Finite Group Databasebeta

← C2×#20266012 #27185939

Subgroups — C2×#20266012 #27185939

Order 12928.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#20266012C2132yesmaximal, direct product factor
2C2×#15692302C216yesmaximal, complemented
3C2×#15692300C219yesmaximal, complemented
4C22×#13108073C219yesmaximal, complemented
5C2×#41×D101C216yesmaximal, complemented
6C2×H(5,𝔽2)×C101C211yesmaximal, characteristic, complemented
7#15692302C22196yescomplemented
8#15692300C221144yescomplemented
9C2×#13108075C2212yescomplemented
10C2×#13108074C22118yescomplemented
11C2×#13108073C22118yescomplemented
12C2×#13108073C22136yescomplemented
13C2×#13108073C221144yescomplemented
14#41×D101C22196yescomplemented
15C22×Dic202C2219yescomplemented
16C2×Q8×D101C2216yescomplemented
17C22×C22⋊D101C22118yescomplemented
18C2×D4×D101C22118yescomplemented
19C23×Dic101C2216yescomplemented
20C22×C4×D101C2219yescomplemented
21H(5,𝔽2)×C101C22116yescomplemented
22C2×#41×C101C2216yescomplemented
23C22×D4×C101C2219yescomplemented
24C23×D101C2319yes—
25#13108075C23116yescomplemented
26#13108074C231144yescomplemented
27#13108073C231144yescomplemented
28#13108073C231288yescomplemented
29C2×Dic202C23118yes—
30C2×Dic202C23172yescomplemented
31Q8×D101C23148yescomplemented
32C2×D404C2316yes—
33C2×C22⋊D101C23136yes—
34C2×C22⋊D101C231144yescomplemented
35D4×D101C231144yescomplemented
36C22×Dic101C2319yes—
37C22×Dic101C23118yes—
38C22×Dic101C23148yescomplemented
39C2×C4×D101C2316yes—
40C2×C4×D101C23118yes—
41C2×C4×D101C23172yescomplemented
42C24×C101C2316yes—
43#41×C101C23148yescomplemented
44C2×Q8×C101C2312yes—
45C2×D4×C101C23118yes—
46C2×D4×C101C23172yescomplemented
47C22×C4×C101C2319yes—
48C23×C101—412no—
49C23×C101C2419yes—
50C23×C101C24124yes—
51C22×D101—418no—
52C22×D101C2416yes—
53C22×D101C24136yes—
54Q8×C101C2418yes—
55D4×C101C24172yes—
56C2×C4×C101C2416yes—
57C2×C4×C101C24136yes—
58C22⋊D101C241144yes—
59C2×Dic101C2411yescharacteristic
60C2×Dic101C2419yes—
61C2×Dic101C24136yes—
62C2×Dic101C24172yes—
63D404C24124yes—
64C4×D101C24124yes—
65C4×D101C24172yes—
66Dic202C24172yes—
67C22×C101—218no—
68C22×C101—424no—
69C22×C101C25118yes—
70C22×C101C2511yescharacteristic
71C2×D101—212no—
72C2×D101—436no—
73C2×D101C25112yes—
74C4×C101C25112yes—
75Dic101C25118yes—
76Dic101C2512yes—
77C2×C101—218no—
78C2×C101#58512yescomplemented
79C2×C101C2611yescharacteristic
80D101—212no—
81C2×#585—1011nomaximal, Sylow
82C101C2×#58511yescharacteristic, complemented, Sylow
83#585—10132no—
84C2×2−1+4—1016no—
85C2×H(5,𝔽2)—1019no—
86C2×H(5,𝔽2)D10111yescharacteristic, complemented
87C22×#41—1016no—
88C22×#41—1019no—
892−1+4—10196no—
90H(5,𝔽2)—101144no—
91H(5,𝔽2)C2×D101116yescomplemented
92C2×#41—10196no—
93C2×#41—1012no—
94C2×#41—10118no—
95C2×#41—10136no—
96C2×#41—101144no—
97C2×#41—10118no—
98C2×#41C2×D10116yescomplemented
99C22×Q8—1016no—
100C22×Q8—1019no—
