Finite Group Databasebeta

← #36337275

Subgroups — #36337275

Order 21376.

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#24924070C212yesmaximal, complemented
2#24923705C212yesmaximal, complemented
3#24923703C214yesmaximal, complemented
4#24923701C211yesmaximal, characteristic, complemented
5#24923698C212yesmaximal, complemented
6D16×D167C212yesmaximal, complemented
7#24923685C211yesmaximal, characteristic, complemented
8C2×D16×C167C211yesmaximal, characteristic, complemented
9#19128877—22no—
10#19128858—22no—
11#19128856—24no—
12#19128854C2212yescomplemented
13#19128851C2214yescomplemented
14D8×D167C2214yescomplemented
15#19128838C2211yescharacteristic
16Dic1336C2211yescharacteristic, complemented
17#19128777C2212yescomplemented
18#19128776C2214yescomplemented
19D2672C2211yescharacteristic, complemented
20#19128773C2214yescomplemented
21#19128772C2211yescharacteristic, complemented
22#19128771C2212yescomplemented
23C16×D167C2212yescomplemented
24C2×D8×C167C2212yescomplemented
25D16×C167C2212yescomplemented
26D16×C167C2212yescomplemented
27C2×C16×C167C2211yescharacteristic, complemented
28#14581950—22no—
29#14581950D411yescharacteristic
30#14581949—24no—
31#14581949—42no—
32C2×C22⋊D167—44no—
33D4×D167—24no—
34D4×D167—42no—
35Dic668C2312yes—
36#14581939—22no—
37#14581938—24no—
38D1336C2312yes—
39#14581935—24no—
40#14581934D411yescharacteristic
41#14581933D412yes—
42C8×D167C2312yes—
43C167⋊C16C2312yes—
44C2×D4×C167—22no—
45D8×C167—22no—
46D8×C167C2314yes—
47C2×C8×C167C2311yescharacteristic
48C16×C167C2311yescharacteristic
49C16×C167C2311yescharacteristic
50C22×D167—44no—
51Dic334—22no—
52Dic334D811yescharacteristic
53D668—22no—
54D668D811yescharacteristic
55C22⋊D167—42no—
56C22⋊D167—44no—
57C22⋊D167—44no—
58C22⋊D167D812yes—
59C2×Dic167—44no—
60C4×D167—42no—
61C4×D167C2×D412yes—
62C167⋊C8C2×D412yes—
63C23×C167—42no—
64D4×C167—24no—
65D4×C167—42no—
66C2×C4×C167C2×D411yescharacteristic
67C8×C167C2×D411yescharacteristic
68C8×C167C2411yescharacteristic
69C22×C167—44no—
70C22×C167—162no—
71C22×C167C2×D811yescharacteristic
72C2×D167—42no—
73C2×D167—164no—
74C2×D167C2×D812yes—
75C4×C167C2×D811yescharacteristic
76C4×C167C22×D411yescharacteristic
77Dic167—42no—
78Dic167C2×D812yes—
79C2×C167—21no—
80C2×C167—84no—
81C2×C167C22×D811yescharacteristic
82D167—22no—
83D167—82no—
84C167#341511yescharacteristic, complemented, Sylow
85#3415—1671nomaximal, Sylow
86#576—1672no—
87#509—1672no—
88#509—1674no—
89#508—1672no—
90#508—1671no—
91C2×D16—1672no—
92C2×D16D16711yescharacteristic, complemented
93#504—1671no—
94H(5,𝔽2)—3342no—
95AGL(1,ℤ/8)—3344no—
96AGL(1,ℤ/8)—3342no—
97#135—1674no—
98#135—1672no—
99C2×D8—1674no—
100C2×D8C2×D16712yescomplemented
101#131—1671no—
102Q32—1671no—
103SD32—1674no—
104SD32—1672no—
105D16—1674no—
106D16—1671no—
107D16C2×D16712yescomplemented
108D16C2×D16712yescomplemented
109M32—1671no—
110M32—1672no—
111C2×C16—1672no—
112C2×C16C2×D16711yescharacteristic, complemented
113#41—3344no—
114#41—3342no—
115#41—6682no—
116#41—1671no—
117C2×D4—22no—
118C2×D4—3344no—
119C2×D4—6682no—
120C2×D4—6684no—
121Q16—1672no—
122SD16—3344no—
123SD16—3342no—
124D8—22no—
125D8—3344no—
126D8—1672no—
127D8C22×D16714yes—
128M16—1671no—
129M16—1672no—
130C2×C8—1672no—
131C2×C8C22×D16711yescharacteristic
132C16—1672no—
133C16C22×D16711yescharacteristic
134C16C22×D16711yescharacteristic
135C23—42no—
136C23—6684no—
137Q8—3342no—
138Q8—1671no—
139D4—24no—
140D4—42no—
141D4—3342no—
142D4—6684no—
143D4—6682no—
144D4—6684no—
145D4—1672no—
146D4—1671no—
147C2×C4—6684no—
148C2×C4—6682no—
149C2×C4—1672no—
150C2×C4D4×D16711yescharacteristic
151C8—1672no—
152C8D4×D16711yescharacteristic
153C8C23×D16711yescharacteristic
154C22—44no—
155C22—162no—
156C22—26724no—
157C22—6682no—
158C22—1672no—
159C22D8×D16711yescharacteristic
160C4—6682no—
161C4—1672no—
162C4D8×D16711yescharacteristic
163C4C2×D4×D16711yescharacteristic
164C2—21no—
165C2—84no—
166C2—13362no—
167C2—3342no—
168C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC2C2×D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
FittingC2×D16×C167C2
socleC2×C167C22×D8
hypercenterC2×D16D167
hypocenterC167#3415
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #24924070C2semidirect
2. #24923705C2semidirect
3. #24923703C2semidirect
4. #24923701C2semidirect
5. #24923698C2semidirect
6. D16×D167C2semidirect
7. #24923685C2semidirect
8. C2×D16×C167C2semidirect
12. #19128854C22semidirect
13. #19128851C22semidirect
14. D8×D167C22semidirect
16. Dic1336C22semidirect
17. #19128777C22semidirect
18. #19128776C22semidirect
19. D2672C22semidirect
20. #19128773C22semidirect
21. #19128772C22semidirect
22. #19128771C22semidirect
23. C16×D167C22semidirect
24. C2×D8×C167C22semidirect
25. D16×C167C22semidirect
26. D16×C167C22semidirect
27. C2×C16×C167C22semidirect
84. C167#3415semidirect
92. C2×D16D167semidirect
100. C2×D8C2×D167semidirect
107. D16C2×D167semidirect
108. D16C2×D167semidirect
112. C2×C16C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249240702
#249237052
#249237032
#249237012
#249236982
D16×D1672
#249236852
C2×D16×C1672
#3415167