Finite Group Databasebeta

← C2×#23782227 #35227839

Subgroups — C2×#23782227 #35227839

Order 20864.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C23×C22⋊D163C212yesmaximal, complemented
2#23782227C2116yesmaximal, direct product factor
3C2×#19104719C214yesmaximal, complemented
4C2×#19104709C214yesmaximal, complemented
5C2×#19104705C214yesmaximal, complemented
6C22×C22⋊C4×C163C211yesmaximal, characteristic, complemented
7C24×D163C2212yescomplemented
8C22×D652C2214yescomplemented
9C22×C22⋊D163—216no—
10C22×C22⋊D163C2214yes—
11C22×C22⋊D163C2214yescomplemented
12C22×C22⋊D163C22116yescomplemented
13C23×Dic163C2212yescomplemented
14#19104719C22132yescomplemented
15#19104709C22132yescomplemented
16#19104705C22132yescomplemented
17C2×C4⋊Dic163C2214yescomplemented
18C2×(C2×D163)⋊C4C2218yescomplemented
19C25×C163C2211yescharacteristic
20C23×C4×C163C2212yescomplemented
21C2×C22⋊C4×C163C2214yescomplemented
22C2×C22⋊C4×C163C2218yescomplemented
23C23×D163—28no—
24C23×D163—48no—
25C23×D163C2314yes—
26C23×D163C2318yescomplemented
27C2×D652—48no—
28C2×D652C23116yescomplemented
29C2×C22⋊D163—216no—
30C2×C22⋊D163—216no—
31C2×C22⋊D163—264no—
32C2×C22⋊D163C23116yes—
33C2×C22⋊D163C23116yescomplemented
34C22×Dic163—28no—
35C22×Dic163C2314yes—
36C22×Dic163C2318yescomplemented
37C4⋊Dic163C23116yescomplemented
38(C2×D163)⋊C4C23132yescomplemented
39C24×C163—44no—
40C24×C163D418yescomplemented
41C24×C163C2312yes—
42C24×C163C2311yescharacteristic
43C24×C163C2314yes—
44C22×C4×C163—44no—
45C22×C4×C163C2314yes—
46C22×C4×C163C2318yescomplemented
47C22⋊C4×C163C23116yescomplemented
48C22×D163—28no—
49C22×D163—216no—
50C22×D163—216no—
51C22×D163—416no—
52C22×D163—432no—
53C22×D163C2418yes—
54D652—416no—
55C22⋊D163—232no—
56C22⋊D163—232no—
57C2×Dic163—28no—
58C2×Dic163—216no—
59C2×Dic163C2418yes—
60C23×C163—24no—
61C23×C163—28no—
62C23×C163—416no—
63C23×C163—48no—
64C23×C163C2×D418yes—
65C23×C163C2×D414yes—
66C23×C163C2×D4116yescomplemented
67C23×C163C2414yes—
68C23×C163C2412yes—
69C23×C163C2411yescharacteristic
70C2×C4×C163—28no—
71C2×C4×C163—48no—
72C2×C4×C163C2418yes—
73C22×C163—28no—
74C22×C163—216no—
75C22×C163—24no—
76C22×C163—28no—
77C22×C163—416no—
78C22×C163C22×D418yes—
79C22×C163C22×D414yes—
80C22×C163C22×D412yes—
81C22×C163H(5,𝔽2)14yes—
82C22×C163C2511yescharacteristic
83C2×D163—28no—
84C2×D163—216no—
85C2×D163—432no—
86C2×D163—416no—
87C4×C163—28no—
88Dic163—28no—
89C2×C163—28no—
90C2×C163—24no—
91C2×C163#53414yescomplemented
92C2×C163C23×D411yescharacteristic
93C2×C163C2×H(5,𝔽2)12yes—
94D163—48no—
95C163C2×#53411yescharacteristic, complemented, Sylow
96C2×#534—1631nomaximal, Sylow
97C23×D4—1632no—
98#534—16316no—
99C2×#123—1634no—
100C2×#121—1634no—
101C2×#120—1634no—
102C22×C22⋊C4D16311yescharacteristic, complemented
103C25—1632no—
104C25C2×D16311yescharacteristic
105C22×D4—32616no—
106C22×D4—1634no—
107C22×D4—1634no—
108C22×D4—16316no—
109C22×D4—1634no—
110C23×C4—1632no—
111C23×C4C2×D16312yescomplemented
