Finite Group Databasebeta

← C2×#23782406 #35228654

Subgroups — C2×#23782406 #35228654

Order 20864.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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#23782406C2116yesmaximal, direct product factor
2C2×#19104732C211yesmaximal, characteristic, complemented
3C2×#19104729C211yesmaximal, characteristic, complemented
4C2×#19104720C213yesmaximal, complemented
5C2×#19104710C213yesmaximal, complemented
6C2×#19104708C213yesmaximal, complemented
7C2×#19104705C213yesmaximal, complemented
8C2×#126×C163C211yesmaximal, characteristic, complemented
9C24×D163C2211yescharacteristic
10C22×D652C2213yes—
11C22×D652C2213yes—
12C22×D652C2213yescomplemented
13C22×C22⋊D163C2213yes—
14C22×C4×D163C2213yescomplemented
15#19104732C2218yescomplemented
16#19104729C2218yescomplemented
17#19104720C22124yescomplemented
18#19104710C22124yescomplemented
19#19104708C22124yescomplemented
20#19104705C22124yescomplemented
21C2×Dic163⋊C4C2213yescomplemented
22C2×(C2×D163)⋊C4C2213yes—
23C2×(C2×D163)⋊C4C2213yes—
24C2×(C2×D163)⋊C4C2213yescomplemented
25#126×C163C2218yescomplemented
26C2×C4⋊C4×C163C2213yescomplemented
27C2×C22⋊C4×C163C2213yes—
28C2×C42×C163C2211yescharacteristic, complemented
29C23×D163—46no—
30C23×D163C2313yes—
31C23×D163C2311yescharacteristic
32C23×D163C2311yescharacteristic
33C23×D163C2314yes—
34C2×D652—46no—
35C2×D652—46no—
36C2×D652—46no—
37C2×D652C23112yes—
38C2×D652C23112yes—
39C2×D652C23112yescomplemented
40C2×C22⋊D163—46no—
41C2×C22⋊D163C23112yes—
42C22×Dic163C2313yes—
43C2×C4×D163—46no—
44C2×C4×D163C23112yescomplemented
45Dic163⋊C4C23112yescomplemented
46(C2×D163)⋊C4C23112yes—
47(C2×D163)⋊C4C23112yes—
48(C2×D163)⋊C4C23112yescomplemented
49C24×C163C2311yescharacteristic
50C22×C4×C163C2313yes—
51C22×C4×C163C2313yes—
52C4⋊C4×C163C23112yescomplemented
53C22⋊C4×C163C23112yes—
54C42×C163C2314yescomplemented
55C22×D163—26no—
56C22×D163—26no—
57C22×D163—26no—
58C22×D163—26no—
59C22×D163—26no—
60C22×D163—412no—
61C22×D163—416no—
62C22×D163C2412yes—
63C22×D163C2412yes—
64C22×D163C2416yes—
65D652—412no—
66D652—412no—
67D652—412no—
68C22⋊D163—412no—
69C2×Dic163—26no—
70C2×Dic163C2416yes—
71C4×D163—412no—
72C23×C163—26no—
73C23×C163C2412yes—
74C23×C163C2411yescharacteristic
75C2×C4×C163—26no—
76C2×C4×C163—26no—
77C2×C4×C163C2416yes—
78C2×C4×C163C2416yes—
79C22×C163—26no—
80C22×C163—44no—
81C22×C163H(5,𝔽2)16yes—
82C22×C163C2511yescharacteristic
83C2×D163—26no—
84C2×D163—26no—
85C2×D163—26no—
86C2×D163—26no—
87C2×D163—26no—
88C2×D163—416no—
89C2×D163—412no—
90C2×D163—44no—
91C2×D163—44no—
92C4×C163—26no—
93C4×C163—26no—
94Dic163—26no—
95C2×C163—42no—
96C2×C163#56014yescomplemented
97C2×C163C2×H(5,𝔽2)13yes—
98D163—46no—
99D163—42no—
100D163—42no—
101C163C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—1631nomaximal, Sylow
103#560—16316no—
104C2×#127—1631no—
105C2×#126—1631no—
106C2×#126D16311yescharacteristic, complemented
107C2×#123—1633no—
108C2×#121—1633no—
109C2×#121—1633no—
110C2×#120—1633no—
111C25—1631no—
112C22×D4—1633no—
113C22×D4—1633no—
114C22×D4—1633no—
115C22×D4—1633no—
