Finite Group Databasebeta

← C2×#26012328 #36368372

Subgroups — C2×#26012328 #36368372

Order 22144.

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#26012328C2132yesmaximal, direct product factor
2C2×#19149727C215yesmaximal, complemented
3C2×#19149725C2110yesmaximal, complemented
4C22×#14583827C215yesmaximal, complemented
5C2×#41×D173C2110yesmaximal, complemented
6C2×2−1+4×C173C211yesmaximal, characteristic, complemented
7#19149727C22180yescomplemented
8#19149725C221160yescomplemented
9C2×#14583827C22110yescomplemented
10C2×#14583827C22120yescomplemented
11C2×#14583827C22180yescomplemented
12C2×#14583826C22130yescomplemented
13C2×#14583825C22110yescomplemented
14#41×D173C221160yescomplemented
15C2×Q8×D173C22110yescomplemented
16C22×D692C22115yescomplemented
17C2×D4×D173C22130yescomplemented
18C22×C4×D173C22115yescomplemented
192−1+4×C173C22116yescomplemented
20C2×#41×C173C22110yescomplemented
21C22×Q8×C173C2215yescomplemented
22C23×D173C23115yes—
23#14583827C23180yescomplemented
24#14583827C231160yescomplemented
25#14583826C231240yescomplemented
26#14583825C23180yescomplemented
27C2×Dic346C23110yes—
28Q8×D173C23180yescomplemented
29C2×D692C23130yes—
30C2×D692C231120yescomplemented
31C2×C22⋊D173C23120yes—
32D4×D173C231240yescomplemented
33C22×Dic173C2315yes—
34C2×C4×D173C23110yes—
35C2×C4×D173C23130yes—
36C2×C4×D173C231120yescomplemented
37#41×C173C23180yescomplemented
38C2×Q8×C173C23110yes—
39C2×Q8×C173C23140yescomplemented
40C2×D4×C173C23110yes—
41C22×C4×C173C23115yes—
42C22×D173—430no—
43C22×D173C24110yes—
44C22×D173C24160yes—
45Dic346C24140yes—
46D692C241120yes—
47C22⋊D173C24180yes—
48C2×Dic173C2411yescharacteristic
49C2×Dic173C2415yes—
50C2×Dic173C24120yes—
51C4×D173C24140yes—
52C4×D173C241120yes—
53C23×C173C2415yes—
54Q8×C173C24140yes—
55D4×C173C24140yes—
56C2×C4×C173C24110yes—
57C2×C4×C173C24160yes—
58C22×C173—210no—
59C22×C173C25110yes—
60C22×C173C2511yescharacteristic
61C2×D173—220no—
62C2×D173—460no—
63C2×D173C25120yes—
64C4×C173C25120yes—
65Dic173C2512yes—
66Dic173C25110yes—
67C2×C173—210no—
68C2×C173#58512yescomplemented
69C2×C173C2611yescharacteristic
70D173—220no—
71C173C2×#58511yescharacteristic, complemented, Sylow
72C2×#585—1731nomaximal, Sylow
73#585—17332no—
74C2×2−1+4—1735no—
75C2×2−1+4D17311yescharacteristic, complemented
76C2×H(5,𝔽2)—17310no—
77C22×#41—17310no—
78C22×#41—1735no—
792−1+4—17380no—
802−1+4C2×D173116yescomplemented
81H(5,𝔽2)—173160no—
82C2×#41—173160no—
83C2×#41—17310no—
84C2×#41—17320no—
85C2×#41—17380no—
86C2×#41—17310no—
87C2×#41—17330no—
88C2×#41C2×D173110yescomplemented
89C22×Q8—17310no—
90C22×Q8C2×D17315yescomplemented
91C22×D4—17330no—
92C22×D4—17315no—
93C23×C4—17315no—
94C24—17315no—
95#41—17380no—
96#41—173160no—
97#41—17380no—
98#41—173240no—
99#41C22×D173180yescomplemented
100C2×Q8—17380no—
101C2×Q8—17310no—
102C2×Q8C22×D173110yes—
103C2×Q8C22×D173140yescomplemented
104C2×D4—17320no—
105C2×D4—173240no—
106C2×D4—17330no—
107C2×D4—173120no—
108C2×D4C22×D173110yes—
109C22×C4—1735no—
110C22×C4—17310no—
111C22×C4—17330no—
112C22×C4—173120no—
113C22×C4C22×D173115yes—
114C23—17310no—
115C23—17360no—
116C23—69230no—
117C23C23×D17315yes—
118Q8—17340no—
119Q8C23×D173140yes—
120D4—17380no—
121D4—173120no—
122D4C23×D173140yes—
123C2×C4—17320no—
124C2×C4—1731no—
125C2×C4—1735no—
126C2×C4—17340no—
127C2×C4—173120no—
128C2×C4C23×D173160yes—
129C2×C4C23×D173110yes—
130C22—210no—
131C22—34620no—
132C22—17320no—
133C22—69260no—
134C22C24×D173110yes—
135C22C24×D17311yescharacteristic
136C4—1732no—
137C4—17310no—
138C4C24×D173120yes—
139C2—210no—
140C2—34620no—
141C2#2601232812yesdirect product factor
142C2C25×D17311yescharacteristic
subgroupG/H
centerC22C24×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC2×2−1+4×C173C2
socleC22×C173C25
hypercenterC2×2−1+4D173
hypocenterC173C2×#585
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #26012328C2direct (N × H)
2. C2×#19149727C2semidirect
3. C2×#19149725C2semidirect
4. C22×#14583827C2semidirect
5. C2×#41×D173C2semidirect
6. C2×2−1+4×C173C2semidirect
7. #19149727C22semidirect
8. #19149725C22semidirect
9. C2×#14583827C22semidirect
10. C2×#14583827C22semidirect
11. C2×#14583827C22semidirect
12. C2×#14583826C22semidirect
13. C2×#14583825C22semidirect
14. #41×D173C22semidirect
15. C2×Q8×D173C22semidirect
16. C22×D692C22semidirect
17. C2×D4×D173C22semidirect
18. C22×C4×D173C22semidirect
19. 2−1+4×C173C22semidirect
20. C2×#41×C173C22semidirect
21. C22×Q8×C173C22semidirect
23. #14583827C23semidirect
24. #14583827C23semidirect
25. #14583826C23semidirect
26. #14583825C23semidirect
28. Q8×D173C23semidirect
30. C2×D692C23semidirect
32. D4×D173C23semidirect
36. C2×C4×D173C23semidirect
37. #41×C173C23semidirect
39. C2×Q8×C173C23semidirect
68. C2×C173#585semidirect
71. C173C2×#585semidirect
75. C2×2−1+4D173semidirect
80. 2−1+4C2×D173semidirect
88. C2×#41C2×D173semidirect
90. C22×Q8C2×D173semidirect
99. #41C22×D173semidirect
103. C2×Q8C22×D173semidirect
141. C2#26012328direct (N × H)

Maximal subgroups

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

subgroupindex
#260123282
C2×#191497272
C2×#191497252
C22×#145838272
C2×#41×D1732
C2×2−1+4×C1732
C2×#585173