Finite Group Databasebeta

← #34098290

Subgroups — #34098290

Order 20096.

106 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 25 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#23757072C211yesmaximal, characteristic, complemented
2#23757022C211yesmaximal, characteristic, complemented
3#456×C157C211yesmaximal, characteristic
4C4⋊#11929434—21no—
5#19097690—21no—
62−1+4×C157C411yescharacteristic
7#137×C157—21no—
8C4⋊Q8×C157C2211yescharacteristic
9#104×C157—21no—
10#101×C157C411yescharacteristic
11#14578938—41no—
12C2×#11929434—41no—
13#41×C157—21no—
14#41×C157—41no—
15#41×C157—41no—
16C2×Q8×C157—41no—
17C2×Q8×C157C2×C411yescharacteristic
18C2×Q8×C157D411yescharacteristic
19Q16×C157—41no—
20SD16×C157—41no—
21M16×C157—21no—
22C4⋊C4×C157—41no—
23C42×C157D411yescharacteristic
24#11929434—81no—
25#11929434—81no—
26Q8×C157—21no—
27Q8×C157—41no—
28Q8×C157—41no—
29Q8×C157—41no—
30D4×C157—21no—
31D4×C157—41no—
32D4×C157—41no—
33C2×C4×C157—21no—
34C2×C4×C157—21no—
35C2×C4×C157—21no—
36C2×C4×C157—41no—
37C2×C4×C157—42no—
38C2×C4×C157C22⋊C411yescharacteristic
39C8×C157—41no—
40C22×C157—41no—
41C22×C157#9911yescharacteristic
42C4×C157—21no—
43C4×C157—21no—
44C4×C157—21no—
45C4×C157—41no—
46C4×C157—41no—
47C4×C157—41no—
48C2×C157—21no—
49C2×C157—81no—
50C2×C157AΣL(1,𝔽16)11yescharacteristic
51C157#140511yescharacteristic, complemented, Sylow
52#1405—1571nomaximal, Sylow
53#456D15711yescharacteristic
54#356—1571no—
55#333—1571no—
562−1+4Dic15711yescharacteristic
57#137—21no—
58C4⋊Q8C2×D15711yescharacteristic
59C4⋊C8—3141no—
60#104—21no—
61#101—3141no—
62#101Dic15711yescharacteristic
63#41—21no—
64#41—41no—
65#41—41no—
66C2×Q8—41no—
67C2×Q8C2×Dic15711yescharacteristic
68C2×Q8C22⋊D15711yescharacteristic
69Q16—41no—
70SD16—41no—
71M16—21no—
72M16—6281no—
73C2×C8—6281no—
74C4⋊C4—41no—
75C42C22⋊D15711yescharacteristic
76Q8—21no—
77Q8—41no—
78Q8—41no—
79Q8—41no—
80D4—21no—
81D4—41no—
82D4—41no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—41no—
87C2×C4—42no—
88C2×C4C22⋊Dic15711yescharacteristic
89C8—41no—
90C8—12561no—
91C8—12561no—
92C22—41no—
93C22#1909768611yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—41no—
100C2—21no—
101C2—81no—
102C2#2375706011yescharacteristic
subgroupG/H
centerC2#23757060
commutatorC2×Q8×C157C2×C4
FrattiniC4⋊Q8C2×D157
Fitting#456×C157C2
socleC2×C157AΣL(1,𝔽16)
prime radical2−1+4×C157C4
cosocleC4⋊Q8×C157C22
hypercenter#456D157
hypocenterC157#1405
LCSG ▷ C2×Q8×C157 ▷ C2×C4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23757072C2semidirect
2. #23757022C2semidirect
51. C157#1405semidirect

Maximal subgroups

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

subgroupindex
#237570722
#237570222
#456×C1572
#1405157