Finite Group Databasebeta

← C2×#127×D167 #36337408

Subgroups — C2×#127×D167 #36337408

Order 21376.

3000 conjugacy classes of subgroups; only the 661 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 661 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
1C22×D4×D167C216yesmaximal, complemented
2C2×#19128829C211yesmaximal, characteristic, complemented
3C2×#19128828C216yesmaximal, complemented
4#127×D167C2116yesmaximal, direct product factor
5C2×C42×D167C211yesmaximal, characteristic, complemented
6C2×#127×C167C211yesmaximal, characteristic, complemented
7C24×D167C2214yes—
8C22×D668C2216yescomplemented
9C22×C22⋊D167C22112yescomplemented
10C2×D4×D167—248no—
11C2×D4×D167C22148yescomplemented
12C22×C4×D167C2213yes—
13#19128829C2218yescomplemented
14#19128828C22148yescomplemented
15C2×C4×Dic167C2213yescomplemented
16C42×D167C2218yescomplemented
17C22×D4×C167C2216yescomplemented
18#127×C167C2218yescomplemented
19C2×C42×C167C2211yescharacteristic, complemented
20C23×D167—248no—
21C23×D167C23116yes—
22C23×D167C2314yes—
23C23×D167C2311yescharacteristic
24C2×D668—224no—
25C2×D668C23124yescomplemented
26C2×C22⋊D167—248no—
27C2×C22⋊D167C23148yescomplemented
28D4×D167—2192no—
29C22×Dic167C2313yes—
30C2×C4×D167D4124yescomplemented
31C2×C4×D167C23112yes—
32C4×Dic167C23112yescomplemented
33C24×C167C2314yes—
34C2×D4×C167—224no—
35C2×D4×C167C23124yescomplemented
36C22×C4×C167C2313yes—
37C42×C167C2314yescomplemented
38C22×D167—224no—
39C22×D167—296no—
40C22×D167—464no—
41C22×D167C2×D4112yes—
42C22×D167C2418yes—
43C22×D167C2412yes—
44D668—248no—
45C22⋊D167—296no—
46C2×Dic167C2×D4112yescomplemented
47C2×Dic167C2416yes—
48C4×D167C2×D4148yescomplemented
49C23×C167—224no—
50C23×C167C2418yes—
51C23×C167C2411yescharacteristic
52D4×C167—248no—
53C2×C4×C167C2×D4112yescomplemented
54C2×C4×C167C2416yes—
55C22×C167—224no—
56C22×C167—416no—
57C22×C167C22×D416yes—
58C22×C167C2511yescharacteristic
59C2×D167—224no—
60C2×D167—416no—
61C2×D167—464no—
62C2×D167#127116yesdirect product factor
63C2×D167C22×D4112yes—
64C4×C167C22×D4112yescomplemented
65Dic167C22×D4112yescomplemented
66C2×C167—48no—
67C2×C167C2×#12714yescomplemented
68C2×C167C23×D413yes—
69D167—48no—
70D167C2×#12718yesdirect product factor
71C167C22×#12711yescharacteristic, complemented, Sylow
72C22×#127—1671nomaximal, Sylow
73C23×D4—1676no—
74C2×#127—1676no—
75C2×#127—16716no—
76C2×#127—1671no—
77C2×#127D16711yescharacteristic, direct product factor
78C22×C42—1671no—
79C25—1674no—
80C22×D4—33448no—
81C22×D4—16712no—
82C22×D4—16748no—
83C22×D4—1676no—
84C22×D4C2×D16716yescomplemented
85C23×C4—1673no—
86#127—16748no—
87#127—1678no—
88#127C2×D16718yesdirect product factor
89C2×C42—1673no—
90C2×C42—1678no—
91C2×C42C2×D16711yescharacteristic, complemented
92C24—33448no—
93C24—16716no—
94C24—1674no—
95C24—1671no—
96C24C22×D16714yes—
97C2×D4—224no—
98C2×D4—334192no—
99C2×D4—33448no—
100C2×D4—33424no—
101C2×D4—16748no—
102C2×D4—16724no—
103C2×D4C22×D167124yescomplemented
104C22×C4—1673no—
105C22×C4—16724no—
106C22×C4—16712no—
107C22×C4C22×D16713yes—
108C42—16712no—
109C42C22×D16714yescomplemented
110C23—224no—
111C23—33496no—
112C23—33424no—
113C23—66864no—
114C23—1678no—
115C23—16712no—
116C23—1672no—
117C23C23×D16718yes—
118C23C23×D16711yescharacteristic
119D4—248no—
120D4—33496no—
121D4—33448no—
122C2×C4—16712no—
123C2×C4—1676no—
124C2×C4—16748no—
125C2×C4D4×D167112yescomplemented
126C2×C4C23×D16716yes—
127C22—224no—
128C22—416no—
129C22—33424no—
130C22—66864no—
131C22—66816no—
132C22—16716no—
133C22—16712no—
134C22C2×D4×D16716yes—
135C22C24×D16711yescharacteristic
136C4—16712no—
137C4C2×D4×D167112yescomplemented
138C2—48no—
139C2—6688no—
140C2—1678no—
141C2#127×D16714yesdirect product factor
142C2C22×D4×D16713yes—
subgroupG/H
centerC23C23×D167
commutator, cosocleC22×C167C25
FrattiniC22C24×D167
FittingC2×#127×C167C2
socleC23×C167C24
hypercenterC2×#127D167
hypocenterC167C22×#127
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D167C2semidirect
2. C2×#19128829C2semidirect
3. C2×#19128828C2semidirect
4. #127×D167C2direct (N × H)
5. C2×C42×D167C2semidirect
6. C2×#127×C167C2semidirect
8. C22×D668C22semidirect
9. C22×C22⋊D167C22semidirect
11. C2×D4×D167C22semidirect
13. #19128829C22semidirect
14. #19128828C22semidirect
15. C2×C4×Dic167C22semidirect
16. C42×D167C22semidirect
17. C22×D4×C167C22semidirect
18. #127×C167C22semidirect
19. C2×C42×C167C22semidirect
25. C2×D668C23semidirect
27. C2×C22⋊D167C23semidirect
30. C2×C4×D167D4semidirect
32. C4×Dic167C23semidirect
35. C2×D4×C167C23semidirect
37. C42×C167C23semidirect
46. C2×Dic167C2×D4semidirect
48. C4×D167C2×D4semidirect
53. C2×C4×C167C2×D4semidirect
62. C2×D167#127direct (N × H)
64. C4×C167C22×D4semidirect
65. Dic167C22×D4semidirect
67. C2×C167C2×#127semidirect
70. D167C2×#127direct (N × H)
71. C167C22×#127semidirect
77. C2×#127D167direct (N × H)
84. C22×D4C2×D167semidirect
88. #127C2×D167direct (N × H)
91. C2×C42C2×D167semidirect
103. C2×D4C22×D167semidirect
109. C42C22×D167semidirect
125. C2×C4D4×D167semidirect
137. C4C2×D4×D167semidirect
141. C2#127×D167direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1672
C2×#191288292
C2×#191288282
#127×D1672
C2×C42×D1672
C2×#127×C1672
C22×#127167