Finite Group Databasebeta

← #37550005

Subgroups — #37550005

Order 23168.

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

normal subgroups only

This group has 238 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
2#26042186C214yesmaximal, complemented
3#26041855C211yesmaximal, characteristic
4#19153895—22no—
5#19153875C2214yes—
6#19153874—22no—
7C2×#15668449C2211yescharacteristic
8C22×Dic362C411yescharacteristic, complemented
9#19153816C412yescomplemented
10#19153804C2212yes—
11#19153800—22no—
12#19153796—22no—
13#117×C181C411yescharacteristic, complemented
14C4⋊#10624382—42no—
15C22⋊#10624382—24no—
16C4×#10624382—42no—
17#15668449—42no—
18C2×Dic362—21no—
19C2×Dic362—42no—
20C2×Dic362C2×C412yes—
21C2×Dic362C2×C411yescharacteristic
22C2×D724C2311yescharacteristic
23C2×C22⋊D181D412yes—
24C22×Dic181—21no—
25C2×C4×D181D412yes—
26C4⋊Dic181—24no—
27#15668426—24no—
28C22×C4×C181C2×C411yescharacteristic
29C4⋊C4×C181—21no—
30C22⋊C4×C181—21no—
31C42×C181C2×C412yes—
32C2×#10624382—44no—
33C22×D181C2×D412yes—
34Dic362—41no—
35Dic362—42no—
36Dic362—42no—
37D724—41no—
38C22⋊D181—42no—
39C2×Dic181—22no—
40C2×Dic181—42no—
41C2×Dic181C22⋊C412yes—
42C4×D181—22no—
43C4×D181—22no—
44C23×C181C22⋊C411yescharacteristic
45C2×C4×C181—22no—
46C2×C4×C181—22no—
47C2×C4×C181C22⋊C411yescharacteristic
48C2×C4×C181C22×C411yescharacteristic
49C22×C181—21no—
50C22×C181—41no—
51C22×C181C2×C22⋊C411yescharacteristic
52C2×D181—22no—
53C2×D181—42no—
54#10624382—84no—
55C4×C181—21no—
56C4×C181—42no—
57C4×C181#9912yes—
58Dic181—22no—
59Dic181—42no—
60C2×C181—21no—
61C2×C181—41no—
62C2×C181C2×#9911yescharacteristic
63D181—42no—
64C181#212911yescharacteristic, complemented, Sylow
65#2129—1811nomaximal, Sylow
66#536—1811no—
67#410—1812no—
68#354—1814no—
69C2×#41—1811no—
70C22×Q8—1811no—
71C4⋊Q8—1812no—
72#124—1812no—
73#123—3622no—
74C22⋊Q8—3622no—
75#117—3622no—
76#117#1062438211yescharacteristic, complemented
77#99—3622no—
78#99—1814no—
79#41—7242no—
80C2×Q8—3621no—
81C2×Q8—1812no—
82C2×Q8—1811no—
83C2×Q8—7242no—
84C2×D4—1812no—
85C2×D4—1811no—
86C22×C4—3621no—
87C22×C4—1812no—
88C22×C4C2×#1062438211yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—3624no—
91C4⋊C4—7242no—
92C22⋊C4—21no—
93C22⋊C4—3624no—
94C22⋊C4—3624no—
95C42—7242no—
96C42C2×#1062438212yes—
97C23—1812no—
98C23C22⋊#1062438211yescharacteristic
99Q8—7241no—
100Q8—7242no—
101Q8—7242no—
102D4—7241no—
103D4—7242no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—3622no—
107C2×C4—3622no—
108C2×C4—3622no—
109C2×C4—1812no—
110C2×C4—7244no—
111C2×C4—7242no—
112C2×C4C22⋊#1062438211yescharacteristic
113C2×C4C22×#1062438211yescharacteristic
114C22—21no—
115C22—41no—
116C22—3622no—
117C22—7242no—
118C22C2×C22⋊#1062438211yescharacteristic
119C4—21no—
120C4—42no—
121C4—3622no—
122C4—14484no—
123C4—7242no—
124C4#1915387512yes—
125C2—21no—
126C2—41no—
127C2—7242no—
128C2C2×#1915387511yescharacteristic
subgroupG/H
centerC2C2×#19153875
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#117×C181C4
socleC2×C181C2×#99
prime radicalC2×#15668449C22
cosocleC2×D724C23
hypercenter#117#10624382
hypocenterC181#2129
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #26042186C2semidirect
8. C22×Dic362C4semidirect
9. #19153816C4semidirect
13. #117×C181C4semidirect
64. C181#2129semidirect
76. #117#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422582
#260421862
#260418552
#2129181