Finite Group Databasebeta

← #34099253

Subgroups — #34099253

Order 20096.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757908C211yesmaximal, characteristic
2#23757645C211yesmaximal, characteristic, complemented
3#23757564C212yesmaximal, complemented
4#23757000C212yesmaximal, complemented
5#331×C157C211yesmaximal, characteristic, complemented
6C2×D4×D157—22no—
7C2×#14578942—22no—
8#19097771C2211yescharacteristic, complemented
9#19097742—22no—
10#19097739—22no—
11C4×D628C2212yes—
12#19097703—22no—
13#19097693—22no—
14C4×#11929434C2211yescharacteristic, complemented
15#127×C157C2211yescharacteristic
16C4⋊C8×C157C2212yescomplemented
17C23×D157—22no—
18C2×Dic314—21no—
19C2×D628D412yes—
20C2×C22⋊D157—24no—
21D4×D157—44no—
22D4×D157—44no—
23C2×C4×D157—22no—
24#14578942—44no—
25C2×#11929434—21no—
26C4⋊Dic157—22no—
27C4⋊Dic157D412yes—
28C22⋊Dic157—22no—
29#14578931—22no—
30C2×D4×C157—22no—
31C2×D4×C157D412yes—
32C2×C8×C157—22no—
33C42×C157C2311yescharacteristic
34C22×D157—22no—
35C22×D157—48no—
36C22×D157—82no—
37Dic314—42no—
38D628—22no—
39D628SD1614yescomplemented
40C22⋊D157—48no—
41C2×Dic157—22no—
42C2×Dic157—41no—
43C4×D157—42no—
44C4×D157—42no—
45#11929434—22no—
46C23×C157—22no—
47D4×C157—42no—
48D4×C157—42no—
49D4×C157—42no—
50C2×C4×C157C2×D412yes—
51C2×C4×C157C2×D411yescharacteristic
52C8×C157—42no—
53C22×C157—44no—
54C22×C157—42no—
55C22×C157#12011yescharacteristic
56C2×D157—24no—
57C2×D157—42no—
58C2×D157—42no—
59C2×D157—88no—
60C4×C157—21no—
61C4×C157C2×SD1612yes—
62C4×C157AGL(1,ℤ/8)12yes—
63Dic157—42no—
64Dic157—81no—
65C2×C157—82no—
66C2×C157#45012yes—
67C2×C157#45311yescharacteristic
68D157—44no—
69C157#162911yescharacteristic, complemented, Sylow
70#1629—1571nomaximal, Sylow
71D42—1571no—
72#492—1571no—
73#476—1572no—
74#331D15711yescharacteristic, complemented
75D4⋊C8—1572no—
76C22×D4—3142no—
77C2×SD16—3142no—
78C4⋊Q8—1571no—
79#127C2×D15711yescharacteristic
80#121—3142no—
81#120—3142no—
82C4×D4—1572no—
83#106—3142no—
84C4⋊C8C2×D15712yescomplemented
85#102—3142no—
86C4×C8—1571no—
87C24—3142no—
88C2×Q8—3141no—
89C2×D4—22no—
90C2×D4—3144no—
91C2×D4—6284no—
92C2×D4—6284no—
93C2×D4—1572no—
94C2×D4D62812yes—
95C22×C4—3142no—
96SD16—6284no—
97C2×C8—22no—
98C2×C8—3141no—
99C4⋊C4—3142no—
100C4⋊C4—1572no—
101C22⋊C4—3142no—
102C22⋊C4—3142no—
103C42C22×D15711yescharacteristic
104C23—22no—
105C23—3142no—
106C23—6288no—
107C23—12562no—
108Q8—6282no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—3142no—
113D4—6288no—
114D4—1574no—
115C2×C4—3142no—
116C2×C4—6282no—
117C2×C4—6282no—
118C2×C4—6281no—
119C2×C4D4×D15712yes—
120C2×C4C2×D62811yescharacteristic
121C8—42no—
122C8—3142no—
123C22—44no—
124C22—42no—
125C22—3144no—
126C22—6282no—
127C22—6282no—
128C22—12568no—
129C22#1909774111yescharacteristic
130C4—21no—
131C4—6282no—
132C4—12561no—
133C4SD16×D15712yes—
134C4#1909779512yes—
135C2—82no—
136C2—6284no—
137C2#2375742512yes—
138C2#2375744311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157#120
prime radical#23757908C2
hypercenter#331D157
hypocenterC157#1629
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #23757645C2semidirect
3. #23757564C2semidirect
4. #23757000C2semidirect
5. #331×C157C2semidirect
8. #19097771C22semidirect
14. C4×#11929434C22semidirect
16. C4⋊C8×C157C22semidirect
39. D628SD16semidirect
69. C157#1629semidirect
74. #331D157semidirect
84. C4⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579082
#237576452
#237575642
#237570002
#331×C1572
#1629157