Finite Group Databasebeta

← #34099220

Subgroups — #34099220

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#23757647C211yesmaximal, characteristic, complemented
3#23757469C212yesmaximal, complemented
4#23757000C212yesmaximal, complemented
5#333×C157C211yesmaximal, characteristic, complemented
6C2×#14578941—22no—
7C2×D1256—22no—
8#19097768C2211yescharacteristic
9#19097760—22no—
10#19097746—22no—
11C4×D628C2212yes—
12#19097729—22no—
13#19097693—22no—
14C4×#11929434C2211yescharacteristic
15C4⋊Q8×C157C2211yescharacteristic
16C4⋊C8×C157C2212yes—
17C2×D628—22no—
18C2×D628—22no—
19C2×D628—21no—
20C2×D628D412yes—
21C2×C4×D157—22no—
22#14578941—44no—
23D1256—44no—
24C2×#11929434—21no—
25C4⋊Dic157D412yes—
26#14578931—24no—
27#14578931—22no—
28C4×Dic157—22no—
29C2×Q8×C157D412yes—
30C2×C8×C157—22no—
31C4⋊C4×C157—22no—
32C42×C157C2311yescharacteristic
33C22×D157—22no—
34C22×D157—41no—
35D628—22no—
36D628—22no—
37D628—42no—
38D628—42no—
39D628—42no—
40D628—82no—
41C2×Dic157—22no—
42C4×D157—42no—
43C4×D157—42no—
44#11929434—22no—
45Q8×C157—42no—
46C2×C4×C157—22no—
47C2×C4×C157C2×D412yes—
48C2×C4×C157C2×D411yescharacteristic
49C8×C157—42no—
50C22×C157#12011yescharacteristic
51C2×D157—42no—
52C2×D157—42no—
53C2×D157—42no—
54C2×D157—82no—
55C2×D157—81no—
56C4×C157—21no—
57C4×C157—42no—
58C4×C157#13512yes—
59C4×C157AGL(1,ℤ/8)12yes—
60Dic157—42no—
61C2×C157#44912yes—
62C2×C157#45311yescharacteristic
63D157—82no—
64D157—161no—
65C157#162411yescharacteristic, complemented, Sylow
66#1624—1571nomaximal, Sylow
67#555—1571no—
68#492—1571no—
69#459—1572no—
70#333D15711yescharacteristic, complemented
71D4⋊C8—1572no—
72C2×SD16—3142no—
73C2×D8—3142no—
74C4⋊Q8C2×D15711yescharacteristic
75#127—1571no—
76#124—3142no—
77#121—3142no—
78C4×D4—1572no—
79#117—3142no—
80C4⋊C8C2×D15712yes—
81#102—3142no—
82C4×C8—1571no—
83C2×Q8D62812yes—
84C2×D4—3142no—
85C2×D4—3142no—
86C2×D4—3141no—
87C2×D4—1572no—
88C22×C4—3142no—
89SD16—6284no—
90D8—6284no—
91C2×C8—22no—
92C2×C8—3141no—
93C4⋊C4—22no—
94C4⋊C4—1572no—
95C22⋊C4—3144no—
96C22⋊C4—3142no—
97C42—3142no—
98C42C22×D15711yescharacteristic
99C23—3142no—
100C23—6281no—
101Q8—42no—
102D4—3142no—
103D4—3142no—
104D4—6282no—
105D4—6282no—
106D4—6282no—
107D4—12562no—
108C2×C4—22no—
109C2×C4—3142no—
110C2×C4—6282no—
111C2×C4—6282no—
112C2×C4D4×D15712yes—
113C2×C4C2×D62811yescharacteristic
114C8—42no—
115C8—3142no—
116C22—6282no—
117C22—6282no—
118C22—6282no—
119C22—12562no—
120C22—12561no—
121C22#1909774111yescharacteristic
122C4—21no—
123C4—42no—
124C4—6282no—
125C4#1909779412yes—
126C4#1909779812yes—
127C2—25121no—
128C2—12562no—
129C2#2375742112yes—
130C2#2375744311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#333×C157C2
socleC22×C157#120
hypercenter#333D157
hypocenterC157#1624
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #23757989C2semidirect
2. #23757647C2semidirect
3. #23757469C2semidirect
4. #23757000C2semidirect
5. #333×C157C2semidirect
65. C157#1624semidirect
70. #333D157semidirect

Maximal subgroups

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

subgroupindex
#237579892
#237576472
#237574692
#237570002
#333×C1572
#1624157