Finite Group Databasebeta

← #36365356

Subgroups — #36365356

Order 22144.

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#26012297C214yesmaximal, complemented
2#26011751C214yesmaximal, complemented
3#26011673C212yesmaximal, complemented
4#26011658C212yesmaximal, complemented
5#26011609C212yesmaximal, complemented
6#492×C173C211yesmaximal, characteristic, complemented
7#19149727C2212yes—
8#19149724C2212yescomplemented
9#19149710C2214yescomplemented
10#19149708—24no—
11#19149706—24no—
12#19149705C2214yes—
13#19149701C2212yes—
14#19149700—24no—
15SD16×D173—24no—
16#19149685C2214yescomplemented
17#19149615C2212yescomplemented
18#19149607C2214yescomplemented
19#19149598C2212yescomplemented
20#19149596C2212yescomplemented
21C2×SD16×C173C2214yescomplemented
22C4⋊Q8×C173C2211yescharacteristic, complemented
23#127×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic, complemented
25#14583827—24no—
26#14583826—42no—
27#14583826C2314yes—
28#14583825—24no—
29Q8×D173—24no—
30C2×C22⋊D173—42no—
31D4×D173—24no—
32Dic692D412yescomplemented
33#14583816—24no—
34#14583815D414yescomplemented
35#14583814—24no—
36#14583813—24no—
37D1384D412yescomplemented
38#14583811—24no—
39#14583810C2314yes—
40#14583809—24no—
41C8×D173—24no—
42C2×Q8×C173C2312yes—
43C2×D4×C173—41no—
44C2×D4×C173C2312yes—
45SD16×C173—24no—
46SD16×C173—24no—
47C2×C8×C173C2312yes—
48C4⋊C4×C173—41no—
49C42×C173C2311yescharacteristic
50C22×D173—42no—
51Dic346—42no—
52Dic346C2×D414yes—
53D692—42no—
54D692C2×D414yes—
55C22⋊D173—24no—
56C22⋊D173—44no—
57C2×Dic173—42no—
58C4×D173—24no—
59C4×D173—44no—
60#13063436—24no—
61C23×C173—22no—
62Q8×C173—24no—
63D4×C173—24no—
64D4×C173—82no—
65C2×C4×C173—21no—
66C2×C4×C173—22no—
67C2×C4×C173C2411yescharacteristic
68C8×C173C2×D412yes—
69C8×C173C2×D412yes—
70C22×C173—42no—
71C22×C173—82no—
72C22×C173H(5,𝔽2)11yescharacteristic
73C2×D173—24no—
74C2×D173—162no—
75C4×C173—22no—
76C4×C173—42no—
77C4×C173C22×D412yes—
78Dic173—24no—
79C2×C173—21no—
80C2×C173—82no—
81C2×C173D4211yescharacteristic
82D173—44no—
83C173#329111yescharacteristic, complemented, Sylow
84#3291—1731nomaximal, Sylow
85#577—1734no—
86#492D17311yescharacteristic, complemented
87#471—1734no—
88#456—1732no—
89#453—1732no—
90#443—1732no—
912−1+4—1732no—
92H(5,𝔽2)—1732no—
93#137—3464no—
94#137—1734no—
95AGL(1,ℤ/8)—3464no—
96AGL(1,ℤ/8)—1734no—
97#135—3464no—
98#135—1732no—
99C2×SD16—3464no—
100C2×SD16C2×D17314yescomplemented
101#131—1734no—
102C4⋊Q8C2×D17311yescharacteristic, complemented
103#127C2×D17311yescharacteristic
104#108—1732no—
105#104—1734no—
106#101—1732no—
107#100—1732no—
108C4×C8C2×D17311yescharacteristic, complemented
109#41—3464no—
110#41—3464no—
111#41—1734no—
112#41—6922no—
113C2×Q8—3464no—
114C2×Q8C22×D17312yes—
115C2×D4—41no—
116C2×D4—3464no—
117C2×D4—6922no—
118C2×D4C22×D17312yes—
119Q16—3464no—
120Q16—1732no—
121SD16—24no—
122SD16—24no—
123SD16—3464no—
124SD16—3464no—
125SD16—1734no—
126D8—3464no—
127D8—1732no—
128M16—3464no—
129M16—1734no—
130C2×C8—3464no—
131C2×C8C22×D17312yes—
132C4⋊C4—41no—
133C42C22×D17311yescharacteristic
134C23—22no—
135C23—6922no—
136Q8—24no—
137Q8—1734no—
138Q8—6922no—
139D4—24no—
140D4—82no—
141D4—3464no—
142D4—1734no—
143D4—6922no—
144D4—6924no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—3464no—
148C2×C4—6924no—
149C2×C4—6922no—
150C2×C4C23×D17311yescharacteristic
151C8—3464no—
152C8D4×D17312yes—
153C8D4×D17312yes—
154C22—42no—
155C22—82no—
156C22—3464no—
157C22—27682no—
158C22#1914972411yescharacteristic
159C4—22no—
160C4—42no—
161C4—3464no—
162C4C2×D4×D17312yes—
163C2—21no—
164C2—82no—
165C2—6924no—
166C2#2601212611yescharacteristic
subgroupG/H
centerC2#26012126
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#492×C173C2
socleC2×C173D42
hypercenter#492D173
hypocenterC173#3291
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #26012297C2semidirect
2. #26011751C2semidirect
3. #26011673C2semidirect
4. #26011658C2semidirect
5. #26011609C2semidirect
6. #492×C173C2semidirect
8. #19149724C22semidirect
9. #19149710C22semidirect
16. #19149685C22semidirect
17. #19149615C22semidirect
18. #19149607C22semidirect
19. #19149598C22semidirect
20. #19149596C22semidirect
21. C2×SD16×C173C22semidirect
22. C4⋊Q8×C173C22semidirect
24. C4×C8×C173C22semidirect
32. Dic692D4semidirect
34. #14583815D4semidirect
37. D1384D4semidirect
83. C173#3291semidirect
86. #492D173semidirect
100. C2×SD16C2×D173semidirect
102. C4⋊Q8C2×D173semidirect
108. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122972
#260117512
#260116732
#260116582
#260116092
#492×C1732
#3291173