Finite Group Databasebeta

← #34102024

Subgroups — #34102024

Order 20096.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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#23757713C211yesmaximal, characteristic
2#23757395C211yesmaximal, characteristic
3#23757379C211yesmaximal, characteristic, complemented
4#23757095C211yesmaximal, characteristic, complemented
5#23757090C211yesmaximal, characteristic, complemented
6#23757050C211yesmaximal, characteristic, complemented
7#360×C157C211yesmaximal, characteristic
8C2×Dic628C2211yescharacteristic
9#19097778—21no—
10#19097736—21no—
11#19097717C412yescomplemented
12#19097715C412yescomplemented
13#19097711C2211yescharacteristic
14#19097707C2211yescharacteristic
15#19097698—21no—
16#19097694—21no—
17#19097681C2211yescharacteristic
18#135×C157C2211yescharacteristic
19M32×C157C2211yescharacteristic
20#106×C157C2211yescharacteristic
21C2×Dic314—21no—
22Dic628—21no—
23Dic628C2×C412yes—
24#14578938—21no—
25#14578938—41no—
26C2×#11929434—41no—
27C2×#11929434D411yescharacteristic
28#14578933—41no—
29C4×Dic157—41no—
30C4×Dic157D411yescharacteristic
31#14578929C2×C412yes—
32#41×C157—21no—
33Q16×C157C2×C411yescharacteristic
34SD16×C157—21no—
35D8×C157C2×C411yescharacteristic
36C2×C8×C157C2311yescharacteristic
37C4⋊C4×C157—21no—
38C16×C157—21no—
39Dic314—22no—
40Dic314—41no—
41C2×Dic157—21no—
42C2×Dic157—41no—
43#11929434—21no—
44#11929434—41no—
45Q8×C157—21no—
46D4×C157—21no—
47D4×C157—41no—
48C2×C4×C157—41no—
49C2×C4×C157—41no—
50C2×C4×C157C2×D411yescharacteristic
51C8×C157C22×C411yescharacteristic
52C8×C157#4111yescharacteristic
53C22×C157—41no—
54C22×C157AGL(1,ℤ/8)11yescharacteristic
55C4×C157—41no—
56C4×C157—81no—
57C4×C157C4×D411yescharacteristic
58C4×C157AGL(1,ℤ/8)11yescharacteristic
59Dic157—41no—
60Dic157—42no—
61C2×C157—21no—
62C2×C157—81no—
63C2×C157#44211yescharacteristic
64C157#218011yescharacteristic, complemented, Sylow
65#2180—1571nomaximal, Sylow
66#510—1571no—
67#444—1571no—
68#441—1571no—
69#365—1571no—
70#362—1571no—
71#360D15711yescharacteristic
72#347—1571no—
73#135C2×D15711yescharacteristic
74C2×Q16—1571no—
75#131—3141no—
76C4×Q8—3141no—
77Q32—1572no—
78SD32—1572no—
79M32—1571no—
80M32C2×D15711yescharacteristic
81#108—1571no—
82#106C2×D15711yescharacteristic
83#104—3141no—
84#103—3141no—
85#97—1571no—
86#41—21no—
87C2×Q8—3141no—
88Q16—3141no—
89Q16—1572no—
90Q16C4×D15711yescharacteristic
91SD16—21no—
92D8C4×D15711yescharacteristic
93M16—3141no—
94M16—6281no—
95C2×C8—6281no—
96C2×C8—1571no—
97C2×C8C22×D15711yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—6281no—
100C42—6281no—
101C42—1571no—
102C16—21no—
103C16—1572no—
104Q8—21no—
105Q8—3142no—
106Q8—6281no—
107D4—21no—
108D4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—3141no—
112C2×C4—6281no—
113C2×C4D4×D15711yescharacteristic
114C8—3141no—
115C8—6281no—
116C8C2×C4×D15711yescharacteristic
117C8#1457895311yescharacteristic
118C22—41no—
119C22#1909779511yescharacteristic
120C4—41no—
121C4—81no—
122C4—6281no—
123C4—6282no—
124C4#1909773111yescharacteristic
125C4#1909779811yescharacteristic
126C2—21no—
127C2—81no—
128C2#2375738211yescharacteristic
subgroupG/H
centerC2#23757382
commutatorC8×C157C22×C4
FrattiniC2×C8C22×D157
Fitting#360×C157C2
socleC2×C157#442
prime radical#135×C157C22
cosocleC2×C8×C157C23
hypercenter#360D157
hypocenterC157#2180
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
3. #23757379C2semidirect
4. #23757095C2semidirect
5. #23757090C2semidirect
6. #23757050C2semidirect
11. #19097717C4semidirect
12. #19097715C4semidirect
64. C157#2180semidirect

Maximal subgroups

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

subgroupindex
#237577132
#237573952
#237573792
#237570952
#237570902
#237570502
#360×C1572
#2180157