Finite Group Databasebeta

← #34099217

Subgroups — #34099217

Order 20096.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757989C211yesmaximal, characteristic, complemented
2#23757465C212yesmaximal, complemented
3#23757022C211yesmaximal, characteristic, complemented
4#23757002C212yesmaximal, complemented
5#492×C157C211yesmaximal, characteristic, complemented
6C2×#14578939—22no—
7#19097760—22no—
8#19097746—22no—
9C4×D628C2212yes—
10#19097729—22no—
11C4⋊#11929434C2212yes—
12#19097692—22no—
13C2×SD16×C157—22no—
14C4⋊Q8×C157C2211yescharacteristic
15#127×C157C2211yescharacteristic
16C4×C8×C157C2211yescharacteristic
17C2×D628—22no—
18C2×D628D412yes—
19C2×C4×D157—22no—
20#14578939—44no—
21C2×#11929434—22no—
22C4⋊Dic157D412yes—
23#14578931—24no—
24#14578931—22no—
25C4×Dic157—22no—
26C2×Q8×C157D412yes—
27C2×D4×C157—21no—
28C2×D4×C157—22no—
29SD16×C157—44no—
30C2×C8×C157—21no—
31C4⋊C4×C157—22no—
32C42×C157C2311yescharacteristic
33C22×D157—22no—
34D628—22no—
35D628—22no—
36D628—82no—
37C2×Dic157—22no—
38C4×D157—42no—
39C4×D157—42no—
40#11929434—42no—
41C23×C157—41no—
42Q8×C157—42no—
43D4×C157—42no—
44D4×C157—42no—
45D4×C157—42no—
46C2×C4×C157—22no—
47C2×C4×C157C2×D412yes—
48C2×C4×C157C2×D411yescharacteristic
49C8×C157—22no—
50C22×C157—82no—
51C22×C157—81no—
52C22×C157#12011yescharacteristic
53C2×D157—42no—
54C2×D157—42no—
55C2×D157—42no—
56C4×C157—21no—
57C4×C157—42no—
58C4×C157#13512yes—
59C4×C157AGL(1,ℤ/8)12yes—
60Dic157—42no—
61C2×C157—161no—
62C2×C157#44912yes—
63C2×C157#45311yescharacteristic
64D157—82no—
65C157#162411yescharacteristic, complemented, Sylow
66#1624—1571nomaximal, Sylow
67#555—1571no—
68#492D15711yescharacteristic, complemented
69#459—1572no—
70#333—1571no—
71D4⋊C8—1572no—
72C2×SD16—22no—
73C2×D8—3142no—
74C4⋊Q8C2×D15711yescharacteristic
75#127C2×D15711yescharacteristic
76#124—3142no—
77#121—3142no—
78C4×D4—1572no—
79#117—3142no—
80C4⋊C8—1572no—
81#102—3142no—
82C4×C8C2×D15711yescharacteristic
83C2×Q8C22⋊D15712yes—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—3142no—
87C2×D4—1572no—
88C22×C4—3142no—
89SD16—44no—
90D8—6284no—
91C2×C8—21no—
92C2×C8—3142no—
93C4⋊C4—22no—
94C4⋊C4—1572no—
95C22⋊C4—3144no—
96C22⋊C4—3142no—
97C42—3142no—
98C42C22×D15711yescharacteristic
99C23—41no—
100C23—3142no—
101Q8—42no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—3142no—
106D4—3142no—
107D4—12562no—
108C2×C4—22no—
109C2×C4—3142no—
110C2×C4—6282no—
111C2×C4—6282no—
112C2×C4D4×D15712yes—
113C2×C4C2×C22⋊D15711yescharacteristic
114C8—22no—
115C8—6282no—
116C22—82no—
117C22—81no—
118C22—6282no—
119C22—6282no—
120C22—6282no—
121C22#1909773911yescharacteristic
122C4—21no—
123C4—42no—
124C4—6282no—
125C4#1909779212yes—
126C4#1909779812yes—
127C2—161no—
128C2—12562no—
129C2#2375741712yes—
130C2#2375744011yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#492×C157C2
socleC22×C157#120
hypercenter#492D157
hypocenterC157#1624
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #23757989C2semidirect
2. #23757465C2semidirect
3. #23757022C2semidirect
4. #23757002C2semidirect
5. #492×C157C2semidirect
65. C157#1624semidirect
68. #492D157semidirect

Maximal subgroups

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

subgroupindex
#237579892
#237574652
#237570222
#237570022
#492×C1572
#1624157