Finite Group Databasebeta

← #37544187

Subgroups — #37544187

Order 23168.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19153849C212yesmaximal, complemented
2C2×#19153844C212yesmaximal, complemented
3#26041829C211yesmaximal, characteristic, complemented
4C2×#19153804C211yesmaximal, characteristic, complemented
5#26041673C212yesmaximal, complemented
6#26041668C212yesmaximal, complemented
7#26041663C214yesmaximal, complemented
8C4×M16×C181C211yesmaximal, characteristic, complemented
9C2×#15668449C2212yes—
10C22×Dic362C2211yescharacteristic
11C22×D724C2211yescharacteristic
12#19153849—28no—
13#19153844—28no—
14C2×Dic724C2214yescomplemented
15C2×#15668438C2214yes—
16C2×#15668438C2214yescomplemented
17C2×D1448C2214yes—
18#19153816C2211yescharacteristic, complemented
19#19153813C2211yescharacteristic, complemented
20#19153804—22no—
21#19153804C2212yes—
22#19153804C2212yes—
23#19153804C2212yescomplemented
24#19153790—22no—
25C4×D724—22no—
26C2×#15668426—22no—
27C2×M16×C181C2212yes—
28C2×C42×C181C2211yescharacteristic
29#97×C181C2212yescomplemented
30C4×C8×C181C2212yescomplemented
31C23×D181—21no—
32#15668449—24no—
33#15668449—24no—
34C2×Dic362—21no—
35C2×Dic362—24no—
36C2×Dic362C2312yes—
37C2×Dic362C2312yes—
38C2×D724—24no—
39C2×D724—22no—
40C2×D724—21no—
41C2×D724C2312yes—
42C2×D724C2312yes—
43C2×C22⋊D181—22no—
44C22×Dic181—21no—
45C2×C4×D181—22no—
46Dic724—28no—
47#15668438—28no—
48#15668438—28no—
49D1448—28no—
50C4⋊Dic181—22no—
51#15668426—24no—
52#15668426—24no—
53#15668426—24no—
54C22×C4×C181D412yes—
55C22×C4×C181C2311yescharacteristic
56M16×C181D418yescomplemented
57C2×C8×C181C2314yes—
58C42×C181D412yes—
59C42×C181C2311yescharacteristic
60C42×C181C2311yescharacteristic
61C22×D181—22no—
62C22×D181—22no—
63C22×D181—44no—
64C22×D181—42no—
65Dic362—22no—
66Dic362—24no—
67Dic362—22no—
68Dic362—42no—
69D724—22no—
70D724—24no—
71D724—22no—
72D724—44no—
73D724—42no—
74C22⋊D181—44no—
75C2×Dic181—22no—
76C2×Dic181—22no—
77C2×Dic181—42no—
78C4×D181—44no—
79C23×C181C2×D411yescharacteristic
80C2×C4×C181—42no—
81C2×C4×C181C2×D412yes—
82C2×C4×C181C2×D412yes—
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D414yes—
85C2×C4×C181C2411yescharacteristic
86C8×C181C2×D418yes—
87C22×C181—22no—
88C22×C181#12712yes—
89C22×C181C22×D411yescharacteristic
90C2×D181—42no—
91C2×D181—42no—
92C2×D181—44no—
93C2×D181—44no—
94C2×D181—84no—
95C4×C181—22no—
96C4×C181—22no—
97C4×C181#12712yes—
98C4×C181C22×D412yes—
99Dic181—42no—
100Dic181—42no—
101C2×C181—22no—
102C2×C181C2×#12711yescharacteristic
103C2×C181#57512yes—
104D181—82no—
105D181—82no—
106C181#315311yescharacteristic, complemented, Sylow
107#3153—1811nomaximal, Sylow
108C2×#137—1812no—
109C2×AGL(1,ℤ/8)—1812no—
110#532—1811no—
111C2×#124—1811no—
112#497—1812no—
113#496—1812no—
