Finite Group Databasebeta

← #37545514

Subgroups — #37545514

Order 23168.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#26042085C214yesmaximal, complemented
2#26041539C214yesmaximal, complemented
3#26041461C212yesmaximal, complemented
4#26041446C212yesmaximal, complemented
5#26041397C212yesmaximal, complemented
6#492×C181C211yesmaximal, characteristic, complemented
7#19153864C2212yes—
8#19153861C2212yescomplemented
9#19153847C2214yescomplemented
10#19153845—24no—
11#19153843—24no—
12#19153842C2214yes—
13#19153838C2212yes—
14#19153837—24no—
15SD16×D181—24no—
16#19153822C2214yescomplemented
17#19153752C2212yescomplemented
18#19153744C2214yescomplemented
19#19153735C2212yescomplemented
20#19153733C2212yescomplemented
21C2×SD16×C181C2214yescomplemented
22C4⋊Q8×C181C2211yescharacteristic, complemented
23#127×C181C2211yescharacteristic
24C4×C8×C181C2211yescharacteristic, complemented
25#15668450—24no—
26#15668449—42no—
27#15668449C2314yes—
28#15668448—24no—
29Q8×D181—24no—
30C2×C22⋊D181—42no—
31D4×D181—24no—
32Dic724D412yescomplemented
33#15668439—24no—
34#15668438D414yescomplemented
35#15668437—24no—
36#15668436—24no—
37D1448D412yescomplemented
38#15668434—24no—
39#15668433C2314yes—
40#15668432—24no—
41C8×D181—24no—
42C2×Q8×C181C2312yes—
43C2×D4×C181—41no—
44C2×D4×C181C2312yes—
45SD16×C181—24no—
46SD16×C181—24no—
47C2×C8×C181C2312yes—
48C4⋊C4×C181—41no—
49C42×C181C2311yescharacteristic
50C22×D181—42no—
51Dic362—42no—
52Dic362C2×D414yes—
53D724—42no—
54D724C2×D414yes—
55C22⋊D181—24no—
56C22⋊D181—44no—
57C2×Dic181—42no—
58C4×D181—24no—
59C4×D181—44no—
60#13089678—24no—
61C23×C181—22no—
62Q8×C181—24no—
63D4×C181—24no—
64D4×C181—82no—
65C2×C4×C181—21no—
66C2×C4×C181—22no—
67C2×C4×C181C2411yescharacteristic
68C8×C181C2×D412yes—
69C8×C181C2×D412yes—
70C22×C181—42no—
71C22×C181—82no—
72C22×C181H(5,𝔽2)11yescharacteristic
73C2×D181—24no—
74C2×D181—162no—
75C4×C181—22no—
76C4×C181—42no—
77C4×C181C22×D412yes—
78Dic181—24no—
79C2×C181—21no—
80C2×C181—82no—
81C2×C181D4211yescharacteristic
82D181—44no—
83C181#329111yescharacteristic, complemented, Sylow
84#3291—1811nomaximal, Sylow
85#577—1814no—
86#492D18111yescharacteristic, complemented
87#471—1814no—
88#456—1812no—
89#453—1812no—
90#443—1812no—
912−1+4—1812no—
92H(5,𝔽2)—1812no—
93#137—3624no—
94#137—1814no—
95AGL(1,ℤ/8)—3624no—
96AGL(1,ℤ/8)—1814no—
97#135—3624no—
98#135—1812no—
99C2×SD16—3624no—
100C2×SD16C2×D18114yescomplemented
101#131—1814no—
102C4⋊Q8C2×D18111yescharacteristic, complemented
103#127C2×D18111yescharacteristic
104#108—1812no—
105#104—1814no—
106#101—1812no—
107#100—1812no—
108C4×C8C2×D18111yescharacteristic, complemented
109#41—3624no—
110#41—3624no—
111#41—1814no—
112#41—7242no—
113C2×Q8—3624no—
114C2×Q8C22×D18112yes—
115C2×D4—41no—
116C2×D4—3624no—
117C2×D4—7242no—
118C2×D4C22×D18112yes—
119Q16—3624no—
120Q16—1812no—
121SD16—24no—
122SD16—24no—
123SD16—3624no—
124SD16—3624no—
125SD16—1814no—
126D8—3624no—
127D8—1812no—
128M16—3624no—
129M16—1814no—
130C2×C8—3624no—
131C2×C8C22×D18112yes—
132C4⋊C4—41no—
133C42C22×D18111yescharacteristic
134C23—22no—
135C23—7242no—
136Q8—24no—
137Q8—1814no—
138Q8—7242no—
139D4—24no—
140D4—82no—
141D4—3624no—
142D4—1814no—
143D4—7242no—
144D4—7244no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—3624no—
148C2×C4—7244no—
149C2×C4—7242no—
150C2×C4C23×D18111yescharacteristic
151C8—3624no—
152C8D4×D18112yes—
153C8D4×D18112yes—
154C22—42no—
155C22—82no—
156C22—28962no—
157C22—3624no—
158C22#1915386111yescharacteristic
159C4—22no—
160C4—42no—
161C4—3624no—
162C4C2×D4×D18112yes—
163C2—21no—
164C2—82no—
165C2—7244no—
166C2#2604191411yescharacteristic
subgroupG/H
centerC2#26041914
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#492×C181C2
socleC2×C181D42
hypercenter#492D181
hypocenterC181#3291
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #26042085C2semidirect
2. #26041539C2semidirect
3. #26041461C2semidirect
4. #26041446C2semidirect
5. #26041397C2semidirect
6. #492×C181C2semidirect
8. #19153861C22semidirect
9. #19153847C22semidirect
16. #19153822C22semidirect
17. #19153752C22semidirect
18. #19153744C22semidirect
19. #19153735C22semidirect
20. #19153733C22semidirect
21. C2×SD16×C181C22semidirect
22. C4⋊Q8×C181C22semidirect
24. C4×C8×C181C22semidirect
32. Dic724D4semidirect
34. #15668438D4semidirect
37. D1448D4semidirect
83. C181#3291semidirect
86. #492D181semidirect
100. C2×SD16C2×D181semidirect
102. C4⋊Q8C2×D181semidirect
108. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420852
#260415392
#260414612
#260414462
#260413972
#492×C1812
#3291181