Finite Group Databasebeta

← C2×#20266014 #27185941

Subgroups — C2×#20266014 #27185941

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#20266014C2132yesmaximal, direct product factor
2C2×#15692303C215yesmaximal, complemented
3C2×#15692301C2110yesmaximal, complemented
4C22×#13108075C215yesmaximal, complemented
5C2×#41×D101C2110yesmaximal, complemented
6C2×2−1+4×C101C211yesmaximal, characteristic, complemented
7#15692303C22180yescomplemented
8#15692301C221160yescomplemented
9C2×#13108075C22110yescomplemented
10C2×#13108075C22120yescomplemented
11C2×#13108075C22180yescomplemented
12C2×#13108074C22130yescomplemented
13C2×#13108073C22110yescomplemented
14#41×D101C221160yescomplemented
15C2×Q8×D101C22110yescomplemented
16C22×D404C22115yescomplemented
17C2×D4×D101C22130yescomplemented
18C22×C4×D101C22115yescomplemented
192−1+4×C101C22116yescomplemented
20C2×#41×C101C22110yescomplemented
21C22×Q8×C101C2215yescomplemented
22C23×D101C23115yes—
23#13108075C23180yescomplemented
24#13108075C231160yescomplemented
25#13108074C231240yescomplemented
26#13108073C23180yescomplemented
27C2×Dic202C23110yes—
28Q8×D101C23180yescomplemented
29C2×D404C23130yes—
30C2×D404C231120yescomplemented
31C2×C22⋊D101C23120yes—
32D4×D101C231240yescomplemented
33C22×Dic101C2315yes—
34C2×C4×D101C23110yes—
35C2×C4×D101C23130yes—
36C2×C4×D101C231120yescomplemented
37#41×C101C23180yescomplemented
38C2×Q8×C101C23110yes—
39C2×Q8×C101C23140yescomplemented
40C2×D4×C101C23110yes—
41C22×C4×C101C23115yes—
42C23×C101C2415yes—
43C22×D101—430no—
44C22×D101C24110yes—
45C22×D101C24160yes—
46Q8×C101C24140yes—
47D4×C101C24140yes—
48C2×C4×C101C24110yes—
49C2×C4×C101C24160yes—
50C22⋊D101C24180yes—
51C2×Dic101C2411yescharacteristic
52C2×Dic101C2415yes—
53C2×Dic101C24120yes—
54D404C241120yes—
55C4×D101C24140yes—
56C4×D101C241120yes—
57Dic202C24140yes—
58C22×C101—210no—
59C22×C101C25110yes—
60C22×C101C2511yescharacteristic
61C2×D101—220no—
62C2×D101—460no—
63C2×D101C25120yes—
64C4×C101C25120yes—
65Dic101C25110yes—
66Dic101C2512yes—
67C2×C101—210no—
68C2×C101#58512yescomplemented
69C2×C101C2611yescharacteristic
70D101—220no—
71C2×#585—1011nomaximal, Sylow
72C101C2×#58511yescharacteristic, complemented, Sylow
73#585—10132no—
74C2×2−1+4—1015no—
75C2×2−1+4D10111yescharacteristic, complemented
76C2×H(5,𝔽2)—10110no—
77C22×#41—10110no—
78C22×#41—1015no—
792−1+4—10180no—
802−1+4C2×D101116yescomplemented
81H(5,𝔽2)—101160no—
82C2×#41—101160no—
83C2×#41—10110no—
84C2×#41—10120no—
85C2×#41—10180no—
86C2×#41—10110no—
87C2×#41—10130no—
88C2×#41C2×D101110yescomplemented
89C22×Q8—10110no—
90C22×Q8C2×D10115yescomplemented
91C22×D4—10130no—
92C22×D4—10115no—
93C23×C4—10115no—
94C24—10115no—
95#41—10180no—
96#41—101160no—
97#41—10180no—
98#41—101240no—
99#41C22×D101180yescomplemented
100C2×Q8—10180no—
101C2×Q8—10110no—
102C2×Q8C22×D101110yes—
103C2×Q8C22×D101140yescomplemented
104C2×D4—10120no—
105C2×D4—101240no—
106C2×D4—10130no—
107C2×D4—101120no—
108C2×D4C22×D101110yes—
109C22×C4—1015no—
110C22×C4—10110no—
111C22×C4—10130no—
112C22×C4—101120no—
113C22×C4C22×D101115yes—
114C23—10160no—
115C23—10110no—
116C23—40430no—
117C23C23×D10115yes—
118Q8—10140no—
119Q8C23×D101140yes—
120D4—10180no—
121D4—101120no—
122D4C23×D101140yes—
123C2×C4—10120no—
124C2×C4—1011no—
125C2×C4—1015no—
126C2×C4—10140no—
127C2×C4—101120no—
128C2×C4C23×D101160yes—
129C2×C4C23×D101110yes—
130C22—210no—
131C22—10120no—
132C22—40460no—
133C22—20220no—
134C22C24×D101110yes—
135C22C24×D10111yescharacteristic
136C4—1012no—
137C4—10110no—
138C4C24×D101120yes—
139C2—210no—
140C2—20220no—
141C2#2026601412yesdirect product factor
142C2C25×D10111yescharacteristic
subgroupG/H
centerC22C24×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC2×2−1+4×C101C2
socleC22×C101C25
hypercenterC2×2−1+4D101
hypocenterC101C2×#585
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20266014C2direct (N × H)
2. C2×#15692303C2semidirect
3. C2×#15692301C2semidirect
4. C22×#13108075C2semidirect
5. C2×#41×D101C2semidirect
6. C2×2−1+4×C101C2semidirect
7. #15692303C22semidirect
8. #15692301C22semidirect
9. C2×#13108075C22semidirect
10. C2×#13108075C22semidirect
11. C2×#13108075C22semidirect
12. C2×#13108074C22semidirect
13. C2×#13108073C22semidirect
14. #41×D101C22semidirect
15. C2×Q8×D101C22semidirect
16. C22×D404C22semidirect
17. C2×D4×D101C22semidirect
18. C22×C4×D101C22semidirect
19. 2−1+4×C101C22semidirect
20. C2×#41×C101C22semidirect
21. C22×Q8×C101C22semidirect
23. #13108075C23semidirect
24. #13108075C23semidirect
25. #13108074C23semidirect
26. #13108073C23semidirect
28. Q8×D101C23semidirect
30. C2×D404C23semidirect
32. D4×D101C23semidirect
36. C2×C4×D101C23semidirect
37. #41×C101C23semidirect
39. C2×Q8×C101C23semidirect
68. C2×C101#585semidirect
72. C101C2×#585semidirect
75. C2×2−1+4D101semidirect
80. 2−1+4C2×D101semidirect
88. C2×#41C2×D101semidirect
90. C22×Q8C2×D101semidirect
99. #41C22×D101semidirect
103. C2×Q8C22×D101semidirect
141. C2#20266014direct (N × H)

Maximal subgroups

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

subgroupindex
#202660142
C2×#156923032
C2×#156923012
C22×#131080752
C2×#41×D1012
C2×2−1+4×C1012
C2×#585101