Finite Group Databasebeta

← #34099250

Subgroups — #34099250

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#23757560C212yesmaximal, complemented
3#23757017C211yesmaximal, characteristic, complemented
4#23757002C212yesmaximal, complemented
5#492×C157C211yesmaximal, characteristic, complemented
6C2×D4×D157—22no—
7#19097742—22no—
8#19097739—22no—
9C4×D628C2212yes—
10#19097702—22no—
11C4⋊#11929434C2212yescomplemented
12#19097691—22no—
13C2×SD16×C157—22no—
14C4⋊Q8×C157C2211yescharacteristic, complemented
15#127×C157C2211yescharacteristic
16C4×C8×C157C2211yescharacteristic, complemented
17C23×D157—22no—
18C2×D628D412yes—
19C2×C22⋊D157—24no—
20D4×D157—44no—
21D4×D157—44no—
22C2×C4×D157—22no—
23C2×#11929434—22no—
24C4⋊Dic157D412yes—
25C22⋊Dic157—22no—
26#14578931—22no—
27C2×Q8×C157—21no—
28C2×D4×C157—22no—
29C2×D4×C157D412yes—
30SD16×C157—44no—
31C2×C8×C157—21no—
32C4⋊C4×C157—22no—
33C42×C157C2311yescharacteristic
34C22×D157—22no—
35C22×D157—48no—
36C22×D157—82no—
37D628—22no—
38D628SD1614yescomplemented
39C22⋊D157—48no—
40C2×Dic157—22no—
41C4×D157—42no—
42C4×D157—42no—
43#11929434—42no—
44C23×C157—22no—
45Q8×C157—42no—
46D4×C157—42no—
47D4×C157—42no—
48D4×C157—42no—
49C2×C4×C157—41no—
50C2×C4×C157C2×D412yes—
51C2×C4×C157C2×D411yescharacteristic
52C8×C157—22no—
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×C157—81no—
62C4×C157C2×SD1612yes—
63C4×C157AGL(1,ℤ/8)12yes—
64Dic157—42no—
65C2×C157—82no—
66C2×C157#45012yes—
67C2×C157#45311yescharacteristic
68D157—44no—
69C157#162911yescharacteristic, complemented, Sylow
70#1629—1571nomaximal, Sylow
71D42—1571no—
72#492D15711yescharacteristic, complemented
73#476—1572no—
74#331—1571no—
75D4⋊C8—1572no—
76C22×D4—3142no—
77C2×SD16—22no—
78C4⋊Q8C2×D15711yescharacteristic, complemented
79#127C2×D15711yescharacteristic
80#121—3142no—
81#120—3142no—
82C4×D4—1572no—
83#106—3142no—
84C4⋊C8—1572no—
85#102—3142no—
86C4×C8C2×D15711yescharacteristic, complemented
87C24—3142no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—3144no—
91C2×D4—6284no—
92C2×D4—6284no—
93C2×D4—1572no—
94C2×D4C22⋊D15712yes—
95C22×C4—3142no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—3142no—
99C4⋊C4—22no—
100C4⋊C4—1572no—
101C22⋊C4—3142no—
102C22⋊C4—3142no—
103C42C22×D15711yescharacteristic
104C23—22no—
105C23—3142no—
106C23—6288no—
107C23—12562no—
108Q8—42no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—3142no—
113D4—6288no—
114D4—1574no—
115C2×C4—41no—
116C2×C4—3142no—
117C2×C4—6282no—
118C2×C4—6282no—
119C2×C4D4×D15712yes—
120C2×C4C2×C22⋊D15711yescharacteristic
121C8—22no—
122C8—6282no—
123C22—44no—
124C22—42no—
125C22—3144no—
126C22—6282no—
127C22—6282no—
128C22—12568no—
129C22#1909773911yescharacteristic
130C4—21no—
131C4—81no—
132C4—6282no—
133C4SD16×D15712yes—
134C4#1909779812yes—
135C2—82no—
136C2—6284no—
137C2#2375742212yes—
138C2#2375744011yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#492×C157C2
socleC22×C157#120
prime radical#23757908C2
hypercenter#492D157
hypocenterC157#1629
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #23757560C2semidirect
3. #23757017C2semidirect
4. #23757002C2semidirect
5. #492×C157C2semidirect
11. C4⋊#11929434C22semidirect
14. C4⋊Q8×C157C22semidirect
16. C4×C8×C157C22semidirect
38. D628SD16semidirect
69. C157#1629semidirect
72. #492D157semidirect
78. C4⋊Q8C2×D157semidirect
86. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579082
#237575602
#237570172
#237570022
#492×C1572
#1629157