Finite Group Databasebeta

← #36366701

Subgroups — #36366701

Order 22144.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#26012293C212yesmaximal, complemented
2#26011928C214yesmaximal, complemented
3#26011926C212yesmaximal, complemented
4#26011924C212yesmaximal, complemented
5C2×D2768C211yesmaximal, characteristic, complemented
6D16×D173C212yesmaximal, complemented
7#26011907C211yesmaximal, characteristic, complemented
8#508×C173C211yesmaximal, characteristic, complemented
9#19149725—22no—
10#19149706—24no—
11#19149704—22no—
12#19149702C2214yescomplemented
13C2×D1384C2212yescomplemented
14D8×D173C2214yescomplemented
15#19149685C2211yescharacteristic
16#19149622C2214yescomplemented
17D2768C2212yescomplemented
18D2768C2212yescomplemented
19#19149620C2214yescomplemented
20#19149619C2211yescharacteristic, complemented
21#19149618C2212yescomplemented
22C16×D173C2212yescomplemented
23#135×C173C2212yescomplemented
24Q32×C173C2211yescharacteristic, complemented
25SD32×C173C2212yescomplemented
26D16×C173C2211yescharacteristic, complemented
27C2×C16×C173C2211yescharacteristic, complemented
28#14583827—24no—
29#14583826—42no—
30#14583826D411yescharacteristic
31C2×D692—22no—
32C2×D692—42no—
33D4×D173—24no—
34D4×D173—44no—
35#14583813—24no—
36D1384—22no—
37D1384C2314yes—
38#14583811—24no—
39#14583810D411yescharacteristic
40#14583809D412yes—
41C8×D173C2312yes—
42#14583801C2312yes—
43#41×C173—22no—
44Q16×C173C2312yes—
45SD16×C173—22no—
46D8×C173C2312yes—
47C2×C8×C173C2311yescharacteristic
48C16×C173C2311yescharacteristic
49C16×C173C2311yescharacteristic
50C22×D173—42no—
51C22×D173—44no—
52Dic346D811yescharacteristic
53D692—24no—
54D692—44no—
55D692—42no—
56D692D811yescharacteristic
57C22⋊D173—44no—
58C22⋊D173D812yes—
59C4×D173—44no—
60C4×D173C2×D412yes—
61#13063436C2×D412yes—
62Q8×C173—22no—
63D4×C173—22no—
64D4×C173—42no—
65C2×C4×C173—42no—
66C2×C4×C173C2×D411yescharacteristic
67C8×C173C2×D411yescharacteristic
68C8×C173C2411yescharacteristic
69C22×C173—42no—
70C22×C173C2×D811yescharacteristic
71C2×D173—44no—
72C2×D173—164no—
73C2×D173—162no—
74C2×D173C2×D812yes—
75C4×C173—42no—
76C4×C173C2×D811yescharacteristic
77C4×C173C22×D411yescharacteristic
78Dic173C2×D812yes—
79C2×C173—21no—
80C2×C173—82no—
81C2×C173C22×D811yescharacteristic
82D173—22no—
83D173—84no—
84C173#341511yescharacteristic, complemented, Sylow
85#3415—1731nomaximal, Sylow
86#576—1732no—
87#509—1732no—
88#509—1734no—
89#508—1732no—
90#508D17311yescharacteristic, complemented
91C2×D16—1732no—
92C2×D16—1731no—
93#504—1731no—
94H(5,𝔽2)—3462no—
95AGL(1,ℤ/8)—3464no—
96AGL(1,ℤ/8)—3462no—
97#135—1734no—
98#135C2×D17312yescomplemented
99C2×D8—1734no—
100C2×D8—1732no—
101#131—1731no—
102Q32C2×D17311yescharacteristic, complemented
103SD32—1734no—
104SD32C2×D17312yescomplemented
105D16—1734no—
106D16—1732no—
107D16—1732no—
108D16C2×D17311yescharacteristic, complemented
109M32—1731no—
110M32—1732no—
111C2×C16—1732no—
112C2×C16C2×D17311yescharacteristic, complemented
113#41—22no—
114#41—3464no—
115#41—1731no—
116#41—6922no—
117C2×D4—3464no—
118C2×D4—3462no—
119C2×D4—6922no—
120C2×D4—6924no—
121Q16C22×D17312yes—
122SD16—22no—
123SD16—3464no—
124D8—3464no—
125D8—3462no—
126D8—1734no—
127D8C22×D17312yes—
128M16—1731no—
129M16—1732no—
130C2×C8—1732no—
131C2×C8C22×D17311yescharacteristic
132C16—1732no—
133C16C22×D17311yescharacteristic
134C16C22×D17311yescharacteristic
135C23—6924no—
136C23—6922no—
137Q8—22no—
138Q8—1731no—
139D4—22no—
140D4—42no—
141D4—3464no—
142D4—1732no—
143D4—1731no—
144D4—6924no—
145D4—6924no—
146D4—6922no—
147C2×C4—42no—
148C2×C4—1732no—
149C2×C4—6924no—
150C2×C4D4×D17311yescharacteristic
151C8—1732no—
152C8D4×D17311yescharacteristic
153C8C23×D17311yescharacteristic
154C22—42no—
155C22—1732no—
156C22—6924no—
157C22—27682no—
158C22—27684no—
159C22D8×D17311yescharacteristic
160C4—42no—
161C4—1732no—
162C4D8×D17311yescharacteristic
163C4C2×D4×D17311yescharacteristic
164C2—21no—
165C2—82no—
166C2—3462no—
167C2—13844no—
168C2C2×D8×D17311yescharacteristic
subgroupG/H
centerC2C2×D8×D173
commutator, cosocleC8×C173C24
FrattiniC8C23×D173
Fitting#508×C173C2
socleC2×C173C22×D8
hypercenter#508D173
hypocenterC173#3415
LCSG ▷ C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #26012293C2semidirect
2. #26011928C2semidirect
3. #26011926C2semidirect
4. #26011924C2semidirect
5. C2×D2768C2semidirect
6. D16×D173C2semidirect
7. #26011907C2semidirect
8. #508×C173C2semidirect
12. #19149702C22semidirect
13. C2×D1384C22semidirect
14. D8×D173C22semidirect
16. #19149622C22semidirect
17. D2768C22semidirect
18. D2768C22semidirect
19. #19149620C22semidirect
20. #19149619C22semidirect
21. #19149618C22semidirect
22. C16×D173C22semidirect
23. #135×C173C22semidirect
24. Q32×C173C22semidirect
25. SD32×C173C22semidirect
26. D16×C173C22semidirect
27. C2×C16×C173C22semidirect
84. C173#3415semidirect
90. #508D173semidirect
98. #135C2×D173semidirect
102. Q32C2×D173semidirect
104. SD32C2×D173semidirect
108. D16C2×D173semidirect
112. C2×C16C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122932
#260119282
#260119262
#260119242
C2×D27682
D16×D1732
#260119072
#508×C1732
#3415173