Finite Group Databasebeta

← #38753161

Subgroups — #38753161

Order 25216.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20263353C212yesmaximal, complemented
2C2×#20263350C212yesmaximal, complemented
3#27161628C211yesmaximal, characteristic, complemented
4#127×D197C211yesmaximal, characteristic, complemented
5#27161469C211yesmaximal, characteristic, complemented
6#27161468C211yesmaximal, characteristic, complemented
7#27161466C211yesmaximal, characteristic, complemented
8#27161454C212yesmaximal, complemented
9#27161450C212yesmaximal, complemented
10#27161031C211yesmaximal, characteristic, complemented
11#496×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C2211yescharacteristic
13C2×#15691357C2211yescharacteristic
14C2×D4×D197—22no—
15C2×D4×D197C2212yes—
16#20263353—28no—
17#20263350—28no—
18C2×#15691347C2212yescomplemented
19C2×#15691346C2212yescomplemented
20C2×#15691345C2212yescomplemented
21C2×D1576C2212yescomplemented
22C2×#15691343C2212yescomplemented
23C2×#15691343C2212yescomplemented
24C2×#15691341C2212yes—
25#20263324C2211yescharacteristic, complemented
26#20263323C2211yescharacteristic
27#20263322—22no—
28#20263322C2212yes—
29#20263322C2211yescharacteristic, complemented
30#20263314C2211yescharacteristic, complemented
31#20263299—22no—
32#20263286—22no—
33C42×D197C2211yescharacteristic
34#20263237C2211yescharacteristic, complemented
35C8×Dic197C2212yescomplemented
36C2×SD16×C197C2212yescomplemented
37C2×D8×C197C2212yescomplemented
38#127×C197C2211yescharacteristic
39#124×C197C2211yescharacteristic, complemented
40#97×C197C2211yescharacteristic, complemented
41C23×D197—22no—
42#15691359—24no—
43#15691357—24no—
44C2×Dic394C2311yescharacteristic
45C2×D788—21no—
46C2×D788—22no—
47C2×D788C2311yescharacteristic
48C2×D788C2312yes—
49C2×C22⋊D197—21no—
50C2×C22⋊D197—22no—
51C2×C22⋊D197—22no—
52C2×C22⋊D197—24no—
53D4×D197—24no—
54D4×D197—24no—
55D4×D197—48no—
56C22×Dic197—21no—
57C2×C4×D197—21no—
58C2×C4×D197D412yes—
59C2×C4×D197C2311yescharacteristic
60#15691347—24no—
61#15691346—24no—
62#15691345—24no—
63D1576—24no—
64#15691343—24no—
65#15691343—24no—
66#15691341D418yescomplemented
67C2×#13096802C2312yes—
68#15691337—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—48no—
84C22×D197—44no—
85C22×D197C2×D411yescharacteristic
86Dic394—22no—
87D788—22no—
88D788—22no—
89D788—22no—
90D788—44no—
91D788—42no—
92C22⋊D197—42no—
93C22⋊D197—44no—
94C22⋊D197—48no—
95C22⋊D197—44no—
96C2×Dic197—21no—
97C2×Dic197—42no—
98C2×Dic197C2×D411yescharacteristic
99C2×Dic197C2×D412yes—
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—22no—
121C2×D197—41no—
122C2×D197—42no—
123C2×D197—44no—
124C2×D197—42no—
125C2×D197—88no—
126C2×D197#12712yes—
127C4×C197—22no—
128C4×C197—41no—
129C4×C197C22×D412yes—
130Dic197—41no—
131Dic197#12712yes—
132Dic197AGL(1,ℤ/8)14yescomplemented
133C2×C197—82no—
134C2×C197—81no—
135C2×C197C2×#12711yescharacteristic
136C2×C197C2×AGL(1,ℤ/8)12yes—
137D197—22no—
138D197—81no—
139D197—82no—
140C197#315111yescharacteristic, complemented, Sylow
141#3151—1971nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1972no—
143C2×AGL(1,ℤ/8)—1972no—
144#532—1971no—
145C2×#127—1971no—
146#496—1971no—
147#496—1971no—
148#496—1971no—
149#496D19711yescharacteristic, complemented
150#493—1972no—
151#492—1972no—
152C4×M16—1971no—
153C2×#41—1971no—
154C2×#41—1971no—
155C22×D4—3942no—
156C22×D4—1972no—
157AGL(1,ℤ/8)—3948no—
