Finite Group Databasebeta

← C2×#20265919 #27185252

Subgroups — C2×#20265919 #27185252

Order 12928.

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#20265919C2116yesmaximal, direct product factor
2C2×#15692244C212yesmaximal, complemented
3C2×#15692243C212yesmaximal, complemented
4C2×#15692241C214yesmaximal, complemented
5C2×#15692225C214yesmaximal, complemented
6C2×#15692223C212yesmaximal, complemented
7C2×#124×C101C211yesmaximal, characteristic, complemented
8C24×D101C2212yescomplemented
9C22×Dic202C2212yescomplemented
10C22×D404C2212yes—
11C22×D404C2212yes—
12C22×C22⋊D101C2214yes—
13#15692244C22116yescomplemented
14#15692243C22116yescomplemented
15#15692241C22132yescomplemented
16#15692225C22132yescomplemented
17#15692223C22116yescomplemented
18C2×C22⋊Dic101C2212yescomplemented
19C2×#13108051C2214yescomplemented
20C2×#13108051C2214yescomplemented
21C2×#13108051C2214yescomplemented
22C2×C4×Dic101C2212yescomplemented
23C22×Q8×C101C2211yescharacteristic, complemented
24C22×D4×C101C2211yescharacteristic
25#124×C101C2218yescomplemented
26C2×C22⋊C4×C101C2214yescomplemented
27C2×C42×C101C2211yescharacteristic, complemented
28C23×D101—48no—
29C23×D101—44no—
30C23×D101C2314yes—
31C23×D101C2318yescomplemented
32C2×Dic202—44no—
33C2×Dic202C2318yescomplemented
34C2×D404—44no—
35C2×D404—44no—
36C2×D404C2318yes—
37C2×D404C2318yes—
38C2×C22⋊D101—48no—
39C2×C22⋊D101C23116yes—
40C22×Dic101C2314yes—
41C22⋊Dic101C2318yescomplemented
42#13108051C23116yescomplemented
43#13108051C23116yescomplemented
44#13108051C23116yescomplemented
45C4×Dic101C2318yescomplemented
46C24×C101C2312yes—
47C2×Q8×C101—42no—
48C2×Q8×C101C2314yescomplemented
49C2×D4×C101—42no—
50C2×D4×C101C2314yes—
51C22×C4×C101C2312yes—
52C22×C4×C101C2312yes—
53C22×C4×C101C2311yescharacteristic
54C22⋊C4×C101C23116yescomplemented
55C42×C101C2314yescomplemented
56C23×C101—24no—
57C23×C101—28no—
58C23×C101C2414yes—
59C23×C101C2411yescharacteristic
60C22×D101—28no—
61C22×D101—28no—
62C22×D101—28no—
63C22×D101—48no—
64C22×D101—416no—
65C22×D101—432no—
66C22×D101C2418yes—
67Q8×C101—44no—
68D4×C101—44no—
69C2×C4×C101—22no—
70C2×C4×C101—24no—
71C2×C4×C101—24no—
72C2×C4×C101C2414yes—
73C2×C4×C101C2412yes—
74C2×C4×C101C2414yes—
75C22⋊D101—416no—
76C2×Dic101—28no—
77C2×Dic101C2418yes—
78D404—48no—
79D404—48no—
80Dic202—48no—
81C22×C101—28no—
82C22×C101—24no—
83C22×C101—48no—
84C22×C101H(5,𝔽2)12yes—
85C22×C101H(5,𝔽2)14yes—
86C22×C101C2511yescharacteristic
87C2×D101—28no—
88C2×D101—28no—
89C2×D101—28no—
90C2×D101—432no—
91C2×D101—416no—
92C4×C101—24no—
93C4×C101—24no—
94C4×C101—22no—
95Dic101—28no—
96C2×C101—44no—
97C2×C101H(3,𝔽4)14yescomplemented
98C2×C101C2×H(5,𝔽2)11yescharacteristic
99C2×C101C2×H(5,𝔽2)12yes—
100D101—48no—
101C2×H(3,𝔽4)—1011nomaximal, Sylow
102C101C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
103H(3,𝔽4)—10116no—
104C2×#124—1012no—
105C2×#124—1014no—
106C2×#124—1012no—
107C2×#124D10111yescharacteristic, complemented
108C2×#120—1012no—
109C2×#120—1014no—
110C25—1012no—
111C22×Q8—1012no—
112C22×Q8C2×D10111yescharacteristic, complemented
113C22×D4—1014no—
114C22×D4—1012no—
115C22×D4—1012no—
116C22×D4C2×D10111yescharacteristic
117#124—10116no—
118#124—10132no—
119#124—10116no—
