Finite Group Databasebeta

← #37532690

Subgroups — #37532690

Order 23168.

282 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 57 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#26041417C212yesmaximal, complemented
2#26041199C211yesmaximal, characteristic, complemented
3#26041013C212yesmaximal, complemented
4#26041005C211yesmaximal, characteristic, complemented
5#537×C181C211yesmaximal, characteristic
6C22×Dic362C2211yescharacteristic
7C2×#15668437—22no—
8C2×#15668428—21no—
9#19153739—22no—
10C22⋊#13089678C2212yes—
11#19153723—21no—
12#19153723C2211yescharacteristic
13C2×#41×C181—21no—
14C22⋊Q8×C181—22no—
15#121×C181—22no—
16#121×C181—21no—
17#121×C181C2212yes—
18C2×C4⋊C4×C181C2211yescharacteristic
19C2×Dic362—22no—
20C2×Dic362—42no—
21C22×Dic181—21no—
22C22×Dic181—21no—
23#15668437—44no—
24C2×#13089678—22no—
25#15668428—42no—
26#41×C181—42no—
27#41×C181—42no—
28C2×Q8×C181—21no—
29C2×D4×C181—21no—
30C2×D4×C181—22no—
31C2×D4×C181—21no—
32C2×D4×C181D412yes—
33C22×C4×C181—22no—
34C22×C4×C181D411yescharacteristic
35C22×C4×C181D411yescharacteristic
36C22×C4×C181C2311yescharacteristic
37C4⋊C4×C181—21no—
38C4⋊C4×C181—22no—
39C4⋊C4×C181D412yes—
40C22⋊C4×C181—22no—
41C22⋊C4×C181—22no—
42Dic362—24no—
43Dic362—42no—
44C2×Dic181—22no—
45C2×Dic181—22no—
46C2×Dic181—41no—
47C2×Dic181—41no—
48C2×Dic181—41no—
49C2×Dic181—41no—
50#13089678—42no—
51C23×C181—22no—
52C23×C181C2×D411yescharacteristic
53Q8×C181—42no—
54D4×C181—42no—
55D4×C181—42no—
56D4×C181—42no—
57D4×C181—42no—
58D4×C181—81no—
59C2×C4×C181—21no—
60C2×C4×C181—21no—
61C2×C4×C181—21no—
62C2×C4×C181—22no—
63C2×C4×C181—41no—
64C2×C4×C181—44no—
65C2×C4×C181—41no—
66C2×C4×C181SD1612yescomplemented
67C2×C4×C181C2×D412yes—
68C22×C181—21no—
69C22×C181—21no—
70C22×C181—42no—
71C22×C181—42no—
72C22×C181—42no—
73C22×C181#12011yescharacteristic
74C22×C181C2×SD1611yescharacteristic
75C22×C181AGL(1,ℤ/8)11yescharacteristic
76C4×C181—22no—
77C4×C181—22no—
78C4×C181—42no—
79C4×C181—41no—
80Dic181—42no—
81Dic181—42no—
82C2×C181—21no—
83C2×C181—21no—
84C2×C181—82no—
85C2×C181#45011yescharacteristic
86C2×C181#45611yescharacteristic
87C2×C181B(4,𝔽2)11yescharacteristic
88C181#160411yescharacteristic, complemented, Sylow
89#1604—1811nomaximal, Sylow
90#537D18111yescharacteristic
91#448—1812no—
92#395—1811no—
93#327—1812no—
94#324—1811no—
95C2×#41—21no—
96C22×Q8—1811no—
97C2×SD16—3622no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C2×D18112yes—
102C2×C4⋊C4—3621no—
103C2×C4⋊C4C2×D18111yescharacteristic
104#103—3622no—
105C22⋊C8—1812no—
106#95—3621no—
107#95—1811no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—3622no—
112C2×Q8—7242no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4C22⋊D18112yes—
117C22×C4—22no—
118C22×C4—3621no—
119C22×C4—3621no—
120C22×C4C22⋊D18111yescharacteristic
121C22×C4C22⋊D18111yescharacteristic
122C22×C4C22×D18111yescharacteristic
123SD16—7244no—
124C2×C8—3622no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—7242no—
128C4⋊C4C22⋊D18112yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×C22⋊D18111yescharacteristic
133Q8—42no—
134Q8—3624no—
135Q8—7242no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—3622no—
149C2×C4—3622no—
150C2×C4—7241no—
151C2×C4—7241no—
152C2×C4—7241no—
153C2×C4—7241no—
154C2×C4#1566843712yescomplemented
155C2×C4C2×C22⋊D18112yes—
156C8—7242no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#1915378511yescharacteristic
163C22C2×#1566843711yescharacteristic
164C22#1915384211yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—41no—
169C4—7242no—
170C4—7242no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2604143011yescharacteristic
175C2#2604146211yescharacteristic
176C2#2604146611yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting, prime radical#537×C181C2
socleC22×C181#120
hypercenter#537D181
hypocenterC181#1604
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26041417C2semidirect
2. #26041199C2semidirect
3. #26041013C2semidirect
4. #26041005C2semidirect
66. C2×C4×C181SD16semidirect
88. C181#1604semidirect
154. C2×C4#15668437semidirect

Maximal subgroups

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

subgroupindex
#260414172
#260411992
#260410132
#260410052
#537×C1812
#1604181