Finite Group Databasebeta

← #37550024

Subgroups — #37550024

Order 23168.

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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#26042266C212yesmaximal
2#26042190C212yesmaximal, complemented
3#26042189C212yesmaximal, complemented
4#26041849C211yesmaximal, characteristic
5C2×#15668458—22no—
6#19153877C2214yes—
7#19153876—22no—
8C2×#15668448C2211yescharacteristic
9C2×Q8×D181C411yescharacteristic
10#19153816C2211yescharacteristic
11#19153814C2211yescharacteristic
12#19153803C411yescharacteristic
13#19153798—22no—
14#19153793—22no—
15#19153773C411yescharacteristic
16#124×C181C411yescharacteristic
17#15668458—24no—
18#15668458—42no—
19C2×#13089685—42no—
20#15668448—42no—
21C2×Dic362—21no—
22C2×Dic362C2×C411yescharacteristic
23C2×Dic362C2311yescharacteristic
24Q8×D181—42no—
25C2×C22⋊D181D412yes—
26C22×Dic181D412yes—
27C2×C4×D181—21no—
28C2×C4×D181C2×C411yescharacteristic
29C4⋊Dic181—22no—
30#15668428—21no—
31#15668428—22no—
32C22⋊Dic181—22no—
33#15668426—21no—
34C4×Dic181C2×C411yescharacteristic
35C2×Q8×C181C2×C411yescharacteristic
36C2×D4×C181C2×C411yescharacteristic
37C22⋊C4×C181—22no—
38C42×C181C2×C411yescharacteristic
39#13089685—44no—
40C22×D181C22⋊C411yescharacteristic
41Dic362—41no—
42Dic362—42no—
43Dic362—41no—
44C22⋊D181—42no—
45C2×Dic181—22no—
46C2×Dic181—22no—
47C2×Dic181—21no—
48C2×Dic181—21no—
49C2×Dic181C22⋊C411yescharacteristic
50C2×Dic181C2×D412yes—
51C4×D181—22no—
52C4×D181—42no—
53C23×C181C22⋊C412yes—
54Q8×C181—41no—
55D4×C181—41no—
56C2×C4×C181—21no—
57C2×C4×C181—21no—
58C2×C4×C181C22×C411yescharacteristic
59C22×C181—22no—
60C22×C181—42no—
61C22×C181C2×C22⋊C411yescharacteristic
62C2×D181—21no—
63C2×D181—41no—
64C4×C181—21no—
65C4×C181—41no—
66C4×C181—41no—
67Dic181—22no—
68Dic181—41no—
69Dic181—41no—
70Dic181#9912yes—
71C2×C181—21no—
72C2×C181—42no—
73C2×C181C2×#9911yescharacteristic
74D181—41no—
75C181#213511yescharacteristic, complemented, Sylow
76#2135—1811nomaximal, Sylow
77#536—1811no—
78#413—1812no—
79#356—1812no—
80#356—1812no—
81C2×#41—1811no—
82C22×Q8—1811no—
83C2×M16—3622no—
84C4⋊Q8—1811no—
85C4⋊Q8—1811no—
86#124—1811no—
87#124#1062438211yescharacteristic
88#123—3622no—
89C22⋊Q8—3622no—
90#117—1811no—
91#101—1814no—
92#100—3622no—
93#41—7242no—
94C2×Q8—3621no—
95C2×Q8—1811no—
96C2×Q8—1811no—
97C2×Q8—7242no—
98C2×Q8C2×#1062438211yescharacteristic
99C2×D4—1812no—
100C2×D4C2×#1062438211yescharacteristic
101C22×C4—3621no—
102C22×C4—1812no—
103C22×C4—1811no—
104M16—3624no—
105M16—7242no—
106C2×C8—7242no—
107C4⋊C4—3621no—
108C4⋊C4—3622no—
109C4⋊C4—3622no—
110C22⋊C4—22no—
111C22⋊C4—3621no—
112C22⋊C4—3622no—
113C42—1811no—
114C42C2×#1062438211yescharacteristic
115C23—1811no—
116C23C22⋊#1062438212yes—
117Q8—41no—
118Q8—7241no—
119Q8—7241no—
120Q8—7242no—
121D4—41no—
122D4—7242no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—3622no—
126C2×C4—3622no—
127C2×C4—3622no—
128C2×C4—3621no—
129C2×C4—3621no—
130C2×C4—1812no—
131C2×C4—1811no—
132C2×C4—7242no—
133C2×C4C22×#1062438211yescharacteristic
134C8—7244no—
135C22—22no—
136C22—42no—
137C22—3621no—
138C22—7241no—
139C22C2×C22⋊#1062438211yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—3622no—
144C4—1812no—
145C4—7241no—
146C4—7241no—
147C2—21no—
148C2—42no—
149C2—7241no—
150C2D181⋊#9911yescharacteristic
subgroupG/H
centerC2D181⋊#99
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#124×C181C4
socleC2×C181C2×#99
prime radicalC2×#15668448C22
cosocleC2×Dic362C23
hypercenter#124#10624382
hypocenterC181#2135
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #26042190C2semidirect
3. #26042189C2semidirect
75. C181#2135semidirect

Maximal subgroups

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

subgroupindex
#260422662
#260421902
#260421892
#260418492
#2135181