Finite Group Databasebeta

← C2×#126×D163 #35227790

Subgroups — C2×#126×D163 #35227790

Order 20864.

1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 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
1C2×#19104730C213yesmaximal, complemented
2C2×#19104729C211yesmaximal, characteristic, complemented
3C2×#19104728C213yesmaximal, complemented
4#126×D163C2116yesmaximal, direct product factor
5C2×C4⋊C4×D163C213yesmaximal, complemented
6C2×C22⋊C4×D163C213yesmaximal
7C2×C42×D163C211yesmaximal, characteristic, complemented
8C2×#126×C163C211yesmaximal, characteristic, complemented
9C24×D163C2211yescharacteristic
10C22×C4×D163C2213yes—
11C22×C4×D163C2213yes—
12#19104730C22124yescomplemented
13#19104729C2218yescomplemented
14#19104728C22124yescomplemented
15C2×C4⋊Dic163C2213yescomplemented
16C2×Dic163⋊C4C2213yescomplemented
17C2×Dic163⋊C4C2213yescomplemented
18C4⋊C4×D163C22124yescomplemented
19C2×C22⋊Dic163C2213yes—
20C2×(C2×D163)⋊C4C2213yes—
21C2×(C2×D163)⋊C4C2213yes—
22C22⋊C4×D163C22124yes—
23C2×C4×Dic163C2213yescomplemented
24C42×D163C2218yescomplemented
25#126×C163C2218yescomplemented
26C2×C4⋊C4×C163C2213yescomplemented
27C2×C22⋊C4×C163C2213yes—
28C2×C42×C163C2211yescharacteristic, complemented
29C23×D163—212no—
30C23×D163C2314yes—
31C23×D163C2311yescharacteristic
32C23×D163C2311yescharacteristic
33C22×Dic163C2313yes—
34C22×Dic163C2313yes—
35C2×C4×D163—212no—
36C2×C4×D163—212no—
37C2×C4×D163C23112yes—
38C2×C4×D163C23112yes—
39C4⋊Dic163C23112yescomplemented
40Dic163⋊C4C23112yescomplemented
41Dic163⋊C4C23112yescomplemented
42C22⋊Dic163C23112yes—
43(C2×D163)⋊C4C23112yes—
44(C2×D163)⋊C4C23112yes—
45C4×Dic163C23112yescomplemented
46C24×C163C2311yescharacteristic
47C22×C4×C163C2313yes—
48C22×C4×C163C2313yes—
49C4⋊C4×C163C23112yescomplemented
50C22⋊C4×C163C23112yes—
51C42×C163C2314yescomplemented
52C22×D163—26no—
53C22×D163—224no—
54C22×D163—416no—
55C22×D163#41112yes—
56C22×D163C2412yes—
57C22×D163C2412yes—
58C2×Dic163—26no—
59C2×Dic163—26no—
60C2×Dic163C2416yes—
61C2×Dic163C2416yes—
62C4×D163—224no—
63C4×D163—224no—
64C23×C163—26no—
65C23×C163C2412yes—
66C23×C163C2411yescharacteristic
67C2×C4×C163—26no—
68C2×C4×C163—26no—
69C2×C4×C163C2416yes—
70C2×C4×C163C2416yes—
71C22×C163—26no—
72C22×C163—44no—
73C22×C163C2×#4116yes—
74C22×C163C2511yescharacteristic
75C2×D163—26no—
76C2×D163—44no—
77C2×D163—416no—
78C2×D163#126116yesdirect product factor
79C2×D163C2×#41112yes—
80C4×C163—26no—
81C4×C163—26no—
82Dic163—26no—
83Dic163—26no—
84C2×C163—42no—
85C2×C163C2×#12614yescomplemented
86C2×C163C22×#4113yes—
87D163—42no—
88D163C2×#12618yesdirect product factor
89C163C22×#12611yescharacteristic, complemented, Sylow
90C22×#126—1631nomaximal, Sylow
91C2×#126—16316no—
92C2×#126—1633no—
93C2×#126—1633no—
94C2×#126—1631no—
95C2×#126D16311yescharacteristic, direct product factor
96C22×C4⋊C4—1633no—
97C22×C22⋊C4—1633no—
98C22×C42—1631no—
99C25—1631no—
100C23×C4—1633no—
101C23×C4—1633no—
