Finite Group Databasebeta

← #31793926

Subgroups — #31793926

Order 17536.

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#22578204C212yesmaximal, complemented
2#22577839C212yesmaximal, complemented
3#22577837C214yesmaximal, complemented
4#22577835C211yesmaximal, characteristic, complemented
5#22577832C212yesmaximal, complemented
6D16×D137C212yesmaximal, complemented
7#22577819C211yesmaximal, characteristic, complemented
8C2×D16×C137C211yesmaximal, characteristic, complemented
9#17984133—22no—
10#17984114—22no—
11#17984112—24no—
12#17984110C2212yescomplemented
13#17984107C2214yescomplemented
14D8×D137C2214yescomplemented
15#17984094C2211yescharacteristic
16Dic1096C2211yescharacteristic, complemented
17#17984033C2212yescomplemented
18#17984032C2214yescomplemented
19D2192C2211yescharacteristic, complemented
20#17984029C2214yescomplemented
21#17984028C2211yescharacteristic, complemented
22#17984027C2212yescomplemented
23C16×D137C2212yescomplemented
24C2×D8×C137C2212yescomplemented
25D16×C137C2212yescomplemented
26D16×C137C2212yescomplemented
27C2×C16×C137C2211yescharacteristic, complemented
28#14555510—22no—
29#14555510D411yescharacteristic
30#14555509—24no—
31#14555509—42no—
32C2×C22⋊D137—44no—
33D4×D137—24no—
34D4×D137—42no—
35Dic548C2312yes—
36#14555499—22no—
37#14555498—24no—
38D1096C2312yes—
39#14555495—24no—
40#14555494D411yescharacteristic
41#14555493D412yes—
42C8×D137C2312yes—
43#14555485C2312yes—
44C2×D4×C137—22no—
45D8×C137—22no—
46D8×C137C2314yes—
47C2×C8×C137C2311yescharacteristic
48C16×C137C2311yescharacteristic
49C16×C137C2311yescharacteristic
50C22×D137—44no—
51Dic274—22no—
52Dic274D811yescharacteristic
53D548—22no—
54D548D811yescharacteristic
55C22⋊D137—42no—
56C22⋊D137—44no—
57C22⋊D137—44no—
58C22⋊D137D812yes—
59C2×Dic137—44no—
60C4×D137—42no—
61C4×D137C2×D412yes—
62#11764040C2×D412yes—
63C23×C137—42no—
64D4×C137—24no—
65D4×C137—42no—
66C2×C4×C137C2×D411yescharacteristic
67C8×C137C2×D411yescharacteristic
68C8×C137C2411yescharacteristic
69C22×C137—44no—
70C22×C137—162no—
71C22×C137C2×D811yescharacteristic
72C2×D137—42no—
73C2×D137—164no—
74C2×D137C2×D812yes—
75C4×C137C2×D811yescharacteristic
76C4×C137C22×D411yescharacteristic
77Dic137—42no—
78Dic137C2×D812yes—
79C2×C137—21no—
80C2×C137—84no—
81C2×C137C22×D811yescharacteristic
82D137—22no—
83D137—82no—
84C137#341511yescharacteristic, complemented, Sylow
85#3415—1371nomaximal, Sylow
86#576—1372no—
87#509—1372no—
88#509—1374no—
89#508—1372no—
90#508—1371no—
91C2×D16—1372no—
92C2×D16D13711yescharacteristic, complemented
93#504—1371no—
94H(5,𝔽2)—2742no—
95AGL(1,ℤ/8)—2744no—
96AGL(1,ℤ/8)—2742no—
97#135—1374no—
98#135—1372no—
99C2×D8—1374no—
100C2×D8C2×D13712yescomplemented
101#131—1371no—
102Q32—1371no—
103SD32—1374no—
104SD32—1372no—
105D16—1374no—
106D16—1371no—
107D16C2×D13712yescomplemented
108D16C2×D13712yescomplemented
109M32—1371no—
110M32—1372no—
111C2×C16—1372no—
112C2×C16C2×D13711yescharacteristic, complemented
113#41—2744no—
114#41—2742no—
115#41—5482no—
116#41—1371no—
117C2×D4—22no—
118C2×D4—2744no—
119C2×D4—5482no—
120C2×D4—5484no—
121Q16—1372no—
122SD16—2744no—
123SD16—2742no—
124D8—22no—
125D8—2744no—
126D8—1372no—
127D8C22×D13714yes—
128M16—1371no—
129M16—1372no—
130C2×C8—1372no—
131C2×C8C22×D13711yescharacteristic
132C16—1372no—
133C16C22×D13711yescharacteristic
134C16C22×D13711yescharacteristic
135C23—42no—
136C23—5484no—
137Q8—2742no—
138Q8—1371no—
139D4—24no—
140D4—42no—
141D4—2742no—
142D4—5484no—
143D4—5482no—
144D4—5484no—
145D4—1372no—
146D4—1371no—
147C2×C4—5484no—
148C2×C4—5482no—
149C2×C4—1372no—
150C2×C4D4×D13711yescharacteristic
151C8—1372no—
152C8D4×D13711yescharacteristic
153C8C23×D13711yescharacteristic
154C22—44no—
155C22—162no—
156C22—5482no—
157C22—1372no—
158C22—21924no—
159C22D8×D13711yescharacteristic
160C4—5482no—
161C4—1372no—
162C4D8×D13711yescharacteristic
163C4C2×D4×D13711yescharacteristic
164C2—21no—
165C2—84no—
166C2—2742no—
167C2—10962no—
168C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
FittingC2×D16×C137C2
socleC2×C137C22×D8
hypercenterC2×D16D137
hypocenterC137#3415
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #22578204C2semidirect
2. #22577839C2semidirect
3. #22577837C2semidirect
4. #22577835C2semidirect
5. #22577832C2semidirect
6. D16×D137C2semidirect
7. #22577819C2semidirect
8. C2×D16×C137C2semidirect
12. #17984110C22semidirect
13. #17984107C22semidirect
14. D8×D137C22semidirect
16. Dic1096C22semidirect
17. #17984033C22semidirect
18. #17984032C22semidirect
19. D2192C22semidirect
20. #17984029C22semidirect
21. #17984028C22semidirect
22. #17984027C22semidirect
23. C16×D137C22semidirect
24. C2×D8×C137C22semidirect
25. D16×C137C22semidirect
26. D16×C137C22semidirect
27. C2×C16×C137C22semidirect
84. C137#3415semidirect
92. C2×D16D137semidirect
100. C2×D8C2×D137semidirect
107. D16C2×D137semidirect
108. D16C2×D137semidirect
112. C2×C16C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782042
#225778392
#225778372
#225778352
#225778322
D16×D1372
#225778192
C2×D16×C1372
#3415137