Finite Group Databasebeta

← #37549180

Subgroups — #37549180

Order 23168.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 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
3#26042134C214yesmaximal, complemented
4C2×#19153803C211yesmaximal, characteristic
5C2×#15668458—21no—
6C4⋊#13089685—21no—
7C22⋊#13089685—21no—
8#19153871C2212yescomplemented
9#19153871C2212yescomplemented
10C4×#13089685—21no—
11C22×Dic362C411yescharacteristic
12#19153803—21no—
13#19153803—21no—
14#19153803C412yes—
15#19153803C2212yes—
16C2×C22⋊Dic181—21no—
17C2×C22⋊Dic181—21no—
18C2×C4×Dic181C2211yescharacteristic
19C22×D4×C181C411yescharacteristic
20#15668458—24no—
21C2×#13089685—22no—
22C2×#13089685—22no—
23C2×Dic362—21no—
24C2×Dic362—42no—
25C2×Dic362C2×C412yes—
26C22×Dic181—21no—
27C22×Dic181D411yescharacteristic
28C22×Dic181D411yescharacteristic
29C22⋊Dic181—22no—
30C22⋊Dic181—22no—
31C22⋊Dic181—22no—
32C22⋊Dic181—22no—
33C4×Dic181C2×C411yescharacteristic
34C4×Dic181D411yescharacteristic
35C4×Dic181D411yescharacteristic
36C4×Dic181C2311yescharacteristic
37C24×C181—21no—
38C2×D4×C181—21no—
39C2×D4×C181—42no—
40C2×D4×C181C2×C412yes—
41C22×C4×C181C2×C411yescharacteristic
42#13089685—24no—
43#13089685—42no—
44Dic362—42no—
45Dic362—42no—
46C2×Dic181—22no—
47C2×Dic181—22no—
48C2×Dic181—41no—
49C2×Dic181—42no—
50C2×Dic181C22⋊C411yescharacteristic
51C2×Dic181C22⋊C411yescharacteristic
52C2×Dic181C2×D411yescharacteristic
53C2×Dic181C2×D411yescharacteristic
54C23×C181—22no—
55C23×C181—42no—
56C23×C181—42no—
57C23×C181—42no—
58C23×C181C22⋊C411yescharacteristic
59D4×C181—42no—
60D4×C181—42no—
61C2×C4×C181—22no—
62C2×C4×C181C22⋊C411yescharacteristic
63C2×C4×C181C22×C411yescharacteristic
64C22×C181—21no—
65C22×C181—21no—
66C22×C181—42no—
67C22×C181—42no—
68C22×C181—42no—
69C22×C181—84no—
70C22×C181#10412yes—
71C22×C181C2×C22⋊C411yescharacteristic
72C4×C181—21no—
73C4×C181#10012yes—
74Dic181—21no—
75Dic181—21no—
76Dic181—21no—
77Dic181—21no—
78Dic181—42no—
79C2×C181—22no—
80C2×C181—82no—
81C2×C181C2×#10011yescharacteristic
82C2×C181C2×#10411yescharacteristic
83C2×C181#42111yescharacteristic
84C181#152711yescharacteristic, complemented, Sylow
85#1527—1811nomaximal, Sylow
86C2×#124—1811no—
87#432—1811no—
88C4×M16—1811no—
89#329—1814no—
90C22×Q8—1811no—
91C22×D4#1062438211yescharacteristic
92C2×M16—3621no—
93#124—3621no—
94#124—3621no—
95#124—1812no—
96#124—1812no—
97C2×C22⋊C4—3621no—
98C2×C22⋊C4—3621no—
99C2×C42—1811no—
100C4⋊C8—3621no—
101C22⋊C8—3621no—
102#97—1812no—
103#97—1812no—
104C4×C8—3621no—
105C24—21no—
106C2×Q8—3621no—
107C2×Q8—1812no—
108C2×Q8—7242no—
109C2×D4—21no—
110C2×D4—42no—
111C2×D4C2×#1062438212yes—
112C22×C4—3621no—
113C22×C4—1811no—
114C22×C4—1811no—
115C22×C4C2×#1062438211yescharacteristic
116M16—3624no—
117C2×C8—3622no—
118C2×C8—3622no—
119C22⋊C4—3622no—
120C22⋊C4—3622no—
121C22⋊C4—3622no—
122C22⋊C4—3622no—
123C42—1811no—
124C42—1811no—
125C42—1811no—
126C42—1811no—
127C23—22no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23C22⋊#1062438211yescharacteristic
132Q8—7242no—
133Q8—7242no—
134D4—42no—
135D4—42no—
136C2×C4—22no—
137C2×C4—3622no—
138C2×C4—3622no—
139C2×C4—1811no—
140C2×C4—1811no—
141C2×C4—1811no—
142C2×C4—1811no—
143C2×C4—7241no—
144C2×C4—7242no—
145C2×C4C22⋊#1062438211yescharacteristic
146C2×C4C22×#1062438211yescharacteristic
147C8—3624no—
148C8—7242no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—84no—
155C22#1915388012yes—
156C22C2×C22⋊#1062438211yescharacteristic
157C4—21no—
158C4—3621no—
159C4—3621no—
160C4—3621no—
161C4—3621no—
162C4—7242no—
163C4#1915387612yes—
164C2—22no—
165C2—82no—
166C2C2×#1915387611yescharacteristic
167C2C2×#1915388011yescharacteristic
168C2#2604227711yescharacteristic
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#1527
LCSG ▷ C2×C4×C181 ▷ C22×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. #26042134C2semidirect
8. #19153871C22semidirect
9. #19153871C22semidirect
84. C181#1527semidirect

Maximal subgroups

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

subgroupindex
#260423002
C4×#156684582
#260421342
C2×#191538032
#1527181