158AGL(1,ℤ/8)—3948no—
159C2×SD16—1972no—
160C2×SD16—1972no—
161C2×SD16—1972no—
162C2×SD16C2×D19712yescomplemented
163C2×D8—1972no—
164C2×D8—1972no—
165C2×D8—1972no—
166C2×D8C2×D19712yescomplemented
167C2×M16—1972no—
168C4⋊Q8—1971no—
169#127—3942no—
170#127—1972no—
171#127—1971no—
172#127—1971no—
173#127C2×D19711yescharacteristic
174#124—1971no—
175#124C2×D19711yescharacteristic, complemented
176#121—3942no—
177C4×D4—3942no—
178C2×C42—1971no—
179#97—1971no—
180#97C2×D19711yescharacteristic, complemented
181C4×C8—1972no—
182C24—3942no—
183#41—3944no—
184#41—3944no—
185C2×Q8—1971no—
186C2×Q8C22×D19711yescharacteristic
187C2×D4—22no—
188C2×D4—7888no—
189C2×D4—3941no—
190C2×D4—3941no—
191C2×D4—3944no—
192C2×D4—3944no—
193C2×D4—3942no—
194C2×D4—3942no—
195C2×D4—3944no—
196C2×D4—3942no—
197C2×D4—1971no—
198C2×D4—1972no—
199C2×D4C22×D19711yescharacteristic
200C2×D4C22×D19712yes—
201C22×C4—3941no—
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—
213M16—1978no—
214C2×C8—1972no—
215C2×C8C22×D19712yes—
216C4⋊C4—3942no—
217C22⋊C4—22no—
218C22⋊C4—3942no—
219C42—1972no—
220C42—1971no—
221C42C22×D19711yescharacteristic
222C23—21no—
223C23—22no—
224C23—7888no—
225C23—7884no—
226C23—3941no—
227C23—3942no—
228C23—1971no—
229Q8—22no—
230Q8—3942no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—7884no—
236D4—7882no—
237D4—7882no—
238D4—7888no—
239D4—7884no—
240D4—7884no—
241D4—3942no—
242D4—3942no—
243D4—3942no—
244C2×C4—21no—
245C2×C4—7882no—
246C2×C4—7882no—
247C2×C4—3944no—
248C2×C4—3941no—
249C2×C4—1972no—
250C2×C4—1971no—
251C2×C4—1974no—
252C2×C4D4×D19712yes—
253C2×C4C23×D19711yescharacteristic
254C8—1974no—
255C8D4×D19714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—7884no—
261C22—7882no—
262C22—7881no—
263C22—7882no—
264C22—15768no—
265C22—3942no—
266C22—1972no—
267C22C2×D4×D19711yescharacteristic
268C4—22no—
269C4—41no—
270C4—7881no—
271C4—1974no—
272C4—1972no—
273C4C2×D4×D19712yes—
274C2—82no—
275C2—81no—
276C2—15762no—
277C2—15761no—
278C2—3942no—
279C2#127×D19711yescharacteristic
280C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#496×C197C2
socleC22×C197C22×D4
hypercenter#496D197
hypocenterC197#3151
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×#20263353C2semidirect
2. C2×#20263350C2semidirect
3. #27161628C2semidirect
4. #127×D197C2semidirect
5. #27161469C2semidirect
6. #27161468C2semidirect
7. #27161466C2semidirect
8. #27161454C2semidirect
9. #27161450C2semidirect
10. #27161031C2semidirect
11. #496×C197C2semidirect
18. C2×#15691347C22semidirect
19. C2×#15691346C22semidirect
20. C2×#15691345C22semidirect
21. C2×D1576C22semidirect
22. C2×#15691343C22semidirect
23. C2×#15691343C22semidirect
25. #20263324C22semidirect
29. #20263322C22semidirect
30. #20263314C22semidirect
34. #20263237C22semidirect
35. C8×Dic197C22semidirect
36. C2×SD16×C197C22semidirect
37. C2×D8×C197C22semidirect
39. #124×C197C22semidirect
40. #97×C197C22semidirect
66. #15691341D4semidirect
103. #13096802C2×D4semidirect
114. C8×C197C2×D4semidirect
132. Dic197AGL(1,ℤ/8)semidirect
140. C197#3151semidirect
149. #496D197semidirect
162. C2×SD16C2×D197semidirect
166. C2×D8C2×D197semidirect
175. #124C2×D197semidirect
180. #97C2×D197semidirect
255. C8D4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
C2×#202633502
#271616282
#127×D1972
#271614692
#271614682
#271614662
#271614542
#271614502
#271610312
#496×C1972
#3151197