Finite Group Databasebeta

← #31793927

Subgroups — #31793927

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#22578205C212yesmaximal, complemented
2#22577840C214yesmaximal, complemented
3#22577838C212yesmaximal, complemented
4#22577836C212yesmaximal, complemented
5C2×D2192C211yesmaximal, characteristic, complemented
6D16×D137C212yesmaximal, complemented
7#22577819C211yesmaximal, characteristic, complemented
8#508×C137C211yesmaximal, characteristic, complemented
9#17984134—22no—
10#17984115—24no—
11#17984113—22no—
12#17984111C2214yescomplemented
13C2×D1096C2212yescomplemented
14D8×D137C2214yescomplemented
15#17984094C2211yescharacteristic
16#17984031C2214yescomplemented
17D2192C2212yescomplemented
18D2192C2212yescomplemented
19#17984029C2214yescomplemented
20#17984028C2211yescharacteristic, complemented
21#17984027C2212yescomplemented
22C16×D137C2212yescomplemented
23#135×C137C2212yescomplemented
24Q32×C137C2211yescharacteristic, complemented
25SD32×C137C2212yescomplemented
26D16×C137C2211yescharacteristic, complemented
27C2×C16×C137C2211yescharacteristic, complemented
28#14555511—24no—
29#14555510—42no—
30#14555510D411yescharacteristic
31C2×D548—22no—
32C2×D548—42no—
33D4×D137—24no—
34D4×D137—44no—
35#14555497—24no—
36D1096—22no—
37D1096C2314yes—
38#14555495—24no—
39#14555494D411yescharacteristic
40#14555493D412yes—
41C8×D137C2312yes—
42#14555485C2312yes—
43#41×C137—22no—
44Q16×C137C2312yes—
45SD16×C137—22no—
46D8×C137C2312yes—
47C2×C8×C137C2311yescharacteristic
48C16×C137C2311yescharacteristic
49C16×C137C2311yescharacteristic
50C22×D137—42no—
51C22×D137—44no—
52Dic274D811yescharacteristic
53D548—24no—
54D548—44no—
55D548—42no—
56D548D811yescharacteristic
57C22⋊D137—44no—
58C22⋊D137D812yes—
59C4×D137—44no—
60C4×D137C2×D412yes—
61#11764040C2×D412yes—
62Q8×C137—22no—
63D4×C137—22no—
64D4×C137—42no—
65C2×C4×C137—42no—
66C2×C4×C137C2×D411yescharacteristic
67C8×C137C2×D411yescharacteristic
68C8×C137C2411yescharacteristic
69C22×C137—42no—
70C22×C137C2×D811yescharacteristic
71C2×D137—44no—
72C2×D137—164no—
73C2×D137—162no—
74C2×D137C2×D812yes—
75C4×C137—42no—
76C4×C137C2×D811yescharacteristic
77C4×C137C22×D411yescharacteristic
78Dic137C2×D812yes—
79C2×C137—21no—
80C2×C137—82no—
81C2×C137C22×D811yescharacteristic
82D137—22no—
83D137—84no—
84C137#341511yescharacteristic, complemented, Sylow
85#3415—1371nomaximal, Sylow
86#576—1372no—
87#509—1372no—
88#509—1374no—
89#508—1372no—
90#508D13711yescharacteristic, complemented
91C2×D16—1372no—
92C2×D16—1371no—
93#504—1371no—
94H(5,𝔽2)—2742no—
95AGL(1,ℤ/8)—2744no—
96AGL(1,ℤ/8)—2742no—
97#135—1374no—
98#135C2×D13712yescomplemented
99C2×D8—1374no—
100C2×D8—1372no—
101#131—1371no—
102Q32C2×D13711yescharacteristic, complemented
103SD32—1374no—
104SD32C2×D13712yescomplemented
105D16—1374no—
106D16—1372no—
107D16—1372no—
108D16C2×D13711yescharacteristic, complemented
109M32—1371no—
110M32—1372no—
111C2×C16—1372no—
112C2×C16C2×D13711yescharacteristic, complemented
113#41—22no—
114#41—2744no—
115#41—5482no—
116#41—1371no—
117C2×D4—2744no—
118C2×D4—2742no—
119C2×D4—5482no—
120C2×D4—5484no—
121Q16C22×D13712yes—
122SD16—22no—
123SD16—2744no—
124D8—2744no—
125D8—2742no—
126D8—1374no—
127D8C22×D13712yes—
128M16—1371no—
129M16—1372no—
130C2×C8—1372no—
131C2×C8C22×D13711yescharacteristic
132C16—1372no—
133C16C22×D13711yescharacteristic
134C16C22×D13711yescharacteristic
135C23—5484no—
136C23—5482no—
137Q8—22no—
138Q8—1371no—
139D4—22no—
140D4—42no—
141D4—2744no—
142D4—5484no—
143D4—5484no—
144D4—5482no—
145D4—1372no—
146D4—1371no—
147C2×C4—42no—
148C2×C4—5484no—
149C2×C4—1372no—
150C2×C4D4×D13711yescharacteristic
151C8—1372no—
152C8D4×D13711yescharacteristic
153C8C23×D13711yescharacteristic
154C22—42no—
155C22—5484no—
156C22—1372no—
157C22—21922no—
158C22—21924no—
159C22D8×D13711yescharacteristic
160C4—42no—
161C4—1372no—
162C4D8×D13711yescharacteristic
163C4C2×D4×D13711yescharacteristic
164C2—21no—
165C2—82no—
166C2—2742no—
167C2—10964no—
168C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC2C2×D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
Fitting#508×C137C2
socleC2×C137C22×D8
hypercenter#508D137
hypocenterC137#3415
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22578205C2semidirect
2. #22577840C2semidirect
3. #22577838C2semidirect
4. #22577836C2semidirect
5. C2×D2192C2semidirect
6. D16×D137C2semidirect
7. #22577819C2semidirect
8. #508×C137C2semidirect
12. #17984111C22semidirect
13. C2×D1096C22semidirect
14. D8×D137C22semidirect
16. #17984031C22semidirect
17. D2192C22semidirect
18. D2192C22semidirect
19. #17984029C22semidirect
20. #17984028C22semidirect
21. #17984027C22semidirect
22. C16×D137C22semidirect
23. #135×C137C22semidirect
24. Q32×C137C22semidirect
25. SD32×C137C22semidirect
26. D16×C137C22semidirect
27. C2×C16×C137C22semidirect
84. C137#3415semidirect
90. #508D137semidirect
98. #135C2×D137semidirect
102. Q32C2×D137semidirect
104. SD32C2×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
#225782052
#225778402
#225778382
#225778362
C2×D21922
D16×D1372
#225778192
#508×C1372
#3415137