D412yes—
104C2×C4×C197C2411yescharacteristic
105C8×C197C2×D412yes—
106C8×C197C2×D412yes—
107C22×C197—42no—
108C22×C197—42no—
109C22×C197—44no—
110C22×C197—44no—
111C22×C197C22×D411yescharacteristic
112C2×D197—22no—
113C2×D197—42no—
114C2×D197—44no—
115C2×D197—88no—
116C2×D197#12712yes—
117C4×C197—22no—
118C4×C197C22×D411yescharacteristic
119C4×C197C22×D411yescharacteristic
120Dic197—22no—
121Dic197—42no—
122Dic197#12712yes—
123C2×C197—82no—
124C2×C197—82no—
125C2×C197C2×#12711yescharacteristic
126C2×C197#57612yes—
127D197—22no—
128D197—82no—
129C197#315511yescharacteristic, complemented, Sylow
130#3155—1971nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1972no—
132C2×#135—1971no—
133C22×D8—1971no—
134#535—1972no—
135#496—1972no—
136#496—1972no—
137#495—1971no—
138#495—1971no—
139#493—1971no—
140#493D19711yescharacteristic, complemented
141#405—1971no—
142C2×#41—1972no—
143C22×D4—1972no—
144AGL(1,ℤ/8)—3948no—
145#135—3944no—
146C2×Q16—1971no—
147C2×SD16—1972no—
148C2×SD16—1972no—
149C2×SD16—1972no—
150C2×D8—3944no—
151C2×D8—1972no—
152C2×D8—1972no—
153C2×D8—1971no—
154C2×D8C2×D19711yescharacteristic
155C2×D8C2×D19711yescharacteristic, complemented
156C2×D8C2×D19712yescomplemented
157C2×M16—1971no—
158C22×C8—1971no—
159#127—1972no—
160#127C2×D19712yescomplemented
161#124—1972no—
162#124—1972no—
163#121—3944no—
164#120—3944no—
165#117—1971no—
166#97—1971no—
167#97—1971no—
168C4×C8—1971no—
169C4×C8C2×D19711yescharacteristic, complemented
170C24—3942no—
171#41—3944no—
172#41—3944no—
173C2×Q8—1972no—
174C2×D4—24no—
175C2×D4—3942no—
176C2×D4—3944no—
177C2×D4—3944no—
178C2×D4—3942no—
179C2×D4—3944no—
180C2×D4—1972no—
181C2×D4C22×D19712yes—
182C2×D4C22×D19712yes—
183C22×C4—3942no—
184C22×C4—1971no—
185Q16—3942no—
186SD16—3944no—
187SD16—3944no—
188SD16—3944no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—3944no—
193D8—3944no—
194D8—3942no—
195M16—1974no—
196C2×C8—1971no—
197C2×C8—1971no—
198C2×C8—1974no—
199C2×C8C22×D19711yescharacteristic
200C2×C8C22×D19711yescharacteristic
201C4⋊C4—1972no—
202C22⋊C4—3944no—
203C22⋊C4—3944no—
204C22⋊C4—1972no—
205C42—1971no—
206C42C22×D19711yescharacteristic
207C23—22no—
208C23—22no—
209C23—7884no—
210C23—15764no—
211C23—3942no—
212C23—1971no—
213Q8—3942no—
214Q8—3942no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—7884no—
221D4—7884no—
222D4—15764no—
223D4—3942no—
224D4—3942no—
225C2×C4—7884no—
226C2×C4—3942no—
227C2×C4—1972no—
228C2×C4—1971no—
229C2×C4—1972no—
230C2×C4—1972no—
231C2×C4D4×D19712yes—
232C2×C4C23×D19711yescharacteristic
233C8—1972no—
234C8—1972no—
235C8D4×D19712yes—
236C8D4×D19712yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—7882no—
242C22—7884no—
243C22—15768no—
244C22—3942no—
245C22—1972no—
246C22C2×D4×D19711yescharacteristic
247C4—22no—
248C4—7882no—
249C4—3942no—
250C4—1972no—
251C4C2×D4×D19711yescharacteristic
252C4C2×D4×D19711yescharacteristic
253C2—82no—
254C2—82no—
255C2—15762no—
256C2—3942no—
257C2#127×D19711yescharacteristic
258C2#2716188012yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#493×C197C2
socleC22×C197C22×D4
hypercenter#493D197
hypocenterC197#3155
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. C2×#20263345C2semidirect
3. C2×D8×D197C2semidirect
4. #27161642C2semidirect
5. #27161467C2semidirect
6. #27161466C2semidirect
7. #27161463C2semidirect
8. #27161462C2semidirect
9. #27161454C2semidirect
10. #27161033C2semidirect
11. #493×C197C2semidirect
16. C2×Dic788C22semidirect
17. C2×#15691347C22semidirect
18. C2×#15691346C22semidirect
19. C2×#15691346C22semidirect
22. C2×#15691343C22semidirect
26. #20263322C22semidirect
27. #20263314C22semidirect
28. #20263313C22semidirect
32. #20263237C22semidirect
33. #20263236C22semidirect
34. C8×Dic197C22semidirect
36. C2×D8×C197C22semidirect
37. C2×D8×C197C22semidirect
38. #127×C197C22semidirect
39. C4×C8×C197C22semidirect
59. #15691341D4semidirect
62. C8×D197D4semidirect
129. C197#3155semidirect
140. #493D197semidirect
155. C2×D8C2×D197semidirect
156. C2×D8C2×D197semidirect
160. #127C2×D197semidirect
169. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
C2×#202633452
C2×D8×D1972
#271616422
#271614672
#271614662
#271614632
#271614622
#271614542
#271610332
#493×C1972
#3155197