Finite Group Databasebeta

← #37544185

Subgroups — #37544185

Order 23168.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153847C212yesmaximal, complemented
2C2×#19153842C212yesmaximal, complemented
3#26041829C211yesmaximal, characteristic, complemented
4#26041674C212yesmaximal, complemented
5#26041669C212yesmaximal, complemented
6#26041661C214yesmaximal, complemented
7C4×#15668433C211yesmaximal, characteristic, complemented
8C2×#124×C181C211yesmaximal, characteristic, complemented
9C2×#15668449C2212yes—
10#19153847—28no—
11#19153842—28no—
12C2×#15668439C2214yescomplemented
13C2×#15668437C2214yes—
14C2×#15668436C2214yescomplemented
15C2×#15668434C2214yes—
16C2×#15668433C2212yes—
17#19153816C2211yescharacteristic, complemented
18#19153813C2211yescharacteristic, complemented
19#19153804C2212yescomplemented
20#19153790—22no—
21C4×D724—22no—
22#19153727C2212yescomplemented
23C4×#13089678C2212yescomplemented
24C22×Q8×C181C2211yescharacteristic
25C22×D4×C181C2211yescharacteristic
26#124×C181—22no—
27#124×C181C2212yes—
28#124×C181C2212yes—
29C2×C22⋊C4×C181—22no—
30C2×C42×C181C2211yescharacteristic
31#15668449—24no—
32#15668449—24no—
33C2×Dic362C2312yes—
34C2×D724—22no—
35C2×D724C2312yes—
36C2×C22⋊D181—22no—
37C2×C4×D181—22no—
38#15668439—28no—
39#15668437—28no—
40#15668436—28no—
41#15668434—28no—
42#15668433D418yescomplemented
43C2×#13089678C2314yes—
44C4⋊Dic181—22no—
45#15668426—24no—
46C24×C181—21no—
47C2×Q8×C181—21no—
48C2×Q8×C181—24no—
49C2×Q8×C181C2312yes—
50C2×D4×C181—24no—
51C2×D4×C181—21no—
52C2×D4×C181C2312yes—
53C22×C4×C181—21no—
54C22×C4×C181D412yes—
55C22×C4×C181C2311yescharacteristic
56C22⋊C4×C181—24no—
57C22⋊C4×C181—24no—
58C42×C181D412yes—
59C42×C181C2311yescharacteristic
60C42×C181C2311yescharacteristic
61C22×D181—22no—
62Dic362—22no—
63Dic362—22no—
64D724—22no—
65D724—22no—
66D724—44no—
67C22⋊D181—44no—
68C2×Dic181—22no—
69C4×D181—44no—
70#13089678C2×D418yes—
71C23×C181—22no—
72C23×C181—44no—
73C23×C181—42no—
74C23×C181C2×D411yescharacteristic
75Q8×C181—24no—
76Q8×C181—42no—
77D4×C181—24no—
78D4×C181—42no—
79C2×C4×C181—22no—
80C2×C4×C181—42no—
81C2×C4×C181—42no—
82C2×C4×C181C2×D414yes—
83C2×C4×C181C2×D412yes—
84C2×C4×C181C2×D412yes—
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2411yescharacteristic
87C22×C181—22no—
88C22×C181—42no—
89C22×C181—44no—
90C22×C181—84no—
91C22×C181#12712yes—
92C22×C181C22×D411yescharacteristic
93C2×D181—42no—
94C2×D181—44no—
95C4×C181—22no—
96C4×C181—22no—
97C4×C181—42no—
98C4×C181#12712yes—
99C4×C181C22×D412yes—
100Dic181—42no—
101C2×C181—22no—
102C2×C181—82no—
103C2×C181C2×#12711yescharacteristic
104C2×C181#57512yes—
105D181—82no—
106C181#315311yescharacteristic, complemented, Sylow
107#3153—1811nomaximal, Sylow
108C2×#137—1812no—
109C2×AGL(1,ℤ/8)—1812no—
110#532—1811no—
111C2×#124D18111yescharacteristic, complemented
112#497—1812no—
113#496—1812no—
114#495—1814no—
115C4×M16—1811no—
116C2×#41—1812no—
117C22×Q8C2×D18111yescharacteristic
118C22×D4C2×D18111yescharacteristic
119#137—3628no—
120AGL(1,ℤ/8)—3628no—
121C2×Q16—1814no—
122C2×SD16—1814no—
123C2×SD16—1814no—
124C2×D8—1814no—
125C2×M16—1812no—
126C4⋊Q8—1811no—
127#127—1811no—
128#124—22no—
129#124—1812no—
130#124C2×D18112yes—
131#124C2×D18112yes—
132#121—3622no—
133C4×D4—3622no—
134C2×C22⋊C4—22no—
135C2×C42C2×D18111yescharacteristic
136#97—1812no—
137C4×C8—1812no—
138C24—21no—
139#41—3624no—
140#41—3624no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—1812no—
144C2×Q8C22×D18112yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3622no—
148C2×D4—3622no—
149C2×D4—1812no—
150C2×D4C22×D18112yes—
151C22×C4—21no—
152C22×C4—3622no—
153C22×C4C22⋊D18112yes—
154C22×C4C22×D18111yescharacteristic
155Q16—3628no—
156SD16—3628no—
157SD16—3628no—
158D8—3628no—
159M16—1818no—
160C2×C8—1814no—
161C4⋊C4—3622no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—3624no—
165C42C22⋊D18112yes—
166C42C22×D18111yescharacteristic
167C42C22×D18111yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—3622no—
172C23C2×C22⋊D18111yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—3622no—
176Q8—3622no—
177D4—24no—
178D4—42no—
179D4—3622no—
180D4—3622no—
181D4—7244no—
182D4—7244no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—3622no—
187C2×C4—7244no—
188C2×C4D4×D18114yes—
189C2×C4C2×C22⋊D18112yes—
190C2×C4C2×C22⋊D18112yes—
191C2×C4C2×C22⋊D18111yescharacteristic
192C2×C4C23×D18111yescharacteristic
193C8—1818no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—7242no—
199C22—7244no—
200C22#1915381212yes—
201C22C22×C22⋊D18111yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—7242no—
206C4#1915381212yes—
207C4C2×D4×D18112yes—
208C2—22no—
209C2—82no—
210C2—14482no—
211C2C2×#1915381211yescharacteristic
212C2#2604207712yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#124×C181C2
socleC22×C181C22×D4
hypercenterC2×#124D181
hypocenterC181#3153
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#19153847C2semidirect
2. C2×#19153842C2semidirect
3. #26041829C2semidirect
4. #26041674C2semidirect
5. #26041669C2semidirect
6. #26041661C2semidirect
7. C4×#15668433C2semidirect
8. C2×#124×C181C2semidirect
12. C2×#15668439C22semidirect
14. C2×#15668436C22semidirect
17. #19153816C22semidirect
18. #19153813C22semidirect
19. #19153804C22semidirect
22. #19153727C22semidirect
23. C4×#13089678C22semidirect
42. #15668433D4semidirect
106. C181#3153semidirect
111. C2×#124D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538472
C2×#191538422
#260418292
#260416742
#260416692
#260416612
C4×#156684332
C2×#124×C1812
#3153181