Finite Group Databasebeta

← C2×#20261477 #27155856

Subgroups — C2×#20261477 #27155856

Order 12416.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 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#20261477C2116yesmaximal, direct product factor
2C2×#15690799C212yesmaximal, complemented
3C2×#15690798C212yesmaximal, complemented
4C2×#15690796C214yesmaximal, complemented
5C2×#15690780C214yesmaximal, complemented
6C2×#15690778C212yesmaximal, complemented
7C2×#124×C97C211yesmaximal, characteristic, complemented
8C24×D97C2212yescomplemented
9C22×Dic194C2212yescomplemented
10C22×D388C2212yes—
11C22×D388C2212yes—
12C22×C22⋊D97C2214yes—
13#15690799C22116yescomplemented
14#15690798C22116yescomplemented
15#15690796C22132yescomplemented
16#15690780C22132yescomplemented
17#15690778C22116yescomplemented
18C2×C22⋊Dic97C2212yescomplemented
19C2×#13095488C2214yescomplemented
20C2×#13095488C2214yescomplemented
21C2×#13095488C2214yescomplemented
22C2×C4×Dic97C2212yescomplemented
23C22×Q8×C97C2211yescharacteristic, complemented
24C22×D4×C97C2211yescharacteristic
25#124×C97C2218yescomplemented
26C2×C22⋊C4×C97C2214yescomplemented
27C2×C42×C97C2211yescharacteristic, complemented
28C23×D97—48no—
29C23×D97—44no—
30C23×D97C2314yes—
31C23×D97C2318yescomplemented
32C2×Dic194—44no—
33C2×Dic194C2318yescomplemented
34C2×D388—44no—
35C2×D388—44no—
36C2×D388C2318yes—
37C2×D388C2318yes—
38C2×C22⋊D97—48no—
39C2×C22⋊D97C23116yes—
40C22×Dic97C2314yes—
41C22⋊Dic97C2318yescomplemented
42#13095488C23116yescomplemented
43#13095488C23116yescomplemented
44#13095488C23116yescomplemented
45C4×Dic97C2318yescomplemented
46C24×C97C2312yes—
47C2×Q8×C97—42no—
48C2×Q8×C97C2314yescomplemented
49C2×D4×C97—42no—
50C2×D4×C97C2314yes—
51C22×C4×C97C2312yes—
52C22×C4×C97C2311yescharacteristic
53C22×C4×C97C2312yes—
54C22⋊C4×C97C23116yescomplemented
55C42×C97C2314yescomplemented
56C23×C97—24no—
57C23×C97—28no—
58C23×C97C2414yes—
59C23×C97C2411yescharacteristic
60C22×D97—28no—
61C22×D97—28no—
62C22×D97—28no—
63C22×D97—48no—
64C22×D97—416no—
65C22×D97—432no—
66C22×D97C2418yes—
67Q8×C97—44no—
68D4×C97—44no—
69C2×C4×C97—22no—
70C2×C4×C97—24no—
71C2×C4×C97—24no—
72C2×C4×C97C2414yes—
73C2×C4×C97C2412yes—
74C2×C4×C97C2414yes—
75C22⋊D97—416no—
76C2×Dic97—28no—
77C2×Dic97C2418yes—
78D388—48no—
79D388—48no—
80Dic194—48no—
81C22×C97—28no—
82C22×C97—24no—
83C22×C97—48no—
84C22×C97H(5,𝔽2)12yes—
85C22×C97H(5,𝔽2)14yes—
86C22×C97C2511yescharacteristic
87C2×D97—28no—
88C2×D97—28no—
89C2×D97—28no—
90C2×D97—432no—
91C2×D97—416no—
92C4×C97—24no—
93C4×C97—24no—
94C4×C97—22no—
95Dic97—28no—
96C2×C97—44no—
97C2×C97H(3,𝔽4)14yescomplemented
98C2×C97C2×H(5,𝔽2)11yescharacteristic
99C2×C97C2×H(5,𝔽2)12yes—
100D97—48no—
101C2×H(3,𝔽4)—971nomaximal, Sylow
102C97C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
103H(3,𝔽4)—9716no—
104C2×#124—972no—
105C2×#124—974no—
106C2×#124—972no—
107C2×#124D9711yescharacteristic, complemented
108C2×#120—972no—
109C2×#120—974no—
110C25—972no—
111C22×Q8—972no—
112C22×Q8C2×D9711yescharacteristic, complemented
113C22×D4—974no—
114C22×D4—972no—
115C22×D4—972no—
116C22×D4C2×D9711yescharacteristic
117#124—9716no—
118#124—9732no—
119#124—9716no—
