Finite Group Databasebeta

← #37549060

Subgroups — #37549060

Order 23168.

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 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#26042300C211yesmaximal, characteristic, complemented
2C4×#15668458C211yesmaximal, characteristic, complemented
3D4⋊#13089685C214yesmaximal, complemented
4C2×D4×Dic181C211yesmaximal, characteristic
5C2×#15668458—21no—
6C4⋊#13089685C2212yescomplemented
7C22⋊#13089685—21no—
8#19153871—21no—
9#19153871—21no—
10C4×#13089685C2212yescomplemented
11C23×Dic181—21no—
12D4×Dic181—21no—
13D4×Dic181—21no—
14D4×Dic181C412yes—
15D4×Dic181C2212yes—
16C2×C4⋊Dic181C411yescharacteristic
17C2×C22⋊Dic181—21no—
18C2×C4×Dic181C2211yescharacteristic
19C22×D4×C181C411yescharacteristic
20#15668458—24no—
21C2×#13089685—22no—
22C2×#13089685—22no—
23C22×Dic181—22no—
24C22×Dic181—22no—
25C22×Dic181—21no—
26C22×Dic181—44no—
27C22×Dic181D411yescharacteristic
28C22×Dic181D411yescharacteristic
29C4⋊Dic181—21no—
30C4⋊Dic181C2×C412yes—
31C22⋊Dic181—22no—
32C22⋊Dic181—22no—
33C4×Dic181C2×C411yescharacteristic
34C4×Dic181D411yescharacteristic
35C4×Dic181D411yescharacteristic
36C4×Dic181C2311yescharacteristic
37C24×C181—21no—
38C2×D4×C181—24no—
39C2×D4×C181—21no—
40C2×D4×C181C2×C412yes—
41C22×C4×C181C2×C411yescharacteristic
42#13089685—24no—
43#13089685—42no—
44C2×Dic181—22no—
45C2×Dic181—22no—
46C2×Dic181—22no—
47C2×Dic181—44no—
48C2×Dic181—42no—
49C2×Dic181—44no—
50C2×Dic181C22⋊C411yescharacteristic
51C2×Dic181C22⋊C411yescharacteristic
52C2×Dic181C2×D411yescharacteristic
53C2×Dic181C2×D411yescharacteristic
54C23×C181—24no—
55C23×C181—22no—
56C23×C181—42no—
57C23×C181—42no—
58C23×C181C22⋊C411yescharacteristic
59D4×C181—24no—
60D4×C181—22no—
61D4×C181M1614yescomplemented
62C2×C4×C181—22no—
63C2×C4×C181C22⋊C411yescharacteristic
64C2×C4×C181C22×C411yescharacteristic
65C22×C181—21no—
66C22×C181—21no—
67C22×C181—24no—
68C22×C181—48no—
69C22×C181—42no—
70C22×C181—42no—
71C22×C181#10412yes—
72C22×C181C2×C22⋊C411yescharacteristic
73C4×C181—21no—
74C4×C181C2×M1612yes—
75Dic181—21no—
76Dic181—21no—
77Dic181—21no—
78Dic181—42no—
79Dic181AGL(1,ℤ/8)12yes—
80C2×C181—22no—
81C2×C181—44no—
82C2×C181#40711yescharacteristic
83C2×C181#41811yescharacteristic
84C2×C181C2×#10411yescharacteristic
85C181#149011yescharacteristic, complemented, Sylow
86#1490—1811nomaximal, Sylow
87C2×C4×D4—1811no—
88#432—1811no—
89C4×M16—1811no—
90D4⋊C8—1814no—
91C22×D4#1062438211yescharacteristic
92C23×C4—3621no—
93C2×M16—3621no—
94C4×D4—3621no—
95C4×D4—3621no—
96C4×D4—1812no—
97C4×D4—1812no—
98C2×C4⋊C4—1811no—
99C2×C22⋊C4—3621no—
100C2×C42—1811no—
101C4⋊C8—1812no—
102C22⋊C8—3621no—
103#97—3621no—
104#97—3621no—
105C4×C8—1812no—
106C24—21no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4C2×#1062438212yes—
110C22×C4—3621no—
111C22×C4—3622no—
112C22×C4—3622no—
113C22×C4—1811no—
114C22×C4—1811no—
115C22×C4—7244no—
116C22×C4C2×#1062438211yescharacteristic
117M16—3624no—
118C2×C8—3622no—
119C2×C8—3622no—
120C4⋊C4—3621no—
121C4⋊C4—1812no—
122C22⋊C4—3622no—
123C22⋊C4—3622no—
124C42—1811no—
125C42—1811no—
126C42—1811no—
127C42—1811no—
128C23—24no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23C22⋊#1062438211yescharacteristic
133D4—24no—
134D4—22no—
135D4#1566845814yescomplemented
136C2×C4—22no—
137C2×C4—3622no—
138C2×C4—3622no—
139C2×C4—3622no—
140C2×C4—1811no—
141C2×C4—1811no—
142C2×C4—1811no—
143C2×C4—1811no—
144C2×C4—7244no—
145C2×C4—7244no—
146C2×C4—7242no—
147C2×C4C22⋊#1062438211yescharacteristic
148C2×C4C22×#1062438211yescharacteristic
149C8—3624no—
150C8—7242no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—48no—
155C22—42no—
156C22—42no—
157C22#1915388012yes—
158C22C2×C22⋊#1062438211yescharacteristic
159C4—21no—
160C4—3621no—
161C4—3621no—
162C4—3621no—
163C4—1812no—
164C4—7242no—
165C4C2×#1566845812yes—
166C2—22no—
167C2—44no—
168C2#2604224911yescharacteristic
169C2#2604227111yescharacteristic
170C2C2×#1915388011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting, prime radicalC22×D4×C181C4
socleC22×C181C2×C22⋊C4
cosocleC4×Dic181C23
hypercenterC22×D4#10624382
hypocenterC181#1490
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26042300C2semidirect
2. C4×#15668458C2semidirect
3. D4⋊#13089685C2semidirect
6. C4⋊#13089685C22semidirect
10. C4×#13089685C22semidirect
61. D4×C181M16semidirect
85. C181#1490semidirect
135. D4#15668458semidirect

Maximal subgroups

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

subgroupindex
#260423002
C4×#156684582
D4⋊#130896852
C2×D4×Dic1812
#1490181