Finite Group Databasebeta

← #36337276

Subgroups — #36337276

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#24924071C212yesmaximal, complemented
2#24923706C214yesmaximal, complemented
3#24923704C212yesmaximal, complemented
4#24923702C212yesmaximal, complemented
5C2×D2672C211yesmaximal, characteristic, complemented
6D16×D167C212yesmaximal, complemented
7#24923685C211yesmaximal, characteristic, complemented
8#508×C167C211yesmaximal, characteristic, complemented
9#19128878—22no—
10#19128859—24no—
11#19128857—22no—
12#19128855C2214yescomplemented
13C2×D1336C2212yescomplemented
14D8×D167C2214yescomplemented
15#19128838C2211yescharacteristic
16#19128775C2214yescomplemented
17D2672C2212yescomplemented
18D2672C2212yescomplemented
19#19128773C2214yescomplemented
20#19128772C2211yescharacteristic, complemented
21#19128771C2212yescomplemented
22C16×D167C2212yescomplemented
23#135×C167C2212yescomplemented
24Q32×C167C2211yescharacteristic, complemented
25SD32×C167C2212yescomplemented
26D16×C167C2211yescharacteristic, complemented
27C2×C16×C167C2211yescharacteristic, complemented
28#14581951—24no—
29#14581950—42no—
30#14581950D411yescharacteristic
31C2×D668—22no—
32C2×D668—42no—
33D4×D167—24no—
34D4×D167—44no—
35#14581937—24no—
36D1336—22no—
37D1336C2314yes—
38#14581935—24no—
39#14581934D411yescharacteristic
40#14581933D412yes—
41C8×D167C2312yes—
42C167⋊C16C2312yes—
43#41×C167—22no—
44Q16×C167C2312yes—
45SD16×C167—22no—
46D8×C167C2312yes—
47C2×C8×C167C2311yescharacteristic
48C16×C167C2311yescharacteristic
49C16×C167C2311yescharacteristic
50C22×D167—42no—
51C22×D167—44no—
52Dic334D811yescharacteristic
53D668—24no—
54D668—44no—
55D668—42no—
56D668D811yescharacteristic
57C22⋊D167—44no—
58C22⋊D167D812yes—
59C4×D167—44no—
60C4×D167C2×D412yes—
61C167⋊C8C2×D412yes—
62Q8×C167—22no—
63D4×C167—22no—
64D4×C167—42no—
65C2×C4×C167—42no—
66C2×C4×C167C2×D411yescharacteristic
67C8×C167C2×D411yescharacteristic
68C8×C167C2411yescharacteristic
69C22×C167—42no—
70C22×C167C2×D811yescharacteristic
71C2×D167—44no—
72C2×D167—164no—
73C2×D167—162no—
74C2×D167C2×D812yes—
75C4×C167—42no—
76C4×C167C2×D811yescharacteristic
77C4×C167C22×D411yescharacteristic
78Dic167C2×D812yes—
79C2×C167—21no—
80C2×C167—82no—
81C2×C167C22×D811yescharacteristic
82D167—22no—
83D167—84no—
84C167#341511yescharacteristic, complemented, Sylow
85#3415—1671nomaximal, Sylow
86#576—1672no—
87#509—1672no—
88#509—1674no—
89#508—1672no—
90#508D16711yescharacteristic, complemented
91C2×D16—1672no—
92C2×D16—1671no—
93#504—1671no—
94H(5,𝔽2)—3342no—
95AGL(1,ℤ/8)—3344no—
96AGL(1,ℤ/8)—3342no—
97#135—1674no—
98#135C2×D16712yescomplemented
99C2×D8—1674no—
100C2×D8—1672no—
101#131—1671no—
102Q32C2×D16711yescharacteristic, complemented
103SD32—1674no—
104SD32C2×D16712yescomplemented
105D16—1674no—
106D16—1672no—
107D16—1672no—
108D16C2×D16711yescharacteristic, complemented
109M32—1671no—
110M32—1672no—
111C2×C16—1672no—
112C2×C16C2×D16711yescharacteristic, complemented
113#41—22no—
114#41—3344no—
115#41—6682no—
116#41—1671no—
117C2×D4—3344no—
118C2×D4—3342no—
119C2×D4—6682no—
120C2×D4—6684no—
121Q16C22×D16712yes—
122SD16—22no—
123SD16—3344no—
124D8—3344no—
125D8—3342no—
126D8—1674no—
127D8C22×D16712yes—
128M16—1671no—
129M16—1672no—
130C2×C8—1672no—
131C2×C8C22×D16711yescharacteristic
132C16—1672no—
133C16C22×D16711yescharacteristic
134C16C22×D16711yescharacteristic
135C23—6684no—
136C23—6682no—
137Q8—22no—
138Q8—1671no—
139D4—22no—
140D4—42no—
141D4—3344no—
142D4—6684no—
143D4—6684no—
144D4—6682no—
145D4—1672no—
146D4—1671no—
147C2×C4—42no—
148C2×C4—6684no—
149C2×C4—1672no—
150C2×C4D4×D16711yescharacteristic
151C8—1672no—
152C8D4×D16711yescharacteristic
153C8C23×D16711yescharacteristic
154C22—42no—
155C22—26722no—
156C22—26724no—
157C22—6684no—
158C22—1672no—
159C22D8×D16711yescharacteristic
160C4—42no—
161C4—1672no—
162C4D8×D16711yescharacteristic
163C4C2×D4×D16711yescharacteristic
164C2—21no—
165C2—82no—
166C2—13364no—
167C2—3342no—
168C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC2C2×D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
Fitting#508×C167C2
socleC2×C167C22×D8
hypercenter#508D167
hypocenterC167#3415
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #24924071C2semidirect
2. #24923706C2semidirect
3. #24923704C2semidirect
4. #24923702C2semidirect
5. C2×D2672C2semidirect
6. D16×D167C2semidirect
7. #24923685C2semidirect
8. #508×C167C2semidirect
12. #19128855C22semidirect
13. C2×D1336C22semidirect
14. D8×D167C22semidirect
16. #19128775C22semidirect
17. D2672C22semidirect
18. D2672C22semidirect
19. #19128773C22semidirect
20. #19128772C22semidirect
21. #19128771C22semidirect
22. C16×D167C22semidirect
23. #135×C167C22semidirect
24. Q32×C167C22semidirect
25. SD32×C167C22semidirect
26. D16×C167C22semidirect
27. C2×C16×C167C22semidirect
84. C167#3415semidirect
90. #508D167semidirect
98. #135C2×D167semidirect
102. Q32C2×D167semidirect
104. SD32C2×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
#249240712
#249237062
#249237042
#249237022
C2×D26722
D16×D1672
#249236852
#508×C1672
#3415167