Finite Group Databasebeta

← #34113273

Subgroups — #34113273

Order 20096.

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#23758074C212yesmaximal, complemented
2#23757709C212yesmaximal, complemented
3#23757707C214yesmaximal, complemented
4#23757705C211yesmaximal, characteristic, complemented
5#23757702C212yesmaximal, complemented
6D16×D157C212yesmaximal, complemented
7#23757689C211yesmaximal, characteristic, complemented
8C2×D16×C157C211yesmaximal, characteristic, complemented
9#19097816—22no—
10#19097797—22no—
11#19097795—24no—
12#19097793C2212yescomplemented
13#19097790C2214yescomplemented
14D8×D157C2214yescomplemented
15#19097777C2211yescharacteristic
16Dic1256C2211yescharacteristic, complemented
17#19097716C2212yescomplemented
18#19097715C2214yescomplemented
19D2512C2211yescharacteristic, complemented
20#19097712C2214yescomplemented
21#19097711C2211yescharacteristic, complemented
22#19097710C2212yescomplemented
23C16×D157C2212yescomplemented
24C2×D8×C157C2212yescomplemented
25D16×C157C2212yescomplemented
26D16×C157C2212yescomplemented
27C2×C16×C157C2211yescharacteristic, complemented
28#14578954—22no—
29#14578954D411yescharacteristic
30#14578953—24no—
31#14578953—42no—
32C2×C22⋊D157—44no—
33D4×D157—24no—
34D4×D157—42no—
35Dic628C2312yes—
36#14578943—22no—
37#14578942—24no—
38D1256C2312yes—
39#14578939—24no—
40#14578938D411yescharacteristic
41#14578937D412yes—
42C8×D157C2312yes—
43#14578929C2312yes—
44C2×D4×C157—22no—
45D8×C157—22no—
46D8×C157C2314yes—
47C2×C8×C157C2311yescharacteristic
48C16×C157C2311yescharacteristic
49C16×C157C2311yescharacteristic
50C22×D157—44no—
51Dic314—22no—
52Dic314D811yescharacteristic
53D628—22no—
54D628D811yescharacteristic
55C22⋊D157—42no—
56C22⋊D157—44no—
57C22⋊D157—44no—
58C22⋊D157D812yes—
59C2×Dic157—44no—
60C4×D157—42no—
61C4×D157C2×D412yes—
62#11929434C2×D412yes—
63C23×C157—42no—
64D4×C157—24no—
65D4×C157—42no—
66C2×C4×C157C2×D411yescharacteristic
67C8×C157C2×D411yescharacteristic
68C8×C157C2411yescharacteristic
69C22×C157—44no—
70C22×C157—162no—
71C22×C157C2×D811yescharacteristic
72C2×D157—42no—
73C2×D157—164no—
74C2×D157C2×D812yes—
75C4×C157C2×D811yescharacteristic
76C4×C157C22×D411yescharacteristic
77Dic157—42no—
78Dic157C2×D812yes—
79C2×C157—21no—
80C2×C157—84no—
81C2×C157C22×D811yescharacteristic
82D157—22no—
83D157—82no—
84C157#341511yescharacteristic, complemented, Sylow
85#3415—1571nomaximal, Sylow
86#576—1572no—
87#509—1572no—
88#509—1574no—
89#508—1572no—
90#508—1571no—
91C2×D16—1572no—
92C2×D16D15711yescharacteristic, complemented
93#504—1571no—
94H(5,𝔽2)—3142no—
95AGL(1,ℤ/8)—3144no—
96AGL(1,ℤ/8)—3142no—
97#135—1574no—
98#135—1572no—
99C2×D8—1574no—
100C2×D8C2×D15712yescomplemented
101#131—1571no—
102Q32—1571no—
103SD32—1574no—
104SD32—1572no—
105D16—1574no—
106D16—1571no—
107D16C2×D15712yescomplemented
108D16C2×D15712yescomplemented
109M32—1571no—
110M32—1572no—
111C2×C16—1572no—
112C2×C16C2×D15711yescharacteristic, complemented
113#41—3144no—
114#41—3142no—
115#41—6282no—
116#41—1571no—
117C2×D4—22no—
118C2×D4—3144no—
119C2×D4—6282no—
120C2×D4—6284no—
121Q16—1572no—
122SD16—3144no—
123SD16—3142no—
124D8—22no—
125D8—3144no—
126D8—1572no—
127D8C22×D15714yes—
128M16—1571no—
129M16—1572no—
130C2×C8—1572no—
131C2×C8C22×D15711yescharacteristic
132C16—1572no—
133C16C22×D15711yescharacteristic
134C16C22×D15711yescharacteristic
135C23—42no—
136C23—6284no—
137Q8—3142no—
138Q8—1571no—
139D4—24no—
140D4—42no—
141D4—3142no—
142D4—6284no—
143D4—6282no—
144D4—6284no—
145D4—1572no—
146D4—1571no—
147C2×C4—6284no—
148C2×C4—6282no—
149C2×C4—1572no—
150C2×C4D4×D15711yescharacteristic
151C8—1572no—
152C8D4×D15711yescharacteristic
153C8C23×D15711yescharacteristic
154C22—44no—
155C22—162no—
156C22—6282no—
157C22—1572no—
158C22—25124no—
159C22D8×D15711yescharacteristic
160C4—6282no—
161C4—1572no—
162C4D8×D15711yescharacteristic
163C4C2×D4×D15711yescharacteristic
164C2—21no—
165C2—84no—
166C2—3142no—
167C2—12562no—
168C2C2×D8×D15711yescharacteristic
subgroupG/H
centerC2C2×D8×D157
commutator, cosocleC8×C157C24
FrattiniC8C23×D157
FittingC2×D16×C157C2
socleC2×C157C22×D8
hypercenterC2×D16D157
hypocenterC157#3415
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #23758074C2semidirect
2. #23757709C2semidirect
3. #23757707C2semidirect
4. #23757705C2semidirect
5. #23757702C2semidirect
6. D16×D157C2semidirect
7. #23757689C2semidirect
8. C2×D16×C157C2semidirect
12. #19097793C22semidirect
13. #19097790C22semidirect
14. D8×D157C22semidirect
16. Dic1256C22semidirect
17. #19097716C22semidirect
18. #19097715C22semidirect
19. D2512C22semidirect
20. #19097712C22semidirect
21. #19097711C22semidirect
22. #19097710C22semidirect
23. C16×D157C22semidirect
24. C2×D8×C157C22semidirect
25. D16×C157C22semidirect
26. D16×C157C22semidirect
27. C2×C16×C157C22semidirect
84. C157#3415semidirect
92. C2×D16D157semidirect
100. C2×D8C2×D157semidirect
107. D16C2×D157semidirect
108. D16C2×D157semidirect
112. C2×C16C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580742
#237577092
#237577072
#237577052
#237577022
D16×D1572
#237576892
C2×D16×C1572
#3415157