Finite Group Databasebeta

← #37536900

Subgroups — #37536900

Order 23168.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#19153814C211yesmaximal, characteristic
2#26041186C211yesmaximal, characteristic
3#26041183C212yesmaximal
4#26041166C212yesmaximal
5#26041163C212yesmaximal
6#26041161C212yesmaximal
7#26041160C212yesmaximal
8#26041146C212yesmaximal
9#391×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11#19153814C418yescomplemented
12C2×C4⋊Dic181C2211yescharacteristic
13C2×#15668428C2212yes—
14C2×#15668428C2212yes—
15C2×#15668428C2212yes—
16C2×C4×Dic181C2212yes—
17C2×C4×Dic181C2211yescharacteristic
18C2×C4×Dic181C2211yescharacteristic
19#19153723C2212yes—
20#19153723C2212yes—
21#19153723C2212yes—
22#19153723C2212yes—
23#19153723C2212yes—
24#19153723C2212yes—
25#19153723C2212yes—
26#19153723C2212yes—
27C22×Q8×C181C2211yescharacteristic
28C2×C4⋊C4×C181C2211yescharacteristic
29C2×C42×C181C2211yescharacteristic
30#95×C181C2212yes—
31#95×C181C2212yes—
32C2×Dic362—41no—
33C2×Dic362—41no—
34C2×Dic362C2×C414yes—
35C22×Dic181C2312yes—
36C22×Dic181C2312yes—
37C22×Dic181C2312yes—
38C22×Dic181C2312yes—
39C4⋊Dic181—41no—
40#15668428—42no—
41#15668428C2×C418yes—
42#15668428C2×C418yes—
43C4×Dic181—41no—
44C4×Dic181—42no—
45C4×Dic181C2×C414yes—
46C2×Q8×C181—41no—
47C2×Q8×C181—41no—
48C2×Q8×C181C2×C414yes—
49C22×C4×C181C2312yes—
50C22×C4×C181C2312yes—
51C22×C4×C181C2311yescharacteristic
52C22×C4×C181C2311yescharacteristic
53C22×C4×C181C2311yescharacteristic
54C4⋊C4×C181—41no—
55C42×C181—41no—
56Dic362—44no—
57C2×Dic181—22no—
58C2×Dic181—22no—
59C2×Dic181—22no—
60C2×Dic181—22no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—22no—
65C2×Dic181—24no—
66C2×Dic181C22×C414yes—
67C2×Dic181C22×C414yes—
68C23×C181C2411yescharacteristic
69Q8×C181—44no—
70C2×C4×C181—21no—
71C2×C4×C181—21no—
72C2×C4×C181—21no—
73C2×C4×C181—21no—
74C2×C4×C181—21no—
75C2×C4×C181—21no—
76C2×C4×C181—22no—
77C2×C4×C181—22no—
78C2×C4×C181—22no—
79C2×C4×C181—24no—
80C2×C4×C181—22no—
81C2×C4×C181C22×C412yes—
82C2×C4×C181C22×C412yes—
83C2×C4×C181C22×C412yes—
84C22×C181C23×C411yescharacteristic
85C22×C181H(5,𝔽2)11yescharacteristic
86C22×C181H(5,𝔽2)11yescharacteristic
87C22×C1812−1+412yes—
88C22×C1812−1+412yes—
89C4×C181—22no—
90C4×C181—22no—
91C4×C181—22no—
92C4×C181—42no—
93C4×C181—42no—
94Dic181—24no—
95Dic181—24no—
96Dic181—42no—
97Dic181—42no—
98C2×C181#51811yescharacteristic
99C2×C181#51912yes—
100C2×C181#56311yescharacteristic
101C2×C181#56311yescharacteristic
102C2×C181#56312yes—
103C181#238111yescharacteristic, complemented, Sylow
104#2381—1811nomaximal, Sylow
105C2×C4⋊Q8—1811no—
106#391—1811no—
107#391—1812no—
108#391D18111yescharacteristic
109#387—1812no—
110#387—1812no—
111#387—1812no—
112#387—1812no—
113#383—1812no—
114C22×Q8—1811no—
115C22×Q8C2×D18111yescharacteristic
116C4⋊Q8—1818no—
117C2×C4⋊C4—1812no—
118C2×C4⋊C4—1812no—
119C2×C4⋊C4—1812no—
120C2×C4⋊C4—1811no—
121C2×C4⋊C4C2×D18111yescharacteristic
122C2×C42—1811no—
123C2×C42—1811no—
124C2×C42—1812no—
125C2×C42C2×D18111yescharacteristic
126#95—1812no—
127#95—1812no—
128#95—1812no—
129#95—1812no—
130#95—1812no—
131#95—1812no—
132#95—1812no—
133#95—1812no—
134#95C2×D18112yes—
135#95C2×D18112yes—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—1814no—
139C2×Q8—7241no—
140C2×Q8—7241no—
141C2×Q8C4×D18114yes—
142C22×C4—1812no—
143C22×C4—1812no—
144C22×C4—1812no—
145C22×C4—1812no—
146C22×C4C22×D18112yes—
147C22×C4C22×D18112yes—
148C22×C4C22×D18111yescharacteristic
149C22×C4C22×D18111yescharacteristic
150C22×C4C22×D18111yescharacteristic
151C4⋊C4—41no—
152C4⋊C4—1818no—
153C4⋊C4—1818no—
154C4⋊C4—7242no—
155C4⋊C4—7241no—
156C42—41no—
157C42—1814no—
158C42—7241no—
159C42—7242no—
160C23C23×D18111yescharacteristic
161Q8—44no—
162Q8—7244no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—3622no—
175C2×C4—3622no—
176C2×C4—3622no—
177C2×C4—3622no—
178C2×C4—3622no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—3624no—
182C2×C4—3622no—
183C2×C4—1814no—
184C2×C4—1814no—
185C2×C4C2×C4×D18112yes—
186C2×C4C2×C4×D18112yes—
187C2×C4C2×C4×D18112yes—
188C22C22×C4×D18111yescharacteristic
189C22#1915386111yescharacteristic
190C22#1915386111yescharacteristic
191C22#1915386312yes—
192C22#1915386412yes—
193C4—22no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—3624no—
199C4—3624no—
200C4—7242no—
201C4—7242no—
202C2#2604175211yescharacteristic
203C2#2604175512yes—
204C2#2604202811yescharacteristic
205C2#2604202911yescharacteristic
206C2#2604203012yes—
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#391×C181C2
socle, prime radical, cosocleC23×C181C24
hypercenter#391D181
hypocenterC181#2381
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #19153814C4semidirect
103. C181#2381semidirect

Maximal subgroups

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

subgroupindex
C2×#191538142
#260411862
#260411832
#260411662
#260411632
#260411612
#260411602
#260411462
#391×C1812
#2381181