112#123—16332no—
113#121—16332no—
114#120—16332no—
115C2×C4⋊C4—1634no—
116C2×C22⋊C4—1638no—
117C2×C22⋊C4C2×D16314yescomplemented
118C2×C22⋊C4C2×D16318yescomplemented
119C24—44no—
120C24—3268no—
121C24—6528no—
122C24—1634no—
123C24—1638no—
124C24D65218yescomplemented
125C24C22×D16312yes—
126C24C22×D16311yescharacteristic
127C24C22×D16314yes—
128C2×D4—32616no—
129C2×D4—32616no—
130C2×D4—32664no—
131C2×D4—6528no—
132C2×D4—16316no—
133C2×D4—16316no—
134C2×D4—16316no—
135C22×C4—44no—
136C22×C4—3268no—
137C22×C4—1638no—
138C22×C4—1634no—
139C22×C4C22×D16314yes—
140C22×C4C22×D16318yescomplemented
141C4⋊C4—16316no—
142C22⋊C4—16332no—
143C22⋊C4C22×D163116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—32616no—
149C23—3268no—
150C23—32616no—
151C23—65216no—
152C23—65232no—
153C23—1638no—
154C23C2×D65218yes—
155C23C2×D65214yes—
156C23C2×D652116yescomplemented
157C23C23×D16314yes—
158C23C23×D16312yes—
159C23C23×D16311yescharacteristic
160D4—32632no—
161D4—32632no—
162D4—65216no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—32616no—
166C2×C4—3268no—
167C2×C4—1638no—
168C2×C4C23×D16318yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—3268no—
175C22—32616no—
176C22—65232no—
177C22—65216no—
178C22C22×D65218yes—
179C22C22×D65214yes—
180C22C22×D65212yes—
181C22#1910478014yes—
182C22C24×D16311yescharacteristic
183C4—28no—
184C4—3268no—
185C2—28no—
186C2—24no—
187C2—6528no—
188C2#2378222714yesdirect product factor
189C2C23×D65211yescharacteristic
190C2C2×#1910478012yes—
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC22×C22⋊C4×C163C2
socleC23×C163C24
hypercenterC22×C22⋊C4D163
hypocenterC163C2×#534
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D163C2semidirect
2. #23782227C2direct (N × H)
3. C2×#19104719C2semidirect
4. C2×#19104709C2semidirect
5. C2×#19104705C2semidirect
6. C22×C22⋊C4×C163C2semidirect
7. C24×D163C22semidirect
8. C22×D652C22semidirect
11. C22×C22⋊D163C22semidirect
12. C22×C22⋊D163C22semidirect
13. C23×Dic163C22semidirect
14. #19104719C22semidirect
15. #19104709C22semidirect
16. #19104705C22semidirect
17. C2×C4⋊Dic163C22semidirect
18. C2×(C2×D163)⋊C4C22semidirect
20. C23×C4×C163C22semidirect
21. C2×C22⋊C4×C163C22semidirect
22. C2×C22⋊C4×C163C22semidirect
26. C23×D163C23semidirect
28. C2×D652C23semidirect
33. C2×C22⋊D163C23semidirect
36. C22×Dic163C23semidirect
37. C4⋊Dic163C23semidirect
38. (C2×D163)⋊C4C23semidirect
40. C24×C163D4semidirect
46. C22×C4×C163C23semidirect
47. C22⋊C4×C163C23semidirect
66. C23×C163C2×D4semidirect
91. C2×C163#534semidirect
95. C163C2×#534semidirect
102. C22×C22⋊C4D163semidirect
111. C23×C4C2×D163semidirect
117. C2×C22⋊C4C2×D163semidirect
118. C2×C22⋊C4C2×D163semidirect
124. C24D652semidirect
140. C22×C4C22×D163semidirect
143. C22⋊C4C22×D163semidirect
156. C23C2×D652semidirect
188. C2#23782227direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1632
#237822272
C2×#191047192
C2×#191047092
C2×#191047052
C22×C22⋊C4×C1632
C2×#534163