116C23×C4—1633no—
117#127—1638no—
118#126—1638no—
119#126C2×D16318yescomplemented
120#123—16324no—
121#121—16324no—
122#121—16324no—
123#120—16324no—
124C2×C4⋊C4—1633no—
125C2×C4⋊C4C2×D16313yescomplemented
126C2×C22⋊C4—1633no—
127C2×C22⋊C4—1633no—
128C2×C22⋊C4—1633no—
129C2×C22⋊C4C2×D16313yes—
130C2×C42C2×D16311yescharacteristic, complemented
131C24—6526no—
132C24—1631no—
133C24—1631no—
134C24—1633no—
135C24—1634no—
136C24C22×D16311yescharacteristic
137C2×D4—6526no—
138C2×D4—6526no—
139C2×D4—6526no—
140C2×D4—6526no—
141C2×D4—16312no—
142C2×D4—16312no—
143C2×D4—16312no—
144C2×D4—16312no—
145C22×C4—6526no—
146C22×C4—1633no—
147C22×C4—16312no—
148C22×C4C22×D16313yes—
149C22×C4C22×D16313yes—
150C4⋊C4—16312no—
151C4⋊C4C22×D163112yescomplemented
152C22⋊C4—16312no—
153C22⋊C4—16312no—
154C22⋊C4—16312no—
155C22⋊C4C22×D163112yes—
156C42C22×D16314yescomplemented
157C23—26no—
158C23—3266no—
159C23—3266no—
160C23—3266no—
161C23—3266no—
162C23—3266no—
163C23—65212no—
164C23—65216no—
165C23—1632no—
166C23—1632no—
167C23—1636no—
168C23C23×D16312yes—
169C23C23×D16311yescharacteristic
170D4—65212no—
171D4—65212no—
172D4—65212no—
173D4—65212no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—3266no—
177C2×C4—65212no—
178C2×C4—1636no—
179C2×C4C23×D16316yes—
180C2×C4C23×D16316yes—
181C22—26no—
182C22—44no—
183C22—3266no—
184C22—3266no—
185C22—3266no—
186C22—3266no—
187C22—3266no—
188C22—65216no—
189C22—6524no—
190C22—6524no—
191C22—65212no—
192C22#1910478116yes—
193C22C24×D16311yescharacteristic
194C4—26no—
195C4—26no—
196C4—3266no—
197C2—42no—
198C2—6522no—
199C2—6522no—
200C2—6526no—
201C2#2378240614yesdirect product factor
202C2C2×#1910478113yes—
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#126×C163C2
socleC23×C163C24
hypercenterC2×#126D163
hypocenterC163C2×#560
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #23782406C2direct (N × H)
2. C2×#19104732C2semidirect
3. C2×#19104729C2semidirect
4. C2×#19104720C2semidirect
5. C2×#19104710C2semidirect
6. C2×#19104708C2semidirect
7. C2×#19104705C2semidirect
8. C2×#126×C163C2semidirect
12. C22×D652C22semidirect
14. C22×C4×D163C22semidirect
15. #19104732C22semidirect
16. #19104729C22semidirect
17. #19104720C22semidirect
18. #19104710C22semidirect
19. #19104708C22semidirect
20. #19104705C22semidirect
21. C2×Dic163⋊C4C22semidirect
24. C2×(C2×D163)⋊C4C22semidirect
25. #126×C163C22semidirect
26. C2×C4⋊C4×C163C22semidirect
28. C2×C42×C163C22semidirect
39. C2×D652C23semidirect
44. C2×C4×D163C23semidirect
45. Dic163⋊C4C23semidirect
48. (C2×D163)⋊C4C23semidirect
52. C4⋊C4×C163C23semidirect
54. C42×C163C23semidirect
96. C2×C163#560semidirect
101. C163C2×#560semidirect
106. C2×#126D163semidirect
119. #126C2×D163semidirect
125. C2×C4⋊C4C2×D163semidirect
130. C2×C42C2×D163semidirect
151. C4⋊C4C22×D163semidirect
156. C42C22×D163semidirect
201. C2#23782406direct (N × H)

Maximal subgroups

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

subgroupindex
#237824062
C2×#191047322
C2×#191047292
C2×#191047202
C2×#191047102
C2×#191047082
C2×#191047052
C2×#126×C1632
C2×#560163