Finite Group Databasebeta

← C2×#20261572 #27156545

Subgroups — C2×#20261572 #27156545

Order 12416.

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#20261572C2132yesmaximal, direct product factor
2C2×#15690858C215yesmaximal, complemented
3C2×#15690856C2110yesmaximal, complemented
4C22×#13095512C215yesmaximal, complemented
5C2×#41×D97C2110yesmaximal, complemented
6C2×2−1+4×C97C211yesmaximal, characteristic, complemented
7#15690858C22180yescomplemented
8#15690856C221160yescomplemented
9C2×#13095512C22110yescomplemented
10C2×#13095512C22120yescomplemented
11C2×#13095512C22180yescomplemented
12C2×#13095511C22130yescomplemented
13C2×#13095510C22110yescomplemented
14#41×D97C221160yescomplemented
15C2×Q8×D97C22110yescomplemented
16C22×D388C22115yescomplemented
17C2×D4×D97C22130yescomplemented
18C22×C4×D97C22115yescomplemented
192−1+4×C97C22116yescomplemented
20C2×#41×C97C22110yescomplemented
21C22×Q8×C97C2215yescomplemented
22C23×D97C23115yes—
23#13095512C23180yescomplemented
24#13095512C231160yescomplemented
25#13095511C231240yescomplemented
26#13095510C23180yescomplemented
27C2×Dic194C23110yes—
28Q8×D97C23180yescomplemented
29C2×D388C23130yes—
30C2×D388C231120yescomplemented
31C2×C22⋊D97C23120yes—
32D4×D97C231240yescomplemented
33C22×Dic97C2315yes—
34C2×C4×D97C23110yes—
35C2×C4×D97C23130yes—
36C2×C4×D97C231120yescomplemented
37#41×C97C23180yescomplemented
38C2×Q8×C97C23110yes—
39C2×Q8×C97C23140yescomplemented
40C2×D4×C97C23110yes—
41C22×C4×C97C23115yes—
42C23×C97C2415yes—
43C22×D97—430no—
44C22×D97C24110yes—
45C22×D97C24160yes—
46Q8×C97C24140yes—
47D4×C97C24140yes—
48C2×C4×C97C24110yes—
49C2×C4×C97C24160yes—
50C22⋊D97C24180yes—
51C2×Dic97C2411yescharacteristic
52C2×Dic97C2415yes—
53C2×Dic97C24120yes—
54D388C241120yes—
55C4×D97C24140yes—
56C4×D97C241120yes—
57Dic194C24140yes—
58C22×C97—210no—
59C22×C97C25110yes—
60C22×C97C2511yescharacteristic
61C2×D97—220no—
62C2×D97—460no—
63C2×D97C25120yes—
64C4×C97C25120yes—
65Dic97C25110yes—
66Dic97C2512yes—
67C2×C97—210no—
68C2×C97#58512yescomplemented
69C2×C97C2611yescharacteristic
70D97—220no—
71C2×#585—971nomaximal, Sylow
72C97C2×#58511yescharacteristic, complemented, Sylow
73#585—9732no—
74C2×2−1+4—975no—
75C2×2−1+4D9711yescharacteristic, complemented
76C2×H(5,𝔽2)—9710no—
77C22×#41—9710no—
78C22×#41—975no—
792−1+4—9780no—
802−1+4C2×D97116yescomplemented
81H(5,𝔽2)—97160no—
82C2×#41—97160no—
83C2×#41—9710no—
84C2×#41—9720no—
85C2×#41—9780no—
86C2×#41—9710no—
87C2×#41—9730no—
88C2×#41C2×D97110yescomplemented
89C22×Q8—9710no—
90C22×Q8C2×D9715yescomplemented
91C22×D4—9730no—
92C22×D4—9715no—
93C23×C4—9715no—
94C24—9715no—
95#41—9780no—
96#41—97160no—
97#41—9780no—
98#41—97240no—
99#41C22×D97180yescomplemented
100C2×Q8—9780no—
101C2×Q8—9710no—
102C2×Q8C22×D97110yes—
103C2×Q8C22×D97140yescomplemented
104C2×D4—9720no—
105C2×D4—97240no—
106C2×D4—9730no—
107C2×D4—97120no—
108C2×D4C22×D97110yes—
109C22×C4—975no—
110C22×C4—9710no—
111C22×C4—9730no—
112C22×C4—97120no—
113C22×C4C22×D97115yes—
114C23—9760no—
115C23—9710no—
116C23—38830no—
117C23C23×D9715yes—
118Q8—9740no—
119Q8C23×D97140yes—
120D4—9780no—
121D4—97120no—
122D4C23×D97140yes—
123C2×C4—9720no—
124C2×C4—971no—
125C2×C4—975no—
126C2×C4—9740no—
127C2×C4—97120no—
128C2×C4C23×D97160yes—
129C2×C4C23×D97110yes—
130C22—210no—
131C22—9720no—
132C22—38860no—
133C22—19420no—
134C22C24×D97110yes—
135C22C24×D9711yescharacteristic
136C4—972no—
137C4—9710no—
138C4C24×D97120yes—
139C2—210no—
140C2—19420no—
141C2#2026157212yesdirect product factor
142C2C25×D9711yescharacteristic
subgroupG/H
centerC22C24×D97
commutator, cosocleC2×C97C26
FrattiniC2C25×D97
FittingC2×2−1+4×C97C2
socleC22×C97C25
hypercenterC2×2−1+4D97
hypocenterC97C2×#585
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20261572C2direct (N × H)
2. C2×#15690858C2semidirect
3. C2×#15690856C2semidirect
4. C22×#13095512C2semidirect
5. C2×#41×D97C2semidirect
6. C2×2−1+4×C97C2semidirect
7. #15690858C22semidirect
8. #15690856C22semidirect
9. C2×#13095512C22semidirect
10. C2×#13095512C22semidirect
11. C2×#13095512C22semidirect
12. C2×#13095511C22semidirect
13. C2×#13095510C22semidirect
14. #41×D97C22semidirect
15. C2×Q8×D97C22semidirect
16. C22×D388C22semidirect
17. C2×D4×D97C22semidirect
18. C22×C4×D97C22semidirect
19. 2−1+4×C97C22semidirect
20. C2×#41×C97C22semidirect
21. C22×Q8×C97C22semidirect
23. #13095512C23semidirect
24. #13095512C23semidirect
25. #13095511C23semidirect
26. #13095510C23semidirect
28. Q8×D97C23semidirect
30. C2×D388C23semidirect
32. D4×D97C23semidirect
36. C2×C4×D97C23semidirect
37. #41×C97C23semidirect
39. C2×Q8×C97C23semidirect
68. C2×C97#585semidirect
72. C97C2×#585semidirect
75. C2×2−1+4D97semidirect
80. 2−1+4C2×D97semidirect
88. C2×#41C2×D97semidirect
90. C22×Q8C2×D97semidirect
99. #41C22×D97semidirect
103. C2×Q8C22×D97semidirect
141. C2#20261572direct (N × H)

Maximal subgroups

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

subgroupindex
#202615722
C2×#156908582
C2×#156908562
C22×#130955122
C2×#41×D972
C2×2−1+4×C972
C2×#58597