Finite Group Databasebeta

← #38754512

Subgroups — #38754512

Order 25216.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#27161884C212yesmaximal, complemented
2#27161881C212yesmaximal, complemented
3#27161456C211yesmaximal, characteristic, complemented
4#27161350C212yesmaximal, complemented
5#27161349C212yesmaximal, complemented
6#27161248C214yesmaximal, complemented
7#27161197C211yesmaximal, characteristic, complemented
8#443×C197C211yesmaximal, characteristic, complemented
9#20263372C2214yescomplemented
10#20263354—24no—
11#20263354C2214yescomplemented
12#20263353—24no—
13#20263351C2214yescomplemented
14#20263349—22no—
15#20263348—22no—
16#20263348C2211yescharacteristic
17C2×D1576—22no—
18C2×D1576C2212yescomplemented
19C2×D1576C2212yescomplemented
20D8×D197—22no—
21#20263332C2212yescomplemented
22#20263323C2212yescomplemented
23#20263262C2211yescharacteristic, complemented
24#20263254C2212yescomplemented
25#20263242C2214yescomplemented
26#135×C197C2211yescharacteristic
27#131×C197C2212yescomplemented
28#108×C197C2211yescharacteristic, complemented
29#104×C197C2212yescomplemented
30C4×C8×C197C2211yescharacteristic, complemented
31#15691359—24no—
32#15691358—24no—
33#15691358C2312yes—
34C2×D788—24no—
35C2×D788—42no—
36C2×D788C2314yes—
37D4×D197—24no—
38D4×D197—44no—
39Dic788D411yescharacteristic, complemented
40#15691347—24no—
41#15691347D412yescomplemented
42#15691345—24no—
43D1576—24no—
44D1576—22no—
45D1576—22no—
46D1576—22no—
47D1576—22no—
48D1576D411yescharacteristic, complemented
49#15691343—24no—
50#15691342C2312yes—
51#15691341—22no—
52C8×D197—22no—
53#41×C197C2312yes—
54Q16×C197D411yescharacteristic, complemented
55SD16×C197D412yescomplemented
56D8×C197D411yescharacteristic, complemented
57M16×C197—22no—
58M16×C197C2312yes—
59C2×C8×C197—22no—
60C2×C8×C197C2311yescharacteristic
61C2×C8×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63C22×D197—24no—
64C22×D197—44no—
65Dic394C2×D412yes—
66D788—24no—
67D788—24no—
68D788—44no—
69D788—84no—
70D788C2×D412yes—
71C22⋊D197—22no—
72C22⋊D197—44no—
73C4×D197—22no—
74C4×D197—44no—
75#13096802—22no—
76Q8×C197C2×D412yes—
77D4×C197—22no—
78D4×C197C2×D412yes—
79C2×C4×C197—21no—
80C2×C4×C197—22no—
81C2×C4×C197C2411yescharacteristic
82C8×C197—22no—
83C8×C197C2×D411yescharacteristic
84C8×C197C2×D411yescharacteristic
85C8×C197C2×D411yescharacteristic
86C8×C197C2×D411yescharacteristic
87C22×C197—22no—
88C22×C197H(5,𝔽2)11yescharacteristic
89C2×D197—22no—
90C2×D197—44no—
91C2×D197—84no—
92C2×D197—164no—
93C4×C197—22no—
94C4×C197—22no—
95C4×C197C22×D411yescharacteristic
96C4×C197C22×D411yescharacteristic
97Dic197—22no—
98C2×C197—21no—
99C2×C197—42no—
100C2×C197D4211yescharacteristic
101D197—42no—
102D197—84no—
103C197#329211yescharacteristic, complemented, Sylow
104#3292—1971nomaximal, Sylow
105#576—1972no—
106#576—1972no—
107#493—1971no—
108#472—1972no—
109#472—1972no—
110#453—1974no—
111#443—1971no—
112#443D19711yescharacteristic, complemented
113H(5,𝔽2)—1974no—
114AGL(1,ℤ/8)—3944no—
115AGL(1,ℤ/8)—3944no—
116AGL(1,ℤ/8)—1974no—
117AGL(1,ℤ/8)—1974no—
118#135—3942no—
119#135—3942no—
