Finite Group Databasebeta

← #37550000

Subgroups — #37550000

Order 23168.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#26042258C212yesmaximal, complemented
2#26042183C214yesmaximal, complemented
3#26041846C211yesmaximal, characteristic
4#19153895—22no—
5#19153875C2214yescomplemented
6#19153874—22no—
7C2×#15668449C2211yescharacteristic
8#19153812C2212yes—
9#19153803C412yescomplemented
10#19153787—22no—
11#19153785—22no—
12#19153775C411yescharacteristic, complemented
13C22×D4×C181C411yescharacteristic, complemented
14C4⋊#10624382—42no—
15C22⋊#10624382—24no—
16C4×#10624382—42no—
17#15668449—42no—
18C2×Dic362C2×C411yescharacteristic
19C2×D724C2311yescharacteristic
20C2×C22⋊D181—24no—
21C2×C22⋊D181D412yes—
22C2×C4×D181D412yes—
23C4⋊Dic181—21no—
24C22⋊Dic181—24no—
25C22⋊Dic181—21no—
26C4×Dic181C2×C412yes—
27C24×C181—21no—
28C2×D4×C181—21no—
29C2×D4×C181—42no—
30C2×D4×C181C2×C412yes—
31C22×C4×C181C2×C411yescharacteristic
32C2×#10624382—44no—
33C22×D181C2×D412yes—
34Dic362—41no—
35D724—41no—
36C22⋊D181—42no—
37C22⋊D181—84no—
38C2×Dic181—22no—
39C2×Dic181C22⋊C412yes—
40C4×D181—22no—
41C4×D181—22no—
42C23×C181—22no—
43C23×C181—42no—
44C23×C181—82no—
45C23×C181C22⋊C411yescharacteristic
46D4×C181—42no—
47D4×C181—42no—
48C2×C4×C181—22no—
49C2×C4×C181C22⋊C411yescharacteristic
50C2×C4×C181C22×C411yescharacteristic
51C22×C181—21no—
52C22×C181—41no—
53C22×C181—44no—
54C22×C181—42no—
55C22×C181—84no—
56C22×C181C2×C22⋊C411yescharacteristic
57C2×D181—22no—
58C2×D181—42no—
59#10624382—84no—
60C4×C181—21no—
61C4×C181#9912yes—
62Dic181—22no—
63Dic181—42no—
64C2×C181—21no—
65C2×C181—41no—
66C2×C181—82no—
67C2×C181C2×#9911yescharacteristic
68D181—42no—
69C181#212811yescharacteristic, complemented, Sylow
70#2128—1811nomaximal, Sylow
71#535—1811no—
72#410—1812no—
73#353—1814no—
74C2×#41—1811no—
75C22×D4#1062438211yescharacteristic, complemented
76#127—1812no—
77#124—1812no—
78#121—3622no—
79#120—3622no—
80#117—3622no—
81#117—1811no—
82#99—3622no—
83#99—1814no—
84C24—21no—
85#41—7242no—
86C2×Q8—1811no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—3624no—
90C2×D4—1812no—
91C2×D4—1811no—
92C2×D4C2×#1062438212yes—
93C22×C4—1812no—
94C22×C4C2×#1062438211yescharacteristic
95C4⋊C4—3621no—
96C4⋊C4—7242no—
97C22⋊C4—3621no—
98C22⋊C4—3624no—
99C22⋊C4—3624no—
100C42—1812no—
101C42—7242no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—1812no—
106C23C22⋊#1062438211yescharacteristic
107Q8—7241no—
108D4—42no—
109D4—42no—
110D4—14484no—
111D4—7241no—
112D4—7242no—
113C2×C4—22no—
114C2×C4—3622no—
115C2×C4—3622no—
116C2×C4—3622no—
117C2×C4—1812no—
118C2×C4—7244no—
119C2×C4C22⋊#1062438211yescharacteristic
120C2×C4C22×#1062438211yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—3622no—
127C22—7242no—
128C22C2×C22⋊#1062438211yescharacteristic
129C4—21no—
130C4—3622no—
131C4—14484no—
132C4—7242no—
133C4#1915387412yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—7242no—
138C2C2×#1915387411yescharacteristic
subgroupG/H
centerC2C2×#19153874
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC22×D4×C181C4
socleC2×C181C2×#99
prime radical#26041846C2
cosocleC2×D724C23
hypercenterC22×D4#10624382
hypocenterC181#2128
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26042258C2semidirect
2. #26042183C2semidirect
5. #19153875C22semidirect
9. #19153803C4semidirect
12. #19153775C4semidirect
13. C22×D4×C181C4semidirect
69. C181#2128semidirect
75. C22×D4#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422582
#260421832
#260418462
#2128181