Finite Group Databasebeta

← #38755849

Subgroups — #38755849

Order 25216.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#27161880C212yesmaximal, complemented
2#27161515C212yesmaximal, complemented
3#27161513C214yesmaximal, complemented
4#27161511C211yesmaximal, characteristic, complemented
5#27161508C212yesmaximal, complemented
6D16×D197C212yesmaximal, complemented
7#27161495C211yesmaximal, characteristic, complemented
8C2×D16×C197C211yesmaximal, characteristic, complemented
9#20263371—22no—
10#20263352—22no—
11#20263350—24no—
12#20263348C2212yescomplemented
13#20263345C2214yescomplemented
14D8×D197C2214yescomplemented
15#20263332C2211yescharacteristic
16Dic1576C2211yescharacteristic, complemented
17#20263271C2212yescomplemented
18#20263270C2214yescomplemented
19D3152C2211yescharacteristic, complemented
20#20263267C2214yescomplemented
21#20263266C2211yescharacteristic, complemented
22#20263265C2212yescomplemented
23C16×D197C2212yescomplemented
24C2×D8×C197C2212yescomplemented
25D16×C197C2212yescomplemented
26D16×C197C2212yescomplemented
27C2×C16×C197C2211yescharacteristic, complemented
28#15691358—22no—
29#15691358D411yescharacteristic
30#15691357—24no—
31#15691357—42no—
32C2×C22⋊D197—44no—
33D4×D197—24no—
34D4×D197—42no—
35Dic788C2312yes—
36#15691347—22no—
37#15691346—24no—
38D1576C2312yes—
39#15691343—24no—
40#15691342D411yescharacteristic
41#15691341D412yes—
42C8×D197C2312yes—
43#15691333C2312yes—
44C2×D4×C197—22no—
45D8×C197—22no—
46D8×C197C2314yes—
47C2×C8×C197C2311yescharacteristic
48C16×C197C2311yescharacteristic
49C16×C197C2311yescharacteristic
50C22×D197—44no—
51Dic394—22no—
52Dic394D811yescharacteristic
53D788—22no—
54D788D811yescharacteristic
55C22⋊D197—42no—
56C22⋊D197—44no—
57C22⋊D197—44no—
58C22⋊D197D812yes—
59C2×Dic197—44no—
60C4×D197—42no—
61C4×D197C2×D412yes—
62#13096802C2×D412yes—
63C23×C197—42no—
64D4×C197—24no—
65D4×C197—42no—
66C2×C4×C197C2×D411yescharacteristic
67C8×C197C2×D411yescharacteristic
68C8×C197C2411yescharacteristic
69C22×C197—44no—
70C22×C197—162no—
71C22×C197C2×D811yescharacteristic
72C2×D197—42no—
73C2×D197—164no—
74C2×D197C2×D812yes—
75C4×C197C2×D811yescharacteristic
76C4×C197C22×D411yescharacteristic
77Dic197—42no—
78Dic197C2×D812yes—
79C2×C197—21no—
80C2×C197—84no—
81C2×C197C22×D811yescharacteristic
82D197—22no—
83D197—82no—
84C197#341511yescharacteristic, complemented, Sylow
85#3415—1971nomaximal, Sylow
86#576—1972no—
87#509—1972no—
88#509—1974no—
89#508—1972no—
90#508—1971no—
91C2×D16—1972no—
92C2×D16D19711yescharacteristic, complemented
93#504—1971no—
94H(5,𝔽2)—3942no—
95AGL(1,ℤ/8)—3944no—
96AGL(1,ℤ/8)—3942no—
97#135—1974no—
98#135—1972no—
99C2×D8—1974no—
100C2×D8C2×D19712yescomplemented
101#131—1971no—
102Q32—1971no—
103SD32—1974no—
104SD32—1972no—
105D16—1974no—
106D16—1971no—
107D16C2×D19712yescomplemented
108D16C2×D19712yescomplemented
109M32—1971no—
110M32—1972no—
111C2×C16—1972no—
112C2×C16C2×D19711yescharacteristic, complemented
113#41—7882no—
114#41—3944no—
115#41—3942no—
116#41—1971no—
117C2×D4—22no—
118C2×D4—7882no—
119C2×D4—7884no—
120C2×D4—3944no—
121Q16—1972no—
122SD16—3944no—
123SD16—3942no—
124D8—22no—
125D8—3944no—
126D8—1972no—
127D8C22×D19714yes—
128M16—1971no—
129M16—1972no—
130C2×C8—1972no—
131C2×C8C22×D19711yescharacteristic
132C16—1972no—
133C16C22×D19711yescharacteristic
134C16C22×D19711yescharacteristic
135C23—42no—
136C23—7884no—
137Q8—3942no—
138Q8—1971no—
139D4—24no—
140D4—42no—
141D4—7884no—
142D4—7882no—
143D4—7884no—
144D4—3942no—
145D4—1972no—
146D4—1971no—
147C2×C4—7884no—
148C2×C4—7882no—
149C2×C4—1972no—
150C2×C4D4×D19711yescharacteristic
151C8—1972no—
152C8D4×D19711yescharacteristic
153C8C23×D19711yescharacteristic
154C22—44no—
155C22—162no—
156C22—7882no—
157C22—31524no—
158C22—1972no—
159C22D8×D19711yescharacteristic
160C4—7882no—
161C4—1972no—
162C4D8×D19711yescharacteristic
163C4C2×D4×D19711yescharacteristic
164C2—21no—
165C2—84no—
166C2—15762no—
167C2—3942no—
168C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
FittingC2×D16×C197C2
socleC2×C197C22×D8
hypercenterC2×D16D197
hypocenterC197#3415
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #27161880C2semidirect
2. #27161515C2semidirect
3. #27161513C2semidirect
4. #27161511C2semidirect
5. #27161508C2semidirect
6. D16×D197C2semidirect
7. #27161495C2semidirect
8. C2×D16×C197C2semidirect
12. #20263348C22semidirect
13. #20263345C22semidirect
14. D8×D197C22semidirect
16. Dic1576C22semidirect
17. #20263271C22semidirect
18. #20263270C22semidirect
19. D3152C22semidirect
20. #20263267C22semidirect
21. #20263266C22semidirect
22. #20263265C22semidirect
23. C16×D197C22semidirect
24. C2×D8×C197C22semidirect
25. D16×C197C22semidirect
26. D16×C197C22semidirect
27. C2×C16×C197C22semidirect
84. C197#3415semidirect
92. C2×D16D197semidirect
100. C2×D8C2×D197semidirect
107. D16C2×D197semidirect
108. D16C2×D197semidirect
112. C2×C16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618802
#271615152
#271615132
#271615112
#271615082
D16×D1972
#271614952
C2×D16×C1972
#3415197