Finite Group Databasebeta

← #37534706

Subgroups — #37534706

Order 23168.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19153795C211yesmaximal, characteristic
2#26041139C211yesmaximal, characteristic
3#26041045C214yesmaximal
4C4×C22⋊C4×C181C211yesmaximal, characteristic
5C22×Dic362C411yescharacteristic, complemented
6#19153795—21no—
7#19153795—21no—
8#19153795—21no—
9#19153795—21no—
10C2×C4⋊Dic181—21no—
11C2×C4⋊Dic181C411yescharacteristic, complemented
12C2×#15668428C412yescomplemented
13C2×C22⋊Dic181C2212yes—
14C2×C22⋊Dic181C2212yes—
15#19153723—21no—
16C23×C4×C181C2211yescharacteristic
17C2×C22⋊C4×C181C2212yes—
18C2×C42×C181—21no—
19#95×C181—21no—
20C2×Dic362—21no—
21C2×Dic362—21no—
22C2×Dic362—42no—
23C22×Dic181—22no—
24C22×Dic181C2×C412yes—
25C22×Dic181C2×C412yes—
26C4⋊Dic181—21no—
27C4⋊Dic181—21no—
28C4⋊Dic181—42no—
29#15668428—22no—
30#15668428—22no—
31C22⋊Dic181—22no—
32C22⋊Dic181—22no—
33C22⋊Dic181—24no—
34C24×C181C2311yescharacteristic
35C22×C4×C181—22no—
36C22×C4×C181—42no—
37C22×C4×C181C2×C411yescharacteristic
38C22×C4×C181C2×C411yescharacteristic
39C22×C4×C181D411yescharacteristic
40C22×C4×C181D411yescharacteristic
41C22×C4×C181D411yescharacteristic
42C22×C4×C181D411yescharacteristic
43C22⋊C4×C181—22no—
44C22⋊C4×C181Q814yescomplemented
45C42×C181—42no—
46Dic362—42no—
47Dic362—42no—
48C2×Dic181—22no—
49C2×Dic181—22no—
50C2×Dic181—42no—
51C2×Dic181—42no—
52C2×Dic181—42no—
53C2×Dic181—42no—
54C2×Dic181—42no—
55C23×C181—41no—
56C23×C181—41no—
57C23×C181C22×C411yescharacteristic
58C23×C181C2×D411yescharacteristic
59C23×C181C2×D411yescharacteristic
60C23×C181C2×Q812yes—
61C23×C181#4112yes—
62C2×C4×C181—22no—
63C2×C4×C181—22no—
64C2×C4×C181—22no—
65C2×C4×C181—24no—
66C2×C4×C181—41no—
67C2×C4×C181—41no—
68C2×C4×C181—41no—
69C2×C4×C181—42no—
70C2×C4×C181—42no—
71C2×C4×C181C22⋊C411yescharacteristic
72C2×C4×C181C22⋊C411yescharacteristic
73C2×C4×C181C22⋊C411yescharacteristic
74C2×C4×C181C22⋊C411yescharacteristic
75C22×C181—21no—
76C22×C181—21no—
77C22×C181—21no—
78C22×C181—21no—
79C22×C181—21no—
80C22×C181—21no—
81C22×C181—42no—
82C22×C181—42no—
83C22×C181C2×C22⋊C411yescharacteristic
84C22×C181C4×Q812yes—
85C22×C181C22⋊Q812yes—
86C22×C181#12411yescharacteristic
87C22×C181C4⋊Q811yescharacteristic
88C4×C181—21no—
89C4×C181—21no—
90C4×C181—21no—
91C4×C181—44no—
92C4×C181#9912yes—
93Dic181—42no—
94Dic181—42no—
95Dic181—82no—
96C2×C181—21no—
97C2×C181—22no—
98C2×C181—22no—
99C2×C181—21no—
100C2×C181#39111yescharacteristic
101C2×C181C2×#9911yescharacteristic
102C2×C181#41011yescharacteristic
103C181#199611yescharacteristic, complemented, Sylow
104#1996—1811nomaximal, Sylow
105C2×C22⋊Q8—1811no—
106#380—1811no—
107C4×C22⋊C4D18111yescharacteristic
108#342—1814no—
109C22×Q8—1811no—
110C23×C4C2×D18111yescharacteristic
111C22⋊Q8—3621no—
112C22⋊Q8—3621no—
113C22⋊Q8—3621no—
114C22⋊Q8—3621no—
115C2×C4⋊C4—3621no—
116C2×C4⋊C4—1812no—
117C2×C4⋊C4—1811no—
118C2×C22⋊C4—1812no—
119C2×C22⋊C4—1812no—
120C2×C22⋊C4C2×D18112yes—
121C2×C42—21no—
122#95—21no—
123#95—3621no—
124C24C22×D18111yescharacteristic
125C2×Q8—3621no—
126C2×Q8—3621no—
127C2×Q8—7242no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—3622no—
131C22×C4—1812no—
132C22×C4—1812no—
133C22×C4C4×D18111yescharacteristic
134C22×C4C4×D18111yescharacteristic
135C22×C4C22⋊D18111yescharacteristic
136C22×C4C22⋊D18111yescharacteristic
137C22×C4D72411yescharacteristic
138C22×C4D72411yescharacteristic
139C4⋊C4—3622no—
140C4⋊C4—3622no—
141C4⋊C4—3621no—
142C4⋊C4—3621no—
143C4⋊C4—7242no—
144C22⋊C4—22no—
145C22⋊C4—3622no—
146C22⋊C4—3622no—
147C22⋊C4—3624no—
148C22⋊C4Dic36214yescomplemented
149C42—42no—
150C23—41no—
151C23—41no—
152C23C2×C4×D18111yescharacteristic
153C23C2×C22⋊D18111yescharacteristic
154C23C2×D72411yescharacteristic
155C23C2×Dic36212yes—
156C23#1566844912yes—
157Q8—7242no—
158Q8—7242no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—3622no—
169C2×C4—3622no—
170C2×C4—7242no—
171C2×C4—7242no—
172C2×C4—7242no—
173C2×C4—7242no—
174C2×C4—7242no—
175C2×C4#1566842611yescharacteristic
176C2×C4#1566842611yescharacteristic
177C2×C4#1566842611yescharacteristic
178C2×C4#1566842611yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22C2×#1566842611yescharacteristic
188C22C4×Dic36212yes—
189C22#1915379512yes—
190C22#1915380411yescharacteristic
191C22#1915381611yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—44no—
196C4—14482no—
197C4—7242no—
198C4—7242no—
199C4#1915373012yes—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2#2604118511yescharacteristic
205C2C2×#1915373011yescharacteristic
206C2#2604125211yescharacteristic
subgroupG/H
centerC22C2×#15668426
commutatorC23×C181C22×C4
FrattiniC24C22×D181
FittingC4×C22⋊C4×C181C2
socleC22×C181C2×C22⋊C4
prime radical, cosocleC24×C181C23
hypercenterC4×C22⋊C4D181
hypocenterC181#1996
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C22×Dic362C4semidirect
11. C2×C4⋊Dic181C4semidirect
12. C2×#15668428C4semidirect
44. C22⋊C4×C181Q8semidirect
103. C181#1996semidirect
148. C22⋊C4Dic362semidirect

Maximal subgroups

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

subgroupindex
C2×#191537952
#260411392
#260410452
C4×C22⋊C4×C1812
#1996181