Finite Group Databasebeta

← #34098848

Subgroups — #34098848

Order 20096.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#127×D157C211yesmaximal, characteristic
2#23757329C212yesmaximal, complemented
3#23757017C211yesmaximal, characteristic, complemented
4#23757016C212yesmaximal, complemented
5#331×C157C211yesmaximal, characteristic, complemented
6C2×D4×D157—22no—
7C2×D4×D157C412yes—
8#19097768C2211yescharacteristic
9#19097767—22no—
10#19097767C412yes—
11C42×D157C2211yescharacteristic
12C4⋊#11929434C2212yescomplemented
13#19097683—22no—
14#19097681—22no—
15#127×C157C2211yescharacteristic
16C4⋊C8×C157C2212yescomplemented
17C23×D157—22no—
18C2×D628—22no—
19C2×D628C2×C412yes—
20C2×C22⋊D157—22no—
21C2×C22⋊D157—24no—
22D4×D157—44no—
23D4×D157—44no—
24D4×D157—44no—
25C2×C4×D157C2×C411yescharacteristic
26C2×C4×D157D412yes—
27C2×#11929434—22no—
28C4×Dic157C2×C411yescharacteristic
29C4×Dic157D412yes—
30C2×D4×C157—22no—
31C2×D4×C157C2×C412yes—
32C2×C8×C157—22no—
33C42×C157C2311yescharacteristic
34C22×D157—22no—
35C22×D157—48no—
36C22×D157—44no—
37C22×D157C22⋊C411yescharacteristic
38D628—42no—
39D628—42no—
40D628—42no—
41C22⋊D157—44no—
42C22⋊D157—48no—
43C2×Dic157C22⋊C411yescharacteristic
44C2×Dic157C22⋊C412yes—
45C4×D157—22no—
46C4×D157—22no—
47C4×D157—22no—
48#11929434—42no—
49C23×C157—22no—
50D4×C157—42no—
51D4×C157—42no—
52D4×C157—42no—
53C2×C4×C157C22×C411yescharacteristic
54C2×C4×C157C2×D412yes—
55C8×C157—42no—
56C22×C157—42no—
57C22×C157—44no—
58C22×C157C2×C22⋊C411yescharacteristic
59C2×D157—22no—
60C2×D157—42no—
61C2×D157—44no—
62C2×D157—88no—
63C2×D157#10012yes—
64C4×C157—21no—
65C4×C157AGL(1,ℤ/8)12yes—
66C4×C157AGL(1,ℤ/8)12yes—
67Dic157—21no—
68Dic157—22no—
69C2×C157—82no—
70C2×C157C2×#10011yescharacteristic
71C2×C157#41812yes—
72D157—22no—
73D157—82no—
74C157#155511yescharacteristic, complemented, Sylow
75#1555—1571nomaximal, Sylow
76C2×#127—1571no—
77#432—1572no—
78#331—1571no—
79#331—1572no—
80#331D15711yescharacteristic, complemented
81C22×D4—3142no—
82C22×D4—1572no—
83#127—3142no—
84#127—1572no—
85#127—1571no—
86#127C2×D15711yescharacteristic
87C2×C42—1571no—
88C4⋊C8—1572no—
89C4⋊C8C2×D15712yescomplemented
90C22⋊C8—3142no—
91#97—3142no—
92C24—3142no—
93C2×D4—22no—
94C2×D4—3142no—
95C2×D4—3144no—
96C2×D4—3142no—
97C2×D4—6284no—
98C2×D4—6284no—
99C2×D4—6284no—
100C2×D4—1572no—
101C2×D4C4×D15712yes—
102C22×C4—1571no—
103C22×C4—1572no—
104C2×C8—22no—
105C2×C8—3142no—
106C42—1571no—
107C42—1572no—
108C42C22×D15711yescharacteristic
109C23—22no—
110C23—3142no—
111C23—6288no—
112C23—6284no—
113C23—1571no—
114D4—42no—
115D4—42no—
116D4—42no—
117D4—6282no—
118D4—6284no—
119D4—6288no—
120D4—6282no—
121D4—6282no—
122C2×C4—3142no—
123C2×C4—3142no—
124C2×C4—3142no—
125C2×C4—1571no—
126C2×C4—1572no—
127C2×C4C2×C4×D15711yescharacteristic
128C2×C4D4×D15712yes—
129C8—42no—
130C8—6282no—
131C22—42no—
132C22—44no—
133C22—3142no—
134C22—6282no—
135C22—6284no—
136C22—1572no—
137C22—12568no—
138C22C22⋊C4×D15711yescharacteristic
139C4—21no—
140C4—3141no—
141C4—3142no—
142C4#1909779512yes—
143C4#1909779812yes—
144C2—82no—
145C2—3142no—
146C2—12562no—
147C2#100×D15711yescharacteristic
148C2#2375727612yes—
subgroupG/H
centerC22C22⋊C4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157C2×C22⋊C4
prime radical#127×D157C2
cosocleC42×C157C23
hypercenter#331D157
hypocenterC157#1555
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #23757329C2semidirect
3. #23757017C2semidirect
4. #23757016C2semidirect
5. #331×C157C2semidirect
12. C4⋊#11929434C22semidirect
16. C4⋊C8×C157C22semidirect
74. C157#1555semidirect
80. #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
#127×D1572
#237573292
#237570172
#237570162
#331×C1572
#1555157