Finite Group Databasebeta

← #34102034

Subgroups — #34102034

Order 20096.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757708C211yesmaximal, characteristic
2#23757386C211yesmaximal, characteristic, complemented
3#23757097C211yesmaximal, characteristic, complemented
4#23757085C211yesmaximal, characteristic, complemented
5#23757083C211yesmaximal, characteristic, complemented
6#23757051C211yesmaximal, characteristic, complemented
7#444×C157C211yesmaximal, characteristic, complemented
8C2×D1256C2211yescharacteristic
9C4×D628—21no—
10#19097714C412yescomplemented
11#19097712C412yescomplemented
12#19097711C2211yescharacteristic
13#19097711C2211yescharacteristic
14#19097703C2211yescharacteristic
15#19097693—21no—
16#135×C157C2211yescharacteristic
17#131×C157—21no—
18#108×C157C2211yescharacteristic
19#104×C157—21no—
20#97×C157C2211yescharacteristic
21C2×D628—21no—
22C2×C4×D157—41no—
23D1256—21no—
24D1256C2×C412yes—
25C4⋊Dic157—21no—
26#14578931—41no—
27#14578929—21no—
28#14578929C2×C412yescomplemented
29#41×C157—21no—
30Q16×C157C2×C411yescharacteristic, complemented
31SD16×C157—21no—
32D8×C157C2×C411yescharacteristic, complemented
33M16×C157—21no—
34M16×C157—41no—
35C2×C8×C157—41no—
36C2×C8×C157D411yescharacteristic
37C2×C8×C157C2311yescharacteristic
38C42×C157D411yescharacteristic
39C22×D157—41no—
40D628—22no—
41D628—41no—
42C2×Dic157—41no—
43C4×D157—82no—
44Q8×C157—21no—
45D4×C157—21no—
46D4×C157—41no—
47C2×C4×C157—21no—
48C2×C4×C157—41no—
49C2×C4×C157C2×D411yescharacteristic
50C8×C157—21no—
51C8×C157—41no—
52C8×C157C22×C411yescharacteristic
53C8×C157#4111yescharacteristic
54C22×C157—41no—
55C22×C157AGL(1,ℤ/8)11yescharacteristic
56C2×D157—42no—
57C2×D157—81no—
58C2×D157—81no—
59C4×C157—41no—
60C4×C157—41no—
61C4×C157C4×D411yescharacteristic
62C4×C157AGL(1,ℤ/8)11yescharacteristic
63Dic157—81no—
64C2×C157—21no—
65C2×C157—81no—
66C2×C157#44211yescharacteristic
67D157—82no—
68C157AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
69AGL(1,ℤ/16)—1571nomaximal, Sylow
70#509—1571no—
71#444D15711yescharacteristic, complemented
72#442—1571no—
73#365—1571no—
74#361—1571no—
75#360—1571no—
76#347—1571no—
77#135C2×D15711yescharacteristic
78C2×D8—1571no—
79#131—21no—
80C4×D4—3141no—
81SD32—1572no—
82D16—1572no—
83M32—1571no—
84M32—1571no—
85#108C2×D15711yescharacteristic
86#106—1571no—
87#104—21no—
88#102—3141no—
89#97C2×D15711yescharacteristic
90#41—21no—
91C2×D4—3141no—
92C22×C4—6281no—
93Q16C4×D15711yescharacteristic, complemented
94SD16—21no—
95D8—3141no—
96D8—1572no—
97D8C4×D15711yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8C22⋊D15711yescharacteristic
102C2×C8C22×D15711yescharacteristic
103C4⋊C4—3141no—
104C22⋊C4—6281no—
105C42C22⋊D15711yescharacteristic
106C16—3141no—
107C16—1572no—
108C23—6281no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—3142no—
113D4—6281no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—6281no—
117C2×C4—12562no—
118C2×C4C2×C22⋊D15711yescharacteristic
119C8—21no—
120C8—41no—
121C8C2×C4×D15711yescharacteristic
122C8#1457895411yescharacteristic
123C22—41no—
124C22—6282no—
125C22—12561no—
126C22—12561no—
127C22#1909779611yescharacteristic
128C4—41no—
129C4—41no—
130C4—12561no—
131C4C4×C22⋊D15711yescharacteristic
132C4#1909779711yescharacteristic
133C2—21no—
134C2—81no—
135C2—12562no—
136C2#2375738311yescharacteristic
subgroupG/H
centerC2#23757383
commutatorC8×C157C22×C4
FrattiniC2×C8C22×D157
Fitting#444×C157C2
socleC2×C157#442
prime radical#23757708C2
cosocleC2×C8×C157C23
hypercenter#444D157
hypocenterC157AGL(1,ℤ/16)
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #23757386C2semidirect
3. #23757097C2semidirect
4. #23757085C2semidirect
5. #23757083C2semidirect
6. #23757051C2semidirect
7. #444×C157C2semidirect
10. #19097714C4semidirect
11. #19097712C4semidirect
28. #14578929C2×C4semidirect
30. Q16×C157C2×C4semidirect
32. D8×C157C2×C4semidirect
68. C157AGL(1,ℤ/16)semidirect
71. #444D157semidirect
93. Q16C4×D157semidirect
97. D8C4×D157semidirect

Maximal subgroups

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

subgroupindex
#237577082
#237573862
#237570972
#237570852
#237570832
#237570512
#444×C1572
AGL(1,ℤ/16)157