102#126—16324no—
103#126—16324no—
104#126—1638no—
105#126C2×D16318yesdirect product factor
106C2×C4⋊C4—1633no—
107C2×C4⋊C4—1633no—
108C2×C4⋊C4—1633no—
109C2×C4⋊C4—16324no—
110C2×C4⋊C4C2×D16313yescomplemented
111C2×C22⋊C4—1633no—
112C2×C22⋊C4—1633no—
113C2×C22⋊C4—1633no—
114C2×C22⋊C4—16324no—
115C2×C22⋊C4C2×D16313yes—
116C2×C42—1633no—
117C2×C42—1638no—
118C2×C42C2×D16311yescharacteristic, complemented
119C24—32612no—
120C24—1634no—
121C24—1631no—
122C24—1631no—
123C24C22×D16311yescharacteristic
124C22×C4—32612no—
125C22×C4—32612no—
126C22×C4—1633no—
127C22×C4—1633no—
128C22×C4—16312no—
129C22×C4—16312no—
130C22×C4C22×D16313yes—
131C22×C4C22×D16313yes—
132C4⋊C4—16312no—
133C4⋊C4—16312no—
134C4⋊C4—16312no—
135C4⋊C4C22×D163112yescomplemented
136C22⋊C4—16312no—
137C22⋊C4—16312no—
138C22⋊C4—16312no—
139C22⋊C4C22×D163112yes—
140C42—16312no—
141C42C22×D16314yescomplemented
142C23—26no—
143C23—32624no—
144C23—3266no—
145C23—65216no—
146C23—1632no—
147C23—16312no—
148C23—1632no—
149C23C23×D16312yes—
150C23C23×D16311yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—32624no—
154C2×C4—32624no—
155C2×C4—3266no—
156C2×C4—3266no—
157C2×C4—1636no—
158C2×C4—1636no—
159C2×C4C23×D16316yes—
160C2×C4C23×D16316yes—
161C22—26no—
162C22—44no—
163C22—3266no—
164C22—65216no—
165C22—6524no—
166C22—16316no—
167C22—16312no—
168C22#41×D16316yes—
169C22C24×D16311yescharacteristic
170C4—26no—
171C4—26no—
172C4—3266no—
173C4—3266no—
174C2—42no—
175C2—6522no—
176C2—1638no—
177C2#126×D16314yesdirect product factor
178C2C2×#41×D16313yes—
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#126×C163C2
socleC23×C163C24
prime radicalC24×D163C22
hypercenterC2×#126D163
hypocenterC163C22×#126
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#19104730C2semidirect
2. C2×#19104729C2semidirect
3. C2×#19104728C2semidirect
4. #126×D163C2direct (N × H)
5. C2×C4⋊C4×D163C2semidirect
7. C2×C42×D163C2semidirect
8. C2×#126×C163C2semidirect
12. #19104730C22semidirect
13. #19104729C22semidirect
14. #19104728C22semidirect
15. C2×C4⋊Dic163C22semidirect
16. C2×Dic163⋊C4C22semidirect
17. C2×Dic163⋊C4C22semidirect
18. C4⋊C4×D163C22semidirect
23. C2×C4×Dic163C22semidirect
24. C42×D163C22semidirect
25. #126×C163C22semidirect
26. C2×C4⋊C4×C163C22semidirect
28. C2×C42×C163C22semidirect
39. C4⋊Dic163C23semidirect
40. Dic163⋊C4C23semidirect
41. Dic163⋊C4C23semidirect
45. C4×Dic163C23semidirect
49. C4⋊C4×C163C23semidirect
51. C42×C163C23semidirect
78. C2×D163#126direct (N × H)
85. C2×C163C2×#126semidirect
88. D163C2×#126direct (N × H)
89. C163C22×#126semidirect
95. C2×#126D163direct (N × H)
105. #126C2×D163direct (N × H)
110. C2×C4⋊C4C2×D163semidirect
118. C2×C42C2×D163semidirect
135. C4⋊C4C22×D163semidirect
141. C42C22×D163semidirect
177. C2#126×D163direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191047302
C2×#191047292
C2×#191047282
#126×D1632
C2×C4⋊C4×D1632
C2×C22⋊C4×D1632
C2×C42×D1632
C2×#126×C1632
C22×#126163