Finite Group Databasebeta

← #37549910

Subgroups — #37549910

Order 23168.

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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#26042221C211yesmaximal, characteristic
2C4×C22⋊#10624382C211yesmaximal, characteristic
3#26042157C214yesmaximal
4C2×#19153792C211yesmaximal, characteristic
5C2×C4⋊#10624382—21no—
6C2×C22⋊#10624382C2212yes—
7C2×C22⋊#10624382C2212yes—
8C2×C4×#10624382—21no—
9(C2×#10624382)⋊C4—21no—
10(C2×#10624382)⋊C4—21no—
11C22×C4×D181C2211yescharacteristic
12#19153792—21no—
13#19153792—21no—
14#19153792—21no—
15#19153792—21no—
16C2×C4⋊Dic181C411yescharacteristic, complemented
17C2×#15668428C412yescomplemented
18C2×#15668426C2212yes—
19C22×Q8×C181C411yescharacteristic, complemented
20C22×#10624382—22no—
21C22×#10624382—22no—
22C4⋊#10624382—42no—
23C22⋊#10624382—22no—
24C22⋊#10624382—24no—
25C22⋊#10624382Q814yescomplemented
26C4×#10624382—42no—
27C23×D181C2311yescharacteristic
28C22×Dic181C2×C412yes—
29C22×Dic181C2×C411yescharacteristic
30C2×C4×D181—42no—
31C2×C4×D181D411yescharacteristic
32C2×C4×D181D411yescharacteristic
33C2×C4×D181D411yescharacteristic
34C2×C4×D181D411yescharacteristic
35C4⋊Dic181—21no—
36C4⋊Dic181—21no—
37#15668428—22no—
38#15668428—22no—
39#15668426—22no—
40#15668426—22no—
41C2×Q8×C181—21no—
42C2×Q8×C181—21no—
43C2×Q8×C181—42no—
44C22×C4×C181C2×C412yes—
45C22×C4×C181C2×C411yescharacteristic
46C2×#10624382—24no—
47C2×#10624382—42no—
48C2×#10624382—42no—
49C2×#10624382—42no—
50C2×#10624382—42no—
51C2×#10624382—42no—
52C22×D181—41no—
53C22×D181—41no—
54C22×D181C2×D411yescharacteristic
55C22×D181C2×D411yescharacteristic
56C22×D181C2×Q812yes—
57C22×D181#4112yes—
58C2×Dic181—22no—
59C2×Dic181—41no—
60C2×Dic181—42no—
61C2×Dic181C22⋊C411yescharacteristic
62C2×Dic181C22⋊C411yescharacteristic
63C4×D181—22no—
64C4×D181—22no—
65C4×D181—41no—
66C4×D181—41no—
67C23×C181C22×C411yescharacteristic
68Q8×C181—42no—
69Q8×C181—42no—
70C2×C4×C181—22no—
71C2×C4×C181—22no—
72C2×C4×C181—42no—
73C2×C4×C181C22⋊C411yescharacteristic
74C2×C4×C181C22⋊C411yescharacteristic
75C22×C181—21no—
76C22×C181—21no—
77C22×C181C2×C22⋊C411yescharacteristic
78C22×C181C4×Q812yes—
79C2×D181—21no—
80C2×D181—21no—
81C2×D181—21no—
82C2×D181—21no—
83C2×D181—42no—
84C2×D181—42no—
85C2×D181C22⋊Q812yes—
86C2×D181#12411yescharacteristic
87C2×D181C4⋊Q811yescharacteristic
88#10624382—44no—
89#10624382—82no—
90C4×C181—21no—
91C4×C181—42no—
92C4×C181#9912yes—
93Dic181—21no—
94Dic181—21no—
95Dic181—42no—
96C2×C181—22no—
97C2×C181#39111yescharacteristic
98C2×C181C2×#9911yescharacteristic
99C2×C181#41011yescharacteristic
100D181—22no—
101D181—21no—
102D181—21no—
103C181#199611yescharacteristic, complemented, Sylow
104#1996—1811nomaximal, Sylow
105C2×C22⋊Q8—1811no—
106#380—1811no—
107C4×C22⋊C4—1811no—
108#342—1814no—
109C22×Q8#1062438211yescharacteristic, complemented
110C23×C4—1811no—
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⋊C4—1812no—
121C2×C42—3621no—
122#95—3621no—
123#95—3621no—
124C24—1811no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—3622no—
129C22×C4—3622no—
130C22×C4—1811no—
131C22×C4—1811no—
132C22×C4—1812no—
133C22×C4—1811no—
134C22×C4—1811no—
135C22×C4—1811no—
136C22×C4—7242no—
137C22×C4C2×#1062438212yes—
138C22×C4C2×#1062438211yescharacteristic
139C4⋊C4—3622no—
140C4⋊C4—3622no—
141C4⋊C4—3621no—
142C4⋊C4—3621no—
143C4⋊C4—7242no—
144C22⋊C4—3622no—
145C22⋊C4—3622no—
146C22⋊C4—3622no—
147C22⋊C4—3624no—
148C22⋊C4—1814no—
149C42—7242no—
150C23—1811no—
151C23—1811no—
152C23—1812no—
153C23—1812no—
154C23—7241no—
155C23—7241no—
156C23C22×#1062438211yescharacteristic
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—3622no—
163C2×C4—3622no—
164C2×C4—3624no—
165C2×C4—3622no—
166C2×C4—1811no—
167C2×C4—1811no—
168C2×C4—7241no—
169C2×C4—7241no—
170C2×C4—7241no—
171C2×C4—7242no—
172C2×C4—7242no—
173C2×C4—7242no—
174C2×C4—7242no—
175C2×C4—7242no—
176C2×C4—7242no—
177C2×C4C22⋊#1062438211yescharacteristic
178C2×C4C22⋊#1062438211yescharacteristic
179C22—21no—
180C22—21no—
181C22—3621no—
182C22—3621no—
183C22—3621no—
184C22—3621no—
185C22—1812no—
186C22—1811no—
187C22—1811no—
188C22—7242no—
189C22—7242no—
190C22C2×C22⋊#1062438211yescharacteristic
191C22Q8×#1062438212yes—
192C4—21no—
193C4—42no—
194C4—3621no—
195C4—3621no—
196C4—14482no—
197C4—7244no—
198C4—7242no—
199C4#1915387412yes—
200C2—22no—
201C2—3621no—
202C2—3622no—
203C2—3621no—
204C2(C2×#10624382)⋊Q811yescharacteristic
205C2C2×#1915387411yescharacteristic
206C2#2604226011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC22×Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radical, cosocleC23×D181C23
hypercenterC22×Q8#10624382
hypocenterC181#1996
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊Dic181C4semidirect
17. C2×#15668428C4semidirect
19. C22×Q8×C181C4semidirect
25. C22⋊#10624382Q8semidirect
103. C181#1996semidirect
109. C22×Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422212
C4×C22⋊#106243822
#260421572
C2×#191537922
#1996181