120#124C2×D9718yescomplemented
121#120—9716no—
122#120—9732no—
123C2×C22⋊C4—972no—
124C2×C22⋊C4—974no—
125C2×C22⋊C4—974no—
126C2×C22⋊C4—974no—
127C2×C22⋊C4C2×D9714yescomplemented
128C2×C42—972no—
129C2×C42C2×D9711yescharacteristic, complemented
130C24—978no—
131C24—974no—
132C24—3884no—
133C24—3888no—
134C24C22×D9712yes—
135C2×Q8—42no—
136C2×Q8—978no—
137C2×Q8—3884no—
138C2×Q8C22×D9714yescomplemented
139C2×D4—42no—
140C2×D4—9716no—
141C2×D4—978no—
142C2×D4—978no—
143C2×D4—3884no—
144C2×D4—3884no—
145C2×D4—3888no—
146C2×D4C22×D9714yes—
147C22×C4—974no—
148C22×C4C22×D9712yes—
149C22×C4C22×D9711yescharacteristic
150C22×C4C22×D9712yes—
151C22⋊C4—978no—
152C22⋊C4—9716no—
153C22⋊C4—9716no—
154C22⋊C4—9716no—
155C22⋊C4C22×D97116yescomplemented
156C42—978no—
157C42C22×D9714yescomplemented
158C23—24no—
159C23—28no—
160C23—978no—
161C23—3888no—
162C23—38816no—
163C23—38832no—
164C23—1948no—
165C23—1948no—
166C23—1948no—
167C23C23×D9714yes—
168C23C23×D9711yescharacteristic
169Q8—44no—
170Q8—3888no—
171D4—44no—
172D4—3888no—
173D4—3888no—
174D4—38816no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—978no—
179C2×C4—1948no—
180C2×C4C23×D9714yes—
181C2×C4C23×D9712yes—
182C2×C4C23×D9714yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—38832no—
187C22—38816no—
188C22—1948no—
189C22—1948no—
190C22—1948no—
191C22#1569085512yes—
192C22#1569085614yes—
193C22C24×D9711yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—1948no—
198C2—44no—
199C2—3888no—
200C2#2026147714yesdirect product factor
201C2C2×#1569085511yescharacteristic
202C2C2×#1569085612yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#124×C97C2
socleC23×C97C24
hypercenterC2×#124D97
hypocenterC97C2×H(3,𝔽4)
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20261477C2direct (N × H)
2. C2×#15690799C2semidirect
3. C2×#15690798C2semidirect
4. C2×#15690796C2semidirect
5. C2×#15690780C2semidirect
6. C2×#15690778C2semidirect
7. C2×#124×C97C2semidirect
8. C24×D97C22semidirect
9. C22×Dic194C22semidirect
13. #15690799C22semidirect
14. #15690798C22semidirect
15. #15690796C22semidirect
16. #15690780C22semidirect
17. #15690778C22semidirect
18. C2×C22⋊Dic97C22semidirect
19. C2×#13095488C22semidirect
20. C2×#13095488C22semidirect
21. C2×#13095488C22semidirect
22. C2×C4×Dic97C22semidirect
23. C22×Q8×C97C22semidirect
25. #124×C97C22semidirect
26. C2×C22⋊C4×C97C22semidirect
27. C2×C42×C97C22semidirect
31. C23×D97C23semidirect
33. C2×Dic194C23semidirect
41. C22⋊Dic97C23semidirect
42. #13095488C23semidirect
43. #13095488C23semidirect
44. #13095488C23semidirect
45. C4×Dic97C23semidirect
48. C2×Q8×C97C23semidirect
54. C22⋊C4×C97C23semidirect
55. C42×C97C23semidirect
97. C2×C97H(3,𝔽4)semidirect
102. C97C2×H(3,𝔽4)semidirect
107. C2×#124D97semidirect
112. C22×Q8C2×D97semidirect
120. #124C2×D97semidirect
127. C2×C22⋊C4C2×D97semidirect
129. C2×C42C2×D97semidirect
138. C2×Q8C22×D97semidirect
155. C22⋊C4C22×D97semidirect
157. C42C22×D97semidirect
200. C2#20261477direct (N × H)

Maximal subgroups

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

subgroupindex
#202614772
C2×#156907992
C2×#156907982
C2×#156907962
C2×#156907802
C2×#156907782
C2×#124×C972
C2×H(3,𝔽4)97