Finite Group Databasebeta

← #37550025

Subgroups — #37550025

Order 23168.

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

normal subgroups only

This group has 230 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#26042265C212yesmaximal
2#26042190C214yesmaximal, complemented
3#26041855C211yesmaximal, characteristic
4D181⋊M16—22no—
5#19153877C2214yes—
6#19153876—22no—
7C2×#15668449C2211yescharacteristic
8C22×Dic362C411yescharacteristic
9#19153816C2212yes—
10#19153804C412yes—
11#19153800—22no—
12#19153796—22no—
13#117×C181C411yescharacteristic
14#15668459—42no—
15#15668458—24no—
16D181⋊C8—42no—
17#15668449—42no—
18C2×Dic362—21no—
19C2×Dic362—42no—
20C2×Dic362C2×C412yes—
21C2×Dic362C2311yescharacteristic
22C2×D724C2×C411yescharacteristic
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—
32#13089685—44no—
33C22×D181C22⋊C412yes—
34Dic362—41no—
35Dic362—42no—
36Dic362—42no—
37D724—41no—
38C22⋊D181—42no—
39C2×Dic181—22no—
40C2×Dic181—42no—
41C2×Dic181C2×D412yes—
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—
54C4×C181—21no—
55C4×C181—42no—
56C4×C181#9912yes—
57Dic181—22no—
58Dic181—42no—
59C2×C181—21no—
60C2×C181—41no—
61C2×C181C2×#9911yescharacteristic
62D181—42no—
63C181#213511yescharacteristic, complemented, Sylow
64#2135—1811nomaximal, Sylow
65#536—1811no—
66#413—1812no—
67#356—1814no—
68C2×#41—1811no—
69C22×Q8—1811no—
70C2×M16—3622no—
71C4⋊Q8—1812no—
72#124—1812no—
73#123—3622no—
74C22⋊Q8—3622no—
75#117#1062438211yescharacteristic
76#101—1814no—
77#100—3622no—
78#41—7242no—
79C2×Q8—3621no—
80C2×Q8—1812no—
81C2×Q8—1811no—
82C2×Q8—7242no—
83C2×D4—1812no—
84C2×D4—1811no—
85C22×C4—3621no—
86C22×C4—1812no—
87C22×C4C2×#1062438211yescharacteristic
88M16—3624no—
89M16—7242no—
90C2×C8—7242no—
91C4⋊C4—21no—
92C4⋊C4—3624no—
93C22⋊C4—21no—
94C22⋊C4—3624no—
95C42C2×#1062438212yes—
96C23—1812no—
97C23C22⋊#1062438211yescharacteristic
98Q8—7241no—
99Q8—7242no—
100Q8—7242no—
101D4—7241no—
102D4—7242no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—3622no—
106C2×C4—3622no—
107C2×C4—3622no—
108C2×C4—1812no—
109C2×C4—7242no—
110C2×C4C22⋊#1062438211yescharacteristic
111C2×C4C22×#1062438211yescharacteristic
112C8—7244no—
113C22—21no—
114C22—41no—
115C22—3622no—
116C22—7242no—
117C22C2×C22⋊#1062438211yescharacteristic
118C4—21no—
119C4—42no—
120C4—3622no—
121C4—7242no—
122C4#1915387512yes—
123C2—21no—
124C2—41no—
125C2—7242no—
126C2C2×#1915387511yescharacteristic
subgroupG/H
centerC2C2×#19153875
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#117×C181C4
socleC2×C181C2×#99
prime radicalC2×#15668449C22
cosocleC2×Dic362C23
hypercenter#117#10624382
hypocenterC181#2135
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. #26042190C2semidirect
63. C181#2135semidirect

Maximal subgroups

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

subgroupindex
#260422652
#260421902
#260418552
#2135181