Finite Group Databasebeta

← #37536439

Subgroups — #37536439

Order 23168.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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
2C2×Q8×Dic181C211yesmaximal, characteristic
3C2×#19153781C211yesmaximal, characteristic
4#26041185C211yesmaximal, characteristic
5#26041166C214yesmaximal
6#26041161C214yesmaximal
7#26041144C211yesmaximal, characteristic
8#26041125C211yesmaximal, characteristic
9#391×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11#19153814C418yescomplemented
12Q8×Dic181—24no—
13#19153781—24no—
14C2×C4⋊Dic181C2211yescharacteristic
15C2×C4⋊Dic181C2212yes—
16C2×#15668428C2212yes—
17C2×#15668428C2214yes—
18C2×C4×Dic181C2212yes—
19C2×C4×Dic181C2212yes—
20C2×C4×Dic181C2211yescharacteristic
21C2×C4×Dic181C2211yescharacteristic
22#19153723C2212yes—
23#19153723C2212yes—
24#19153723C2214yes—
25#19153723C2214yes—
26C22×Q8×C181C2211yescharacteristic
27C2×C4⋊C4×C181C2211yescharacteristic
28C2×C42×C181C2211yescharacteristic
29#95×C181C2214yes—
30C2×Dic362—22no—
31C2×Dic362—22no—
32C2×Dic362C2×C414yes—
33C22×Dic181C2312yes—
34C22×Dic181C2312yes—
35C22×Dic181C2312yes—
36C22×Dic181C2311yescharacteristic
37C22×Dic181C2311yescharacteristic
38C4⋊Dic181—24no—
39C4⋊Dic181—22no—
40#15668428—24no—
41#15668428C2×C4116yes—
42C4×Dic181—24no—
43C4×Dic181—24no—
44C4×Dic181C2×C414yes—
45C4×Dic181D414yes—
46C2×Q8×C181—22no—
47C2×Q8×C181—22no—
48C2×Q8×C181C2×C414yes—
49C22×C4×C181C2312yes—
50C22×C4×C181C2312yes—
51C22×C4×C181C2312yes—
52C22×C4×C181C2311yescharacteristic
53C4⋊C4×C181—22no—
54C42×C181—22no—
55Dic362—28no—
56C2×Dic181—22no—
57C2×Dic181—22no—
58C2×Dic181—22no—
59C2×Dic181—22no—
60C2×Dic181—22no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—24no—
64C2×Dic181C22×C414yes—
65C2×Dic181C22×C414yes—
66C2×Dic181C2×D412yes—
67C2×Dic181C2×D412yes—
68C23×C181C2411yescharacteristic
69Q8×C181—28no—
70C2×C4×C181—22no—
71C2×C4×C181—22no—
72C2×C4×C181—22no—
73C2×C4×C181—22no—
74C2×C4×C181—24no—
75C2×C4×C181—22no—
76C2×C4×C181—22no—
77C2×C4×C181C22×C414yes—
78C2×C4×C181C22×C412yes—
79C2×C4×C181C2×D412yes—
80C2×C4×C181#4112yes—
81C22×C181C23×C411yescharacteristic
82C22×C181C22×D411yescharacteristic
83C22×C181C2×#4111yescharacteristic
84C22×C1812−1+412yes—
85C22×C1812−1+412yes—
86C4×C181—24no—
87C4×C181—42no—
88C4×C181—42no—
89C4×C181C4×D414yes—
90Dic181—24no—
91Dic181—24no—
92Dic181—22no—
93Dic181—22no—
94Dic181—42no—
95C2×C181C2×C4×D411yescharacteristic
96C2×C181#51912yes—
97C2×C181#53611yescharacteristic
98C2×C181#53611yescharacteristic
99C2×C181#54112yes—
100C181#232711yescharacteristic, complemented, Sylow
101#2327—1811nomaximal, Sylow
102C2×C4⋊Q8—1811no—
103C2×C4×Q8—1811no—
104C2×C4×Q8—1811no—
105#391—1811no—
106#391D18111yescharacteristic
107#387—1814no—
108#387—1814no—
109#382—1811no—
110#376—1811no—
111C22×Q8—1811no—
112C22×Q8C2×D18111yescharacteristic
113C4⋊Q8—1818no—
114C4×Q8—3624no—
115C4×Q8—3624no—
116C2×C4⋊C4—1812no—
117C2×C4⋊C4—1811no—
118C2×C4⋊C4—1812no—
119C2×C4⋊C4—1814no—
120C2×C4⋊C4C2×D18111yescharacteristic
121C2×C42—1812no—
122C2×C42—1812no—
123C2×C42—1811no—
124C2×C42—1811no—
125C2×C42C2×D18111yescharacteristic
126#95—1812no—
127#95—1812no—
128#95—1814no—
129#95—1814no—
130#95C2×D18114yes—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—3622no—
134C2×Q8—3622no—
135C2×Q8—1814no—
136C2×Q8C4×D18114yes—
137C22×C4—1812no—
138C22×C4—1812no—
139C22×C4—1811no—
140C22×C4—1811no—
141C22×C4—1812no—
142C22×C4C22×D18112yes—
143C22×C4C22×D18112yes—
144C22×C4C22×D18112yes—
145C22×C4C22×D18111yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—3624no—
148C4⋊C4—3624no—
149C4⋊C4—3622no—
150C4⋊C4—18116no—
151C42—22no—
152C42—3624no—
153C42—3624no—
154C42—1814no—
155C42—1814no—
156C23C23×D18111yescharacteristic
157Q8—28no—
158Q8—3628no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—3622no—
167C2×C4—3622no—
168C2×C4—3622no—
169C2×C4—3622no—
170C2×C4—3622no—
171C2×C4—3622no—
172C2×C4—3622no—
173C2×C4—3624no—
174C2×C4—1814no—
175C2×C4—1814no—
176C2×C4—1812no—
177C2×C4—1812no—
178C2×C4C2×C4×D18114yes—
179C2×C4C2×C4×D18112yes—
180C2×C4D4×D18112yes—
181C2×C4#1566844812yes—
182C22C22×C4×D18111yescharacteristic
183C22C2×D4×D18111yescharacteristic
184C22C2×#1566844811yescharacteristic
185C22#1915386312yes—
186C22#1915386412yes—
187C4—24no—
188C4—42no—
189C4—42no—
190C4—3624no—
191C4—3624no—
192C4—3622no—
193C4—3622no—
194C4—7242no—
195C4#1915377614yes—
196C2C2×#1915377611yescharacteristic
197C2#2604175512yes—
198C2#2604184911yescharacteristic
199C2#2604185411yescharacteristic
200C2#2604188912yes—
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#391×C181C2
socle, prime radical, cosocleC23×C181C24
hypercenter#391D181
hypocenterC181#2327
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #19153814C4semidirect
100. C181#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#191538142
C2×Q8×Dic1812
C2×#191537812
#260411852
#260411662
#260411612
#260411442
#260411252
#391×C1812
#2327181