Finite Group Databasebeta

← #23757459

Subgroups — #23757459

Order 10048.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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#19097816C211yesmaximal, characteristic, complemented
2#19097740C212yesmaximal, complemented
3#19097686C211yesmaximal, characteristic, complemented
4#19097685C212yesmaximal, complemented
5#120×C157C211yesmaximal, characteristic, complemented
6#14578954—21no—
7#14578953—22no—
8C2×C22⋊D157—22no—
9C2×C22⋊D157—21no—
10C2×C22⋊D157C2212yes—
11D4×D157—22no—
12C22⋊Dic157—22no—
13C22⋊Dic157C2212yescomplemented
14C24×C157C2211yescharacteristic, complemented
15C2×D4×C157—21no—
16C2×D4×C157C2211yescharacteristic
17C22⋊C4×C157—21no—
18C22⋊C4×C157C2211yescharacteristic, complemented
19C22×D157—21no—
20C22×D157D412yescomplemented
21Dic314—21no—
22D628—21no—
23C22⋊D157—22no—
24C22⋊D157—22no—
25C22⋊D157—22no—
26C22⋊D157—44no—
27C2×Dic157—21no—
28C2×Dic157—22no—
29C2×Dic157D412yescomplemented
30C4×D157—22no—
31C23×C157—22no—
32C23×C157—21no—
33C23×C157—42no—
34C23×C157D411yescharacteristic, complemented
35C23×C157C2311yescharacteristic
36D4×C157—22no—
37D4×C157—42no—
38C2×C4×C157—21no—
39C2×C4×C157D411yescharacteristic, complemented
40C22×C157—22no—
41C22×C157—24no—
42C22×C157—21no—
43C22×C157—41no—
44C22×C157—41no—
45C22×C157—42no—
46C22×C157—41no—
47C22×C157—41no—
48C22×C157C2×D412yes—
49C22×C157C2×D411yescharacteristic
50C2×D157—22no—
51C2×D157—42no—
52C2×D157—81no—
53C4×C157—21no—
54C4×C157—41no—
55Dic157—22no—
56Dic157—42no—
57C2×C157—22no—
58C2×C157—21no—
59C2×C157—41no—
60C2×C157—42no—
61C2×C157#12011yescharacteristic
62D157—42no—
63C157B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1571nomaximal, Sylow
65H(5,𝔽2)—1571no—
66#120—1572no—
67#120D15711yescharacteristic, complemented
68#99—1571no—
69#99—1572no—
70C24C2×D15711yescharacteristic, complemented
71#41—3142no—
72#41—3141no—
73C2×D4—21no—
74C2×D4—3142no—
75C2×D4—3141no—
76C2×D4—3142no—
77C2×D4—1572no—
78C2×D4C2×D15711yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—3142no—
81C22⋊C4—1572no—
82C22⋊C4C2×D15711yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—3141no—
87C23—1572no—
88C23C22⋊D15711yescharacteristic, complemented
89C23C22×D15711yescharacteristic
90Q8—3141no—
91D4—22no—
92D4—42no—
93D4—3141no—
94D4—3142no—
95D4—3142no—
96D4—3142no—
97D4—6284no—
98C2×C4—21no—
99C2×C4—3142no—
100C2×C4—3141no—
101C2×C4—3142no—
102C2×C4—1572no—
103C2×C4C22⋊D15711yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—3142no—
113C22—6282no—
114C22—12561no—
115C22D4×D15712yes—
116C22C2×C22⋊D15711yescharacteristic
117C4—21no—
118C4—41no—
119C4—3142no—
120C4—6282no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—6282no—
126C2#1909773911yescharacteristic
subgroupG/H
centerC2#19097739
commutator, cosocleC23×C157C23
FrattiniC23C22×D157
Fitting#120×C157C2
socleC2×C157#120
hypercenter#120D157
hypocenterC157B(4,𝔽2)
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #19097816C2semidirect
2. #19097740C2semidirect
3. #19097686C2semidirect
4. #19097685C2semidirect
5. #120×C157C2semidirect
13. C22⋊Dic157C22semidirect
14. C24×C157C22semidirect
18. C22⋊C4×C157C22semidirect
20. C22×D157D4semidirect
29. C2×Dic157D4semidirect
34. C23×C157D4semidirect
39. C2×C4×C157D4semidirect
63. C157B(4,𝔽2)semidirect
67. #120D157semidirect
70. C24C2×D157semidirect
82. C22⋊C4C2×D157semidirect
88. C23C22⋊D157semidirect
103. C2×C4C22⋊D157semidirect