101C22×D4—10118no—
102C22×D4—10118no—
103C22×D4C2×D10119yescomplemented
104C23×C4—1016no—
105C23×C4—1019no—
106C24—1019no—
107C24C22×D10116yes—
108#41—10116no—
109#41—101144no—
110#41—101288no—
111#41—101144no—
112#41C22×D101148yescomplemented
113C2×Q8—10148no—
114C2×Q8—10118no—
115C2×Q8—10172no—
116C2×Q8C22×D10112yes—
117C2×D4—10136no—
118C2×D4—101144no—
119C2×D4—101144no—
120C2×D4—1016no—
121C2×D4C22×D101118yes—
122C2×D4C22×D101172yescomplemented
123C22×C4—1019no—
124C22×C4—10118no—
125C22×C4—10148no—
126C22×C4—1016no—
127C22×C4—10118no—
128C22×C4—10172no—
129C22×C4C22×D10119yes—
130C23—412no—
131C23—10136no—
132C23—1016no—
133C23—40418no—
134C23C23×D10119yes—
135C23C23×D101124yes—
136Q8—10172no—
137Q8C23×D10118yes—
138D4—101144no—
139D4—10124no—
140D4C23×D101172yes—
141C2×C4—10136no—
142C2×C4—10172no—
143C2×C4—1011no—
144C2×C4—1019no—
145C2×C4—10124no—
146C2×C4—10172no—
147C2×C4C23×D101136yes—
148C2×C4C23×D10116yes—
149C22—218no—
150C22—424no—
151C22—10112no—
152C22—40436no—
153C22—20212no—
154C22C24×D101118yes—
155C22C24×D10111yescharacteristic
156C4—1012no—
157C4—10118no—
158C4C24×D101112yes—
159C2—218no—
160C2—20212no—
161C2#2026601212yesdirect product factor
162C2C25×D10111yescharacteristic
subgroupG/H
centerC22C24×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC2×H(5,𝔽2)×C101C2
socleC22×C101C25
hypercenterC2×H(5,𝔽2)D101
hypocenterC101C2×#585
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20266012C2direct (N × H)
2. C2×#15692302C2semidirect
3. C2×#15692300C2semidirect
4. C22×#13108073C2semidirect
5. C2×#41×D101C2semidirect
6. C2×H(5,𝔽2)×C101C2semidirect
7. #15692302C22semidirect
8. #15692300C22semidirect
9. C2×#13108075C22semidirect
10. C2×#13108074C22semidirect
11. C2×#13108073C22semidirect
12. C2×#13108073C22semidirect
13. C2×#13108073C22semidirect
14. #41×D101C22semidirect
15. C22×Dic202C22semidirect
16. C2×Q8×D101C22semidirect
17. C22×C22⋊D101C22semidirect
18. C2×D4×D101C22semidirect
19. C23×Dic101C22semidirect
20. C22×C4×D101C22semidirect
21. H(5,𝔽2)×C101C22semidirect
22. C2×#41×C101C22semidirect
23. C22×D4×C101C22semidirect
25. #13108075C23semidirect
26. #13108074C23semidirect
27. #13108073C23semidirect
28. #13108073C23semidirect
30. C2×Dic202C23semidirect
31. Q8×D101C23semidirect
34. C2×C22⋊D101C23semidirect
35. D4×D101C23semidirect
38. C22×Dic101C23semidirect
41. C2×C4×D101C23semidirect
43. #41×C101C23semidirect
46. C2×D4×C101C23semidirect
78. C2×C101#585semidirect
82. C101C2×#585semidirect
86. C2×H(5,𝔽2)D101semidirect
91. H(5,𝔽2)C2×D101semidirect
98. C2×#41C2×D101semidirect
103. C22×D4C2×D101semidirect
112. #41C22×D101semidirect
122. C2×D4C22×D101semidirect
161. C2#20266012direct (N × H)

Maximal subgroups

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

subgroupindex
#202660122
C2×#156923022
C2×#156923002
C22×#131080732
C2×#41×D1012
C2×H(5,𝔽2)×C1012
C2×#585101