Finite Group Databasebeta

← C2×#20261297 #27155039

Subgroups — C2×#20261297 #27155039

Order 12416.

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⋊D97C212yesmaximal, complemented
2#20261297C2116yesmaximal, direct product factor
3C2×#15690794C214yesmaximal, complemented
4C2×#15690784C214yesmaximal, complemented
5C2×#15690780C214yesmaximal, complemented
6C22×C22⋊C4×C97C211yesmaximal, characteristic, complemented
7C24×D97C2212yescomplemented
8C22×D388C2214yescomplemented
9C22×C22⋊D97—216no—
10C22×C22⋊D97C2214yes—
11C22×C22⋊D97C2214yescomplemented
12C22×C22⋊D97C22116yescomplemented
13C23×Dic97C2212yescomplemented
14#15690794C22132yescomplemented
15#15690784C22132yescomplemented
16#15690780C22132yescomplemented
17C2×C4⋊Dic97C2214yescomplemented
18C2×#13095488C2218yescomplemented
19C25×C97C2211yescharacteristic
20C23×C4×C97C2212yescomplemented
21C2×C22⋊C4×C97C2214yescomplemented
22C2×C22⋊C4×C97C2218yescomplemented
23C23×D97—28no—
24C23×D97—48no—
25C23×D97C2314yes—
26C23×D97C2318yescomplemented
27C2×D388—48no—
28C2×D388C23116yescomplemented
29C2×C22⋊D97—216no—
30C2×C22⋊D97—216no—
31C2×C22⋊D97—264no—
32C2×C22⋊D97C23116yes—
33C2×C22⋊D97C23116yescomplemented
34C22×Dic97—28no—
35C22×Dic97C2314yes—
36C22×Dic97C2318yescomplemented
37C4⋊Dic97C23116yescomplemented
38#13095488C23132yescomplemented
39C24×C97—44no—
40C24×C97D418yescomplemented
41C24×C97C2312yes—
42C24×C97C2311yescharacteristic
43C24×C97C2314yes—
44C22×C4×C97—44no—
45C22×C4×C97C2314yes—
46C22×C4×C97C2318yescomplemented
47C22⋊C4×C97C23116yescomplemented
48C23×C97—24no—
49C23×C97—28no—
50C23×C97—416no—
51C23×C97—48no—
52C23×C97C2×D418yes—
53C23×C97C2×D414yes—
54C23×C97C2×D4116yescomplemented
55C23×C97C2414yes—
56C23×C97C2412yes—
57C23×C97C2411yescharacteristic
58C22×D97—28no—
59C22×D97—216no—
60C22×D97—216no—
61C22×D97—416no—
62C22×D97—432no—
63C22×D97C2418yes—
64C2×C4×C97—28no—
65C2×C4×C97—48no—
66C2×C4×C97C2418yes—
67C22⋊D97—232no—
68C22⋊D97—232no—
69C2×Dic97—28no—
70C2×Dic97—216no—
71C2×Dic97C2418yes—
72D388—416no—
73C22×C97—216no—
74C22×C97—28no—
75C22×C97—24no—
76C22×C97—28no—
77C22×C97—416no—
78C22×C97C22×D418yes—
79C22×C97C22×D414yes—
80C22×C97C22×D412yes—
81C22×C97H(5,𝔽2)14yes—
82C22×C97C2511yescharacteristic
83C2×D97—28no—
84C2×D97—216no—
85C2×D97—432no—
86C2×D97—416no—
87C4×C97—28no—
88Dic97—28no—
89C2×C97—28no—
90C2×C97—24no—
91C2×C97#53414yescomplemented
92C2×C97C23×D411yescharacteristic
93C2×C97C2×H(5,𝔽2)12yes—
94D97—48no—
95C2×#534—971nomaximal, Sylow
96C97C2×#53411yescharacteristic, complemented, Sylow
97C23×D4—972no—
98#534—9716no—
99C2×#123—974no—
100C2×#121—974no—
101C2×#120—974no—
102C22×C22⋊C4D9711yescharacteristic, complemented
103C25—972no—
104C25C2×D9711yescharacteristic
105C22×D4—974no—
106C22×D4—974no—
107C22×D4—9716no—
108C22×D4—974no—
109C22×D4—19416no—
110C23×C4—972no—
111C23×C4C2×D9712yescomplemented
112#123—9732no—