120#124C2×D10118yescomplemented
121#120—10116no—
122#120—10132no—
123C2×C22⋊C4—1012no—
124C2×C22⋊C4—1014no—
125C2×C22⋊C4—1014no—
126C2×C22⋊C4—1014no—
127C2×C22⋊C4C2×D10114yescomplemented
128C2×C42—1012no—
129C2×C42C2×D10111yescharacteristic, complemented
130C24—1018no—
131C24—1014no—
132C24—4044no—
133C24—4048no—
134C24C22×D10112yes—
135C2×Q8—42no—
136C2×Q8—1018no—
137C2×Q8—4044no—
138C2×Q8C22×D10114yescomplemented
139C2×D4—42no—
140C2×D4—10116no—
141C2×D4—1018no—
142C2×D4—1018no—
143C2×D4—4044no—
144C2×D4—4044no—
145C2×D4—4048no—
146C2×D4C22×D10114yes—
147C22×C4—1014no—
148C22×C4C22×D10112yes—
149C22×C4C22×D10111yescharacteristic
150C22×C4C22×D10112yes—
151C22⋊C4—1018no—
152C22⋊C4—10116no—
153C22⋊C4—10116no—
154C22⋊C4—10116no—
155C22⋊C4C22×D101116yescomplemented
156C42—1018no—
157C42C22×D10114yescomplemented
158C23—24no—
159C23—28no—
160C23—1018no—
161C23—4048no—
162C23—40416no—
163C23—40432no—
164C23—2028no—
165C23—2028no—
166C23—2028no—
167C23C23×D10114yes—
168C23C23×D10111yescharacteristic
169Q8—44no—
170Q8—4048no—
171D4—44no—
172D4—4048no—
173D4—4048no—
174D4—40416no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—1018no—
179C2×C4—2028no—
180C2×C4C23×D10114yes—
181C2×C4C23×D10112yes—
182C2×C4C23×D10114yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—40432no—
187C22—40416no—
188C22—2028no—
189C22—2028no—
190C22—2028no—
191C22#1569230012yes—
192C22#1569230114yes—
193C22C24×D10111yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—2028no—
198C2—44no—
199C2—4048no—
200C2#2026591914yesdirect product factor
201C2C2×#1569230011yescharacteristic
202C2C2×#1569230112yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#124×C101C2
socleC23×C101C24
hypercenterC2×#124D101
hypocenterC101C2×H(3,𝔽4)
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20265919C2direct (N × H)
2. C2×#15692244C2semidirect
3. C2×#15692243C2semidirect
4. C2×#15692241C2semidirect
5. C2×#15692225C2semidirect
6. C2×#15692223C2semidirect
7. C2×#124×C101C2semidirect
8. C24×D101C22semidirect
9. C22×Dic202C22semidirect
13. #15692244C22semidirect
14. #15692243C22semidirect
15. #15692241C22semidirect
16. #15692225C22semidirect
17. #15692223C22semidirect
18. C2×C22⋊Dic101C22semidirect
19. C2×#13108051C22semidirect
20. C2×#13108051C22semidirect
21. C2×#13108051C22semidirect
22. C2×C4×Dic101C22semidirect
23. C22×Q8×C101C22semidirect
25. #124×C101C22semidirect
26. C2×C22⋊C4×C101C22semidirect
27. C2×C42×C101C22semidirect
31. C23×D101C23semidirect
33. C2×Dic202C23semidirect
41. C22⋊Dic101C23semidirect
42. #13108051C23semidirect
43. #13108051C23semidirect
44. #13108051C23semidirect
45. C4×Dic101C23semidirect
48. C2×Q8×C101C23semidirect
54. C22⋊C4×C101C23semidirect
55. C42×C101C23semidirect
97. C2×C101H(3,𝔽4)semidirect
102. C101C2×H(3,𝔽4)semidirect
107. C2×#124D101semidirect
112. C22×Q8C2×D101semidirect
120. #124C2×D101semidirect
127. C2×C22⋊C4C2×D101semidirect
129. C2×C42C2×D101semidirect
138. C2×Q8C22×D101semidirect
155. C22⋊C4C22×D101semidirect
157. C42C22×D101semidirect
200. C2#20265919direct (N × H)

Maximal subgroups

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

subgroupindex
#202659192
C2×#156922442
C2×#156922432
C2×#156922412
C2×#156922252
C2×#156922232
C2×#124×C1012
C2×H(3,𝔽4)101