114#495—1814no—
115C4×M16D18111yescharacteristic, complemented
116C2×#41—1812no—
117C22×Q8—1811no—
118C22×D4—1811no—
119#137—3628no—
120AGL(1,ℤ/8)—3628no—
121C2×Q16—1814no—
122C2×SD16—1814no—
123C2×SD16—1814no—
124C2×D8—1814no—
125C2×M16C2×D18112yes—
126C4⋊Q8—1811no—
127#127—1811no—
128#124—3622no—
129#124—1812no—
130#124—1812no—
131#124—1812no—
132#121—3622no—
133C4×D4—3622no—
134C2×C22⋊C4—3622no—
135C2×C42C2×D18111yescharacteristic
136#97C2×D18112yescomplemented
137C4×C8C2×D18112yescomplemented
138C24—3621no—
139#41—3624no—
140#41—3624no—
141C2×Q8—3621no—
142C2×Q8—3624no—
143C2×Q8—1812no—
144C2×Q8—1812no—
145C2×D4—3622no—
146C2×D4—3624no—
147C2×D4—3622no—
148C2×D4—3621no—
149C2×D4—1812no—
150C2×D4—1812no—
151C22×C4—3621no—
152C22×C4—3622no—
153C22×C4D72412yes—
154C22×C4C22×D18111yescharacteristic
155Q16—3628no—
156SD16—3628no—
157SD16—3628no—
158D8—3628no—
159M16D72418yescomplemented
160C2×C8C22×D18114yes—
161C4⋊C4—3622no—
162C22⋊C4—3624no—
163C22⋊C4—3624no—
164C22⋊C4—3624no—
165C42D72412yes—
166C42C22×D18111yescharacteristic
167C42C22×D18111yescharacteristic
168C23—3622no—
169C23—3622no—
170C23—7244no—
171C23—7242no—
172C23C2×D72411yescharacteristic
173Q8—3622no—
174Q8—3624no—
175Q8—3622no—
176Q8—7242no—
177D4—3622no—
178D4—3624no—
179D4—3622no—
180D4—7244no—
181D4—7244no—
182D4—7242no—
183C2×C4—42no—
184C2×C4—3622no—
185C2×C4—3622no—
186C2×C4—7242no—
187C2×C4—7244no—
188C2×C4C2×D72412yes—
189C2×C4C2×D72412yes—
190C2×C4C2×D72411yescharacteristic
191C2×C4C2×D72414yes—
192C2×C4C23×D18111yescharacteristic
193C8C2×D72418yes—
194C22—22no—
195C22—14484no—
196C22—7242no—
197C22—7242no—
198C22—7244no—
199C22—7244no—
200C22#1915381312yes—
201C22C22×D72411yescharacteristic
202C4—22no—
203C4—22no—
204C4—7242no—
205C4—7242no—
206C4#1915381312yes—
207C4C22×D72412yes—
208C2—22no—
209C2—14482no—
210C2—14482no—
211C2C2×#1915381311yescharacteristic
212C2#2604207912yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×M16×C181C2
socleC22×C181C22×D4
hypercenterC4×M16D181
hypocenterC181#3153
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#19153849C2semidirect
2. C2×#19153844C2semidirect
3. #26041829C2semidirect
4. C2×#19153804C2semidirect
5. #26041673C2semidirect
6. #26041668C2semidirect
7. #26041663C2semidirect
8. C4×M16×C181C2semidirect
14. C2×Dic724C22semidirect
16. C2×#15668438C22semidirect
18. #19153816C22semidirect
19. #19153813C22semidirect
23. #19153804C22semidirect
29. #97×C181C22semidirect
30. C4×C8×C181C22semidirect
56. M16×C181D4semidirect
106. C181#3153semidirect
115. C4×M16D181semidirect
136. #97C2×D181semidirect
137. C4×C8C2×D181semidirect
159. M16D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C2×#191538442
#260418292
C2×#191538042
#260416732
#260416682
#260416632
C4×M16×C1812
#3153181