Finite Group Databasebeta

← C2×#14556174 #17986830

Subgroups — C2×#14556174 #17986830

Order 4544.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 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#14556174C2116yesmaximal, direct product factor
2C2×#11765866C212yesmaximal, complemented
3C2×#11765865C213yesmaximal, complemented
4C22×D284C213yesmaximal, complemented
5C2×D4×D71C216yesmaximal, complemented
6C2×#41×C71C211yesmaximal, characteristic, complemented
7C23×D71C2216yescomplemented
8#11765866C22116yescomplemented
9#11765865C22124yescomplemented
10C2×Dic142C2211yescharacteristic, complemented
11C2×D284C2216yescomplemented
12C2×D284C2213yescomplemented
13C2×D284C22124yescomplemented
14C2×C22⋊D71C2216yescomplemented
15D4×D71C22148yescomplemented
16C2×C4×D71C2216yescomplemented
17#41×C71C2218yescomplemented
18C2×Q8×C71C2211yescharacteristic, complemented
19C2×D4×C71C2213yescomplemented
20C22×C4×C71C2213yescomplemented
21C23×C71C2313yes—
22C22×D71—412no—
23C22×D71C2316yes—
24C22×D71C23124yescomplemented
25Q8×C71C2314yescomplemented
26D4×C71C23112yescomplemented
27C2×C4×C71C2311yescharacteristic
28C2×C4×C71C2313yes—
29C2×C4×C71C23112yescomplemented
30C22⋊D71C23124yescomplemented
31C2×Dic71C2312yes—
32D284C23124yescomplemented
33D284C23112yescomplemented
34C4×D71C23124yescomplemented
35Dic142C2314yescomplemented
36C22×C71—26no—
37C22×C71C2416yes—
38C22×C71C2411yescharacteristic
39C2×D71—212no—
40C2×D71—424no—
41C2×D71C24112yes—
42C4×C71C2412yes—
43C4×C71C2416yes—
44Dic71C2414yes—
45C2×C71—26no—
46C2×C71H(5,𝔽2)12yescomplemented
47C2×C71C2511yescharacteristic
48D71—212no—
49C71C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—711nomaximal, Sylow
51H(5,𝔽2)—7116no—
52C2×#41—712no—
53C2×#41—713no—
54C2×#41D7111yescharacteristic, complemented
55C22×D4—716no—
56C22×D4—713no—
57C24—716no—
58#41—7116no—
59#41—7124no—
60#41C2×D7118yescomplemented
61C2×Q8—711no—
62C2×Q8C2×D7111yescharacteristic, complemented
63C2×D4—716no—
64C2×D4—7148no—
65C2×D4—716no—
66C2×D4—713no—
67C2×D4—7124no—
68C2×D4C2×D7113yescomplemented
69C22×C4—716no—
70C22×C4C2×D7113yescomplemented
71C23—28412no—
72C23—7124no—
73C23—716no—
74C23C22×D7113yes—
75Q8—714no—
76Q8C22×D7114yescomplemented
77D4—7124no—
78D4—7124no—
79D4—7112no—
80D4C22×D71112yescomplemented
81C2×C4—712no—
82C2×C4—7124no—
83C2×C4C22×D7113yes—
84C2×C4C22×D7111yescharacteristic
85C2×C4C22×D71112yescomplemented
86C22—26no—
87C22—28424no—
88C22—7112no—
89C22—14212no—
90C22C23×D7116yes—
91C22C23×D7111yescharacteristic
92C4—714no—
93C4C23×D7116yes—
94C4C23×D7112yes—
95C2—26no—
96C2—14212no—
97C2#1455617412yesdirect product factor
98C2C24×D7111yescharacteristic
subgroupG/H
centerC22C23×D71
commutator, cosocleC2×C71C25
FrattiniC2C24×D71
FittingC2×#41×C71C2
socleC22×C71C24
hypercenterC2×#41D71
hypocenterC71C2×H(5,𝔽2)
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #14556174C2direct (N × H)
2. C2×#11765866C2semidirect
3. C2×#11765865C2semidirect
4. C22×D284C2semidirect
5. C2×D4×D71C2semidirect
6. C2×#41×C71C2semidirect
7. C23×D71C22semidirect
8. #11765866C22semidirect
9. #11765865C22semidirect
10. C2×Dic142C22semidirect
11. C2×D284C22semidirect
12. C2×D284C22semidirect
13. C2×D284C22semidirect
14. C2×C22⋊D71C22semidirect
15. D4×D71C22semidirect
16. C2×C4×D71C22semidirect
17. #41×C71C22semidirect
18. C2×Q8×C71C22semidirect
19. C2×D4×C71C22semidirect
20. C22×C4×C71C22semidirect
24. C22×D71C23semidirect
25. Q8×C71C23semidirect
26. D4×C71C23semidirect
29. C2×C4×C71C23semidirect
30. C22⋊D71C23semidirect
32. D284C23semidirect
33. D284C23semidirect
34. C4×D71C23semidirect
35. Dic142C23semidirect
46. C2×C71H(5,𝔽2)semidirect
49. C71C2×H(5,𝔽2)semidirect
54. C2×#41D71semidirect
60. #41C2×D71semidirect
62. C2×Q8C2×D71semidirect
68. C2×D4C2×D71semidirect
70. C22×C4C2×D71semidirect
76. Q8C22×D71semidirect
80. D4C22×D71semidirect
85. C2×C4C22×D71semidirect
97. C2#14556174direct (N × H)

Maximal subgroups

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

subgroupindex
#145561742
C2×#117658662
C2×#117658652
C22×D2842
C2×D4×D712
C2×#41×C712
C2×H(5,𝔽2)71