120#135—1971no—
121#135C2×D19711yescharacteristic
122C2×D8—3942no—
123C2×D8—3942no—
124C2×D8—1972no—
125C2×D8—1972no—
126#131—1972no—
127#131C2×D19712yescomplemented
128#127—1972no—
129#108—1971no—
130#108C2×D19711yescharacteristic, complemented
131#104—1972no—
132#104C2×D19712yescomplemented
133#100—1974no—
134C4×C8C2×D19711yescharacteristic, complemented
135#41—3944no—
136#41—3944no—
137#41—1972no—
138#41C22×D19712yes—
139C2×D4—7882no—
140C2×D4—7884no—
141C2×D4—3944no—
142C2×D4—3944no—
143C2×D4—1974no—
144Q16—1971no—
145Q16D78811yescharacteristic, complemented
146SD16—3944no—
147SD16—3944no—
148SD16—1972no—
149SD16D78812yescomplemented
150D8—3944no—
151D8—3944no—
152D8—3942no—
153D8—3942no—
154D8—3942no—
155D8—3942no—
156D8—1971no—
157D8D78811yescharacteristic, complemented
158M16—22no—
159M16—3942no—
160M16—1972no—
161M16C22×D19712yes—
162C2×C8—22no—
163C2×C8—3942no—
164C2×C8C22×D19711yescharacteristic
165C2×C8C22×D19711yescharacteristic
166C42C22×D19711yescharacteristic
167C23—7884no—
168C23—3944no—
169Q8—1972no—
170Q8C2×D78812yes—
171D4—22no—
172D4—7884no—
173D4—7884no—
174D4—15764no—
175D4—3942no—
176D4—3944no—
177D4—3944no—
178D4—1972no—
179D4C2×D78812yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—7884no—
183C2×C4—3942no—
184C2×C4C23×D19711yescharacteristic
185C8—22no—
186C8—3942no—
187C8D4×D19711yescharacteristic
188C8D4×D19711yescharacteristic
189C8C2×D78811yescharacteristic
190C8C2×D78811yescharacteristic
191C22—22no—
192C22—7884no—
193C22—15764no—
194C22—31524no—
195C22—3942no—
196C22#2026337211yescharacteristic
197C4—22no—
198C4—22no—
199C4—3942no—
200C4C2×D4×D19711yescharacteristic
201C4C22×D78811yescharacteristic
202C2—21no—
203C2—42no—
204C2—7882no—
205C2—15764no—
206C2D4×D78811yescharacteristic
subgroupG/H
centerC2D4×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#443×C197C2
socleC2×C197D42
hypercenter#443D197
hypocenterC197#3292
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #27161884C2semidirect
2. #27161881C2semidirect
3. #27161456C2semidirect
4. #27161350C2semidirect
5. #27161349C2semidirect
6. #27161248C2semidirect
7. #27161197C2semidirect
8. #443×C197C2semidirect
9. #20263372C22semidirect
11. #20263354C22semidirect
13. #20263351C22semidirect
18. C2×D1576C22semidirect
19. C2×D1576C22semidirect
21. #20263332C22semidirect
22. #20263323C22semidirect
23. #20263262C22semidirect
24. #20263254C22semidirect
25. #20263242C22semidirect
27. #131×C197C22semidirect
28. #108×C197C22semidirect
29. #104×C197C22semidirect
30. C4×C8×C197C22semidirect
39. Dic788D4semidirect
41. #15691347D4semidirect
48. D1576D4semidirect
54. Q16×C197D4semidirect
55. SD16×C197D4semidirect
56. D8×C197D4semidirect
103. C197#3292semidirect
112. #443D197semidirect
127. #131C2×D197semidirect
130. #108C2×D197semidirect
132. #104C2×D197semidirect
134. C4×C8C2×D197semidirect
145. Q16D788semidirect
149. SD16D788semidirect
157. D8D788semidirect

Maximal subgroups

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

subgroupindex
#271618842
#271618812
#271614562
#271613502
#271613492
#271612482
#271611972
#443×C1972
#3292197