Finite Group Databasebeta

← #34099191

Subgroups — #34099191

Order 20096.

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

normal subgroups only

This group has 336 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, complemented
2#23757649C211yesmaximal, characteristic, complemented
3#23757469C212yesmaximal, complemented
4#23757000C212yesmaximal, complemented
5#331×C157C211yesmaximal, characteristic, complemented
6C2×D4×D157—22no—
7C2×D1256—22no—
8C2×#14578939—22no—
9#19097768C2211yescharacteristic, complemented
10#19097742—22no—
11#19097739—22no—
12C4×D628C2212yes—
13#19097693—22no—
14C4×#11929434C2211yescharacteristic, complemented
15#127×C157C2211yescharacteristic
16C4⋊C8×C157C2212yescomplemented
17C23×D157—22no—
18C2×D628—22no—
19C2×D628—21no—
20C2×D628D412yes—
21C2×C22⋊D157—24no—
22D4×D157—44no—
23D4×D157—44no—
24C2×C4×D157—22no—
25D1256—44no—
26#14578939—44no—
27C2×#11929434—21no—
28C4⋊Dic157D412yes—
29C22⋊Dic157—22no—
30#14578931—22no—
31C2×D4×C157—22no—
32C2×D4×C157D412yes—
33C2×C8×C157—22no—
34C42×C157C2311yescharacteristic
35C22×D157—22no—
36C22×D157—48no—
37C22×D157—41no—
38C22×D157—82no—
39D628—22no—
40D628—42no—
41D628—42no—
42D628—42no—
43D628D814yescomplemented
44C22⋊D157—48no—
45C2×Dic157—22no—
46C4×D157—42no—
47C4×D157—42no—
48#11929434—22no—
49C23×C157—22no—
50D4×C157—42no—
51D4×C157—42no—
52D4×C157—42no—
53C2×C4×C157C2×D412yes—
54C2×C4×C157C2×D411yescharacteristic
55C8×C157—42no—
56C22×C157—44no—
57C22×C157—42no—
58C22×C157#12011yescharacteristic
59C2×D157—24no—
60C2×D157—42no—
61C2×D157—42no—
62C2×D157—88no—
63C2×D157—82no—
64C2×D157—81no—
65C4×C157—21no—
66C4×C157C2×D812yes—
67C4×C157AGL(1,ℤ/8)12yes—
68Dic157—42no—
69C2×C157—82no—
70C2×C157#44712yes—
71C2×C157#45311yescharacteristic
72D157—44no—
73D157—161no—
74C157#161911yescharacteristic, complemented, Sylow
75#1619—1571nomaximal, Sylow
76D42—1571no—
77#493—1571no—
78#459—1572no—
79#331D15711yescharacteristic, complemented
80D4⋊C8—1572no—
81C22×D4—3142no—
82C2×D8—3142no—
83C2×D8—3142no—
84#127—1571no—
85#127C2×D15711yescharacteristic
86#121—3142no—
87#120—3142no—
88C4×D4—1572no—
89C4⋊C8C2×D15712yescomplemented
90#102—3142no—
91C4×C8—1571no—
92C24—3142no—
93C2×D4—22no—
94C2×D4—3144no—
95C2×D4—3142no—
96C2×D4—3141no—
97C2×D4—6284no—
98C2×D4—6284no—
99C2×D4—1572no—
100C2×D4D62812yes—
101C22×C4—3142no—
102D8—6284no—
103D8—6284no—
104C2×C8—22no—
105C2×C8—3141no—
106C4⋊C4—1572no—
107C22⋊C4—3142no—
108C22⋊C4—3142no—
109C42C22×D15711yescharacteristic
110C23—22no—
111C23—3142no—
112C23—6288no—
113C23—6281no—
114C23—12562no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—3142no—
119D4—6282no—
120D4—6288no—
121D4—6282no—
122D4—6282no—
123D4—1574no—
124C2×C4—3142no—
125C2×C4—6282no—
126C2×C4—6282no—
127C2×C4D4×D15712yes—
128C2×C4C2×D62811yescharacteristic
129C8—42no—
130C8—3142no—
131C22—44no—
132C22—42no—
133C22—3144no—
134C22—6282no—
135C22—6282no—
136C22—12568no—
137C22—12562no—
138C22—12561no—
139C22#1909774111yescharacteristic
140C4—21no—
141C4—6282no—
142C4D8×D15712yes—
143C4#1909779812yes—
144C2—82no—
145C2—6284no—
146C2—25121no—
147C2#2375740812yes—
148C2#2375744311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157#120
hypercenter#331D157
hypocenterC157#1619
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #23757908C2semidirect
2. #23757649C2semidirect
3. #23757469C2semidirect
4. #23757000C2semidirect
5. #331×C157C2semidirect
9. #19097768C22semidirect
14. C4×#11929434C22semidirect
16. C4⋊C8×C157C22semidirect
43. D628D8semidirect
74. C157#1619semidirect
79. #331D157semidirect
89. C4⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579082
#237576492
#237574692
#237570002
#331×C1572
#1619157