Finite Group Databasebeta

← #37548955

Subgroups — #37548955

Order 23168.

264 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 76 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#26042296C212yesmaximal, complemented
2#26042124C212yesmaximal, complemented
3#26042123C212yesmaximal
4C2×#19153774C211yesmaximal, characteristic
5#19153883—22no—
6C22⋊#13089685C2214yes—
7C4×#13089685—22no—
8C22×C4×D181C411yescharacteristic
9#19153774—22no—
10#19153774—22no—
11C2×C4⋊Dic181C2211yescharacteristic
12C2×#15668426C411yescharacteristic
13C2×#15668426C2211yescharacteristic
14C2×C4×Dic181C411yescharacteristic
15C2×C4×Dic181C2211yescharacteristic
16C2×C4⋊C4×C181C411yescharacteristic
17C2×#13089685—24no—
18C23×D181C2×C411yescharacteristic
19C22×Dic181C2×C411yescharacteristic
20C22×Dic181C2×C411yescharacteristic
21C22×Dic181C2311yescharacteristic
22C2×C4×D181—21no—
23C2×C4×D181—21no—
24C2×C4×D181—22no—
25C2×C4×D181C814yescomplemented
26C4⋊Dic181—22no—
27#15668426—22no—
28#15668426—22no—
29C4×Dic181—21no—
30C4×Dic181—21no—
31C4×Dic181D412yes—
32C4×Dic181D412yes—
33C22×C4×C181C2×C411yescharacteristic
34C22×C4×C181C2×C411yescharacteristic
35C22×C4×C181C2×C411yescharacteristic
36C4⋊C4×C181—22no—
37#13089685—44no—
38C22×D181—21no—
39C22×D181—21no—
40C22×D181—41no—
41C22×D181—41no—
42C22×D181C2×C812yes—
43C2×Dic181—21no—
44C2×Dic181—41no—
45C2×Dic181—41no—
46C2×Dic181C22⋊C411yescharacteristic
47C2×Dic181C22⋊C411yescharacteristic
48C2×Dic181C2×C812yes—
49C2×Dic181M1612yes—
50C2×Dic181C2×D412yes—
51C4×D181—24no—
52C4×D181—24no—
53C23×C181C22×C411yescharacteristic
54C2×C4×C181—21no—
55C2×C4×C181—21no—
56C2×C4×C181—21no—
57C2×C4×C181—41no—
58C2×C4×C181—41no—
59C2×C4×C181C22⋊C412yes—
60C2×C4×C181C2×C812yes—
61C22×C181—21no—
62C22×C181—21no—
63C22×C181C2×C22⋊C411yescharacteristic
64C22×C181C22×C811yescharacteristic
65C22×C181C2×M1611yescharacteristic
66C2×D181—22no—
67C2×D181—42no—
68C2×D181—41no—
69C2×D181—41no—
70C2×D181—42no—
71C4×C181—22no—
72C4×C181—22no—
73C4×C181—41no—
74Dic181—22no—
75Dic181—41no—
76Dic181C22⋊C812yes—
77Dic181C22⋊C812yes—
78C2×C181—21no—
79C2×C181—21no—
80C2×C181C2×C22⋊C811yescharacteristic
81C2×C181#41011yescharacteristic
82C2×C181#41311yescharacteristic
83D181—42no—
84C181#145811yescharacteristic, complemented, Sylow
85#1458—1811nomaximal, Sylow
86C2×#117—1811no—
87#431—1812no—
88#324—1812no—
89#323—1812no—
90C23×C4—1811no—
91#117—3622no—
92#117—3622no—
93C2×C4⋊C4—1811no—
94C2×C4⋊C4#1062438211yescharacteristic
95C2×C22⋊C4—1811no—
96C2×C22⋊C4—1811no—
97C2×C42—1811no—
98C2×C42—1811no—
99C4⋊C8—3622no—
100C22⋊C8—1814no—
101C4×C8—3622no—
102C24—1811no—
103C22×C4—3621no—
104C22×C4—3621no—
105C22×C4—3622no—
106C22×C4—1814no—
107C22×C4—1811no—
108C22×C4—1811no—
109C22×C4—1811no—
110C22×C4C2×#1062438211yescharacteristic
111C22×C4C2×#1062438211yescharacteristic
112C22×C4C2×#1062438211yescharacteristic
113C2×C8—3624no—
114C4⋊C4—22no—
115C4⋊C4—3622no—
116C22⋊C4—3622no—
117C22⋊C4—3622no—
118C42—3621no—
119C42—3621no—
120C42—1812no—
121C42—1812no—
122C23—3621no—
123C23—3621no—
124C23—1812no—
125C23—7241no—
126C23—7241no—
127C23C22×#1062438211yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—3624no—
134C2×C4—3624no—
135C2×C4—3621no—
136C2×C4—1812no—
137C2×C4—1811no—
138C2×C4—1811no—
139C2×C4—1812no—
140C2×C4—1812no—
141C2×C4—7241no—
142C2×C4—7241no—
143C2×C4C22⋊#1062438212yes—
144C2×C4D181⋊C812yes—
145C8—7244no—
146C22—21no—
147C22—21no—
148C22—3622no—
149C22—7242no—
150C22—7242no—
151C22—7241no—
152C22—7241no—
153C22C2×C22⋊#1062438211yescharacteristic
154C22C2×D181⋊C811yescharacteristic
155C22D181⋊M1611yescharacteristic
156C4—22no—
157C4—22no—
158C4—41no—
159C4—3622no—
160C4—1812no—
161C4—1812no—
162C4—7241no—
163C2—21no—
164C2—21no—
165C2—7242no—
166C2D181⋊(C22⋊C8)11yescharacteristic
167C2#2604226011yescharacteristic
168C2#2604226611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC22×C181C22×C8
FrattiniC23C22×#10624382
FittingC2×C4⋊C4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC23×D181C2×C4
cosocleC22×Dic181C23
hypercenterC2×C4⋊C4#10624382
hypocenterC181#1458
LCSG ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #26042296C2semidirect
2. #26042124C2semidirect
25. C2×C4×D181C8semidirect
84. C181#1458semidirect

Maximal subgroups

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

subgroupindex
#260422962
#260421242
#260421232
C2×#191537742
#1458181