113#121—9732no—
114#120—9732no—
115C2×C4⋊C4—974no—
116C2×C22⋊C4—978no—
117C2×C22⋊C4C2×D9714yescomplemented
118C2×C22⋊C4C2×D9718yescomplemented
119C24—44no—
120C24—978no—
121C24—974no—
122C24—3888no—
123C24—1948no—
124C24D38818yescomplemented
125C24C22×D9712yes—
126C24C22×D9711yescharacteristic
127C24C22×D9714yes—
128C2×D4—9716no—
129C2×D4—9716no—
130C2×D4—9716no—
131C2×D4—3888no—
132C2×D4—19416no—
133C2×D4—19416no—
134C2×D4—19464no—
135C22×C4—44no—
136C22×C4—978no—
137C22×C4—974no—
138C22×C4—1948no—
139C22×C4C22×D9714yes—
140C22×C4C22×D9718yescomplemented
141C4⋊C4—9716no—
142C22⋊C4—9732no—
143C22⋊C4C22×D97116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—978no—
149C23—38832no—
150C23—38816no—
151C23—19416no—
152C23—1948no—
153C23—19416no—
154C23C2×D38818yes—
155C23C2×D38814yes—
156C23C2×D388116yescomplemented
157C23C23×D9714yes—
158C23C23×D9712yes—
159C23C23×D9711yescharacteristic
160D4—38816no—
161D4—19432no—
162D4—19432no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—978no—
166C2×C4—19416no—
167C2×C4—1948no—
168C2×C4C23×D9718yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—38832no—
175C22—38816no—
176C22—1948no—
177C22—19416no—
178C22C22×D38818yes—
179C22C22×D38814yes—
180C22C22×D38812yes—
181C22#1569085514yes—
182C22C24×D9711yescharacteristic
183C4—28no—
184C4—1948no—
185C2—28no—
186C2—24no—
187C2—3888no—
188C2#2026129714yesdirect product factor
189C2C23×D38811yescharacteristic
190C2C2×#1569085512yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC22×C22⋊C4×C97C2
socleC23×C97C24
hypercenterC22×C22⋊C4D97
hypocenterC97C2×#534
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D97C2semidirect
2. #20261297C2direct (N × H)
3. C2×#15690794C2semidirect
4. C2×#15690784C2semidirect
5. C2×#15690780C2semidirect
6. C22×C22⋊C4×C97C2semidirect
7. C24×D97C22semidirect
8. C22×D388C22semidirect
11. C22×C22⋊D97C22semidirect
12. C22×C22⋊D97C22semidirect
13. C23×Dic97C22semidirect
14. #15690794C22semidirect
15. #15690784C22semidirect
16. #15690780C22semidirect
17. C2×C4⋊Dic97C22semidirect
18. C2×#13095488C22semidirect
20. C23×C4×C97C22semidirect
21. C2×C22⋊C4×C97C22semidirect
22. C2×C22⋊C4×C97C22semidirect
26. C23×D97C23semidirect
28. C2×D388C23semidirect
33. C2×C22⋊D97C23semidirect
36. C22×Dic97C23semidirect
37. C4⋊Dic97C23semidirect
38. #13095488C23semidirect
40. C24×C97D4semidirect
46. C22×C4×C97C23semidirect
47. C22⋊C4×C97C23semidirect
54. C23×C97C2×D4semidirect
91. C2×C97#534semidirect
96. C97C2×#534semidirect
102. C22×C22⋊C4D97semidirect
111. C23×C4C2×D97semidirect
117. C2×C22⋊C4C2×D97semidirect
118. C2×C22⋊C4C2×D97semidirect
124. C24D388semidirect
140. C22×C4C22×D97semidirect
143. C22⋊C4C22×D97semidirect
156. C23C2×D388semidirect
188. C2#20261297direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
#202612972
C2×#156907942
C2×#156907842
C2×#156907802
C22×C22⋊C4×C972
C2×#53497