Finite Group Databasebeta

← #34098868

Subgroups — #34098868

Order 20096.

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

normal subgroups only

This group has 254 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#23757821C211yesmaximal, characteristic
2#23757295C212yesmaximal, complemented
3#23757021C212yesmaximal, complemented
4#23757017C211yesmaximal, characteristic, complemented
5#331×C157C211yesmaximal, characteristic, complemented
6C2×#14578953C412yes—
7C2×#14578937—22no—
8#19097771C2211yescharacteristic
9#19097758C412yes—
10#19097742—22no—
11D4×Dic157—22no—
12C42×D157C2211yescharacteristic
13C4⋊#11929434C2212yes—
14#19097700—22no—
15#127×C157C2211yescharacteristic
16C4⋊C8×C157C2212yes—
17#14578953—44no—
18C2×Dic314C2×C412yes—
19C2×C22⋊D157—22no—
20C22×Dic157—22no—
21C2×C4×D157C2×C411yescharacteristic
22C2×C4×D157D412yes—
23#14578937—44no—
24C2×#11929434—22no—
25C4⋊Dic157—22no—
26C22⋊Dic157—24no—
27C4×Dic157C2×C411yescharacteristic
28C4×Dic157D412yes—
29C2×D4×C157—22no—
30C2×D4×C157C2×C412yes—
31C2×C8×C157—22no—
32C42×C157C2311yescharacteristic
33C22×D157C22⋊C411yescharacteristic
34Dic314—42no—
35C22⋊D157—44no—
36C2×Dic157—22no—
37C2×Dic157—44no—
38C2×Dic157C22⋊C412yes—
39C2×Dic157C22⋊C411yescharacteristic
40C4×D157—22no—
41C4×D157—22no—
42C4×D157—22no—
43#11929434—42no—
44C23×C157—22no—
45D4×C157—42no—
46D4×C157—42no—
47D4×C157—42no—
48C2×C4×C157C22×C411yescharacteristic
49C2×C4×C157C2×D412yes—
50C8×C157—42no—
51C22×C157—42no—
52C22×C157—44no—
53C22×C157C2×C22⋊C411yescharacteristic
54C2×D157—21no—
55C2×D157—22no—
56C4×C157—21no—
57C4×C157AGL(1,ℤ/8)12yes—
58C4×C157#13712yes—
59Dic157—22no—
60Dic157—42no—
61Dic157#10012yes—
62C2×C157—82no—
63C2×C157C2×#10011yescharacteristic
64C2×C157#41712yes—
65D157—41no—
66C157#156011yescharacteristic, complemented, Sylow
67#1560—1571nomaximal, Sylow
68#532—1571no—
69#423—1572no—
70#333—1572no—
71#331—1571no—
72#331D15711yescharacteristic, complemented
73C2×#41—1572no—
74C2×M16—3142no—
75C4⋊Q8—1571no—
76#127C2×D15711yescharacteristic
77#124—1572no—
78#121—3142no—
79C4×D4—3142no—
80C2×C42—1571no—
81C4⋊C8—3142no—
82C4⋊C8—1572no—
83C4⋊C8C2×D15712yes—
84#41—6284no—
85C2×Q8—1572no—
86C2×D4—22no—
87C2×D4—3142no—
88C2×D4C4×D15712yes—
89C22×C4—3142no—
90C22×C4—1571no—
91C22×C4—1572no—
92M16—6284no—
93C2×C8—22no—
94C2×C8—3142no—
95C4⋊C4—3142no—
96C22⋊C4—3144no—
97C42—1571no—
98C42—1572no—
99C42C22×D15711yescharacteristic
100C23—22no—
101C23—1571no—
102Q8—6282no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—6284no—
107C2×C4—3142no—
108C2×C4—3142no—
109C2×C4—3142no—
110C2×C4—3142no—
111C2×C4—6284no—
112C2×C4—1572no—
113C2×C4—1571no—
114C2×C4C2×C4×D15711yescharacteristic
115C2×C4D4×D15712yes—
116C8—42no—
117C8—6282no—
118C22—42no—
119C22—44no—
120C22—3141no—
121C22—3142no—
122C22C22⋊C4×D15711yescharacteristic
123C4—21no—
124C4—3142no—
125C4—6282no—
126C4—1572no—
127C4#1909779512yes—
128C4#1909780012yes—
129C2—82no—
130C2—6281no—
131C2#100×D15711yescharacteristic
132C2#2375727112yes—
subgroupG/H
centerC22C22⋊C4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757821C2
cosocleC42×C157C23
hypercenter#331D157
hypocenterC157#1560
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. #23757295C2semidirect
3. #23757021C2semidirect
4. #23757017C2semidirect
5. #331×C157C2semidirect
66. C157#1560semidirect
72. #331D157semidirect

Maximal subgroups

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

subgroupindex
#237578212
#237572952
#237570212
#237570172
#331×C1572
#1560157