Finite Group Databasebeta

← #37532689

Subgroups — #37532689

Order 23168.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#26041860C211yesmaximal, characteristic
2#26041418C212yesmaximal, complemented
3#26041012C212yesmaximal, complemented
4#26041005C211yesmaximal, characteristic, complemented
5#395×C181C211yesmaximal, characteristic, complemented
6C2×#15668449—21no—
7C2×#15668436—22no—
8#19153796—22no—
9#19153791—22no—
10#19153790—21no—
11#19153790C2212yes—
12#19153740—22no—
13C22⋊#13089678C2212yes—
14C22×Q8×C181C2211yescharacteristic
15C2×C4⋊C4×C181—21no—
16C2×C4⋊C4×C181C2211yescharacteristic
17#95×C181—21no—
18#95×C181C2211yescharacteristic
19#15668449—42no—
20#15668449—42no—
21C2×Dic362—21no—
22C2×D724—21no—
23C2×D724—21no—
24C2×D724D412yes—
25C2×C22⋊D181—22no—
26C2×C4×D181—22no—
27#15668436—44no—
28C2×#13089678—22no—
29C4⋊Dic181—21no—
30C4⋊Dic181D412yes—
31#15668426—22no—
32#15668426—22no—
33C2×Q8×C181—22no—
34C2×Q8×C181—42no—
35C22×C4×C181—21no—
36C22×C4×C181—21no—
37C22×C4×C181D411yescharacteristic
38C22×C4×C181D411yescharacteristic
39C22×C4×C181C2311yescharacteristic
40C4⋊C4×C181—22no—
41C4⋊C4×C181—42no—
42C22×D181—22no—
43Dic362—42no—
44D724—42no—
45D724—42no—
46D724—81no—
47C22⋊D181—42no—
48C22⋊D181—42no—
49C2×Dic181—22no—
50C4×D181—44no—
51#13089678—42no—
52C23×C181C2×D411yescharacteristic
53Q8×C181—24no—
54Q8×C181—42no—
55C2×C4×C181—21no—
56C2×C4×C181—21no—
57C2×C4×C181—22no—
58C2×C4×C181—22no—
59C2×C4×C181—21no—
60C2×C4×C181—41no—
61C2×C4×C181—41no—
62C2×C4×C181—41no—
63C2×C4×C181—41no—
64C2×C4×C181—41no—
65C2×C4×C181—41no—
66C2×C4×C181SD1612yescomplemented
67C2×C4×C181C2×D412yes—
68C22×C181—21no—
69C22×C181—21no—
70C22×C181#12011yescharacteristic
71C22×C181C2×SD1611yescharacteristic
72C22×C181AGL(1,ℤ/8)11yescharacteristic
73C2×D181—42no—
74C2×D181—42no—
75C2×D181—42no—
76C4×C181—22no—
77C4×C181—22no—
78C4×C181—42no—
79C4×C181—42no—
80C4×C181—41no—
81Dic181—42no—
82C2×C181—21no—
83C2×C181—21no—
84C2×C181#45011yescharacteristic
85C2×C181#45611yescharacteristic
86C2×C181B(4,𝔽2)11yescharacteristic
87D181—82no—
88C181#160411yescharacteristic, complemented, Sylow
89#1604—1811nomaximal, Sylow
90#537—1811no—
91#448—1812no—
92#395D18111yescharacteristic, complemented
93#327—1812no—
94#324—1811no—
95C2×#41—3621no—
96C22×Q8C2×D18111yescharacteristic
97C2×SD16—3622no—
98C22⋊Q8—3622no—
99#121—3622no—
100#121—3621no—
101#121—1812no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C2×D18111yescharacteristic
104#103—3622no—
105C22⋊C8—1812no—
106#95—21no—
107#95C2×D18111yescharacteristic
108#41—7242no—
109#41—7242no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—3621no—
113C2×D4—3621no—
114C2×D4—3621no—
115C2×D4—3622no—
116C2×D4—1812no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—3622no—
120C22×C4C22⋊D18111yescharacteristic
121C22×C4C22⋊D18111yescharacteristic
122C22×C4C22×D18111yescharacteristic
123SD16—7244no—
124C2×C8—3622no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—3621no—
128C4⋊C4—1812no—
129C22⋊C4—3622no—
130C22⋊C4—3622no—
131C23—3622no—
132C23C2×C22⋊D18111yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—7242no—
136D4—14481no—
137D4—7242no—
138D4—7242no—
139D4—7242no—
140D4—7242no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3622no—
153C2×C4—7244no—
154C2×C4#1566843612yescomplemented
155C2×C4D4×D18112yes—
156C8—7242no—
157C22—21no—
158C22—21no—
159C22—7242no—
160C22—7242no—
161C22—7242no—
162C22#1915378411yescharacteristic
163C22C2×#1566843611yescharacteristic
164C22#1915384111yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—42no—
169C4—41no—
170C4—7242no—
171C2—21no—
172C2—21no—
173C2—14482no—
174C2#2604142911yescharacteristic
175C2#2604146111yescharacteristic
176C2#2604146511yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#395×C181C2
socleC22×C181#120
prime radical#26041860C2
hypercenter#395D181
hypocenterC181#1604
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #26041418C2semidirect
3. #26041012C2semidirect
4. #26041005C2semidirect
5. #395×C181C2semidirect
66. C2×C4×C181SD16semidirect
88. C181#1604semidirect
92. #395D181semidirect
154. C2×C4#15668436semidirect

Maximal subgroups

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

subgroupindex
#260418602
#260414182
#260410122
#260410052
#395×C1812
#1604181