Finite Group Databasebeta

← C5×#1159⋊C3 #13132560

Subgroups — C5×#1159⋊C3 #13132560

Order 1680.

160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C5×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C5×(C22⋊D7)⋊C3C212yesmaximal, complemented
3C2×C5×ΓL(1,𝔽8)⋊C4C212yesmaximal, complemented
4C4×C5×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
5C5×ΓL(1,𝔽8)⋊Q8C211yesmaximal, characteristic, complemented
6C5×#1159C311yesmaximal, characteristic, complemented, Hall
7C22×C5×ΓL(1,𝔽8)C2212yes—
8C2×C5×AGL(1,𝔽7)C2211yescharacteristic
9C4×C5×ΓL(1,𝔽8)C2211yescharacteristic
10C5×ΓL(1,𝔽8)⋊C4C2212yes—
11C5×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
12#1159⋊C3C511yesmaximal, characteristic, direct product factor, Hall
13C5×C7×D4C2×C311yescharacteristic, complemented
14C5×C22⋊D7C2×C312yescomplemented
15C2×C5×Dic7C2×C312yescomplemented
16C4×C5×D7C2×C311yescharacteristic, complemented
17C5×Dic14C2×C311yescharacteristic, complemented
18C3×C5×#41—71nomaximal, Hall
19C2×C5×ΓL(1,𝔽8)—22no—
20C2×C5×ΓL(1,𝔽8)C2311yescharacteristic
21C5×AGL(1,𝔽7)—21no—
22D4×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
23(C22⋊D7)⋊C3C2×C512yescomplemented
24C2×ΓL(1,𝔽8)⋊C4C2×C512yescomplemented
25C4×AGL(1,𝔽7)C2×C511yescharacteristic, complemented
26ΓL(1,𝔽8)⋊Q8C2×C511yescharacteristic, complemented
27C22×C5×C7C22×C312yes—
28C2×C5×D7C22×C311yescharacteristic
29C4×C5×C7C22×C311yescharacteristic
30C5×Dic7C22×C312yes—
31C5×Dic7C22×C311yescharacteristic
32C3×C5×Q8—71no—
33C3×C5×D4—72no—
34C3×C5×D4—71no—
35C2×C3×C4×C5—71no—
36C2×C3×C4×C5—72no—
37#1159C3×C511yescharacteristic, complemented, Hall
38C5×ΓL(1,𝔽8)#4111yescharacteristic, complemented, Hall
39C22×ΓL(1,𝔽8)C22×C512yes—
40C2×AGL(1,𝔽7)C22×C511yescharacteristic
41C4×ΓL(1,𝔽8)C22×C511yescharacteristic
42ΓL(1,𝔽8)⋊C4C22×C512yes—
43ΓL(1,𝔽8)⋊C4C22×C511yescharacteristic
44C5×#41—71noHall
45C2×C5×C7—22no—
46C2×C5×C7C23×C311yescharacteristic
47C5×D7—21no—
48C22×C3×C5—71no—
49C22×C3×C5—72no—
50C3×C4×C5—71no—
51C3×C4×C5—72no—
52C3×C4×C5—71no—
53C7×D4C2×C3×C511yescharacteristic, complemented
54C22⋊D7C2×C3×C512yescomplemented
55C2×Dic7C2×C3×C512yescomplemented
56C4×D7C2×C3×C511yescharacteristic, complemented
57Dic14C2×C3×C511yescharacteristic, complemented
58C3×#41—71noHall
59C2×ΓL(1,𝔽8)—22no—
60C2×ΓL(1,𝔽8)C23×C511yescharacteristic
61AGL(1,𝔽7)—21no—
62C5×Q8—71no—
63C5×D4—72no—
64C5×D4AGL(1,𝔽7)11yescharacteristic, complemented
65C2×C4×C5—72no—
66C2×C4×C5—71no—
67C5×C7C3×#4111yescharacteristic, complemented, Hall
68C2×C3×C5—71no—
69C2×C3×C5—142no—
70C2×C3×C5—141no—
71C22×C7C22×C3×C512yes—
72C2×D7C22×C3×C511yescharacteristic
73C4×C7C22×C3×C511yescharacteristic
74Dic7C22×C3×C512yes—
75Dic7C22×C3×C511yescharacteristic
76C3×Q8—71no—
77C3×D4—72no—
78C3×D4—71no—
79C2×C3×C4—71no—
80C2×C3×C4—72no—
81ΓL(1,𝔽8)C5×#4111yescharacteristic, complemented, Hall
82C22×C5—71no—
83C22×C5C2×AGL(1,𝔽7)12yes—
84C4×C5—71no—
85C4×C5—72no—
86C4×C5C2×AGL(1,𝔽7)11yescharacteristic
87#41—71noSylow
88C3×C5—71noHall
89C2×C7—22no—
90C2×C7C23×C3×C511yescharacteristic
91D7—21no—
92C22×C3—71no—
93C22×C3—72no—
94C3×C4—71no—
95C3×C4—72no—
96C3×C4—71no—
97C2×C5—22no—
98C2×C5—141no—
99C2×C5C22×AGL(1,𝔽7)11yescharacteristic
100Q8—71no—
101D4—72no—
102D4C5×AGL(1,𝔽7)11yescharacteristic, complemented
103C2×C4—72no—
104C2×C4—71no—
105C7C3×C5×#4111yescharacteristic, complemented, Sylow
106C2×C3—71no—
107C2×C3—142no—
108C2×C3—141no—
109C5#1159⋊C311yescharacteristic, direct product factor, Sylow
110C22—71no—
111C22C2×C5×AGL(1,𝔽7)12yes—
112C4—71no—
113C4—72no—
114C4C2×C5×AGL(1,𝔽7)11yescharacteristic
115C3—71noSylow
116C2—22no—
117C2—141no—
118C2C22×C5×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2×C5C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
FittingC5×C7×D4C2×C3
socleC2×C5×C7C23×C3
hypercenterC5×D4AGL(1,𝔽7)
hypocenterC7C3×C5×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C5×D4

Complements

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

NH ≅ G/Ntype
1. C5×D4×ΓL(1,𝔽8)C2semidirect
2. C5×(C22⋊D7)⋊C3C2semidirect
3. C2×C5×ΓL(1,𝔽8)⋊C4C2semidirect
4. C4×C5×AGL(1,𝔽7)C2semidirect
5. C5×ΓL(1,𝔽8)⋊Q8C2semidirect
6. C5×#1159C3semidirect
12. #1159⋊C3C5direct (N × H)
13. C5×C7×D4C2×C3semidirect
14. C5×C22⋊D7C2×C3semidirect
15. C2×C5×Dic7C2×C3semidirect
16. C4×C5×D7C2×C3semidirect
17. C5×Dic14C2×C3semidirect
22. D4×ΓL(1,𝔽8)C2×C5semidirect
23. (C22⋊D7)⋊C3C2×C5semidirect
24. C2×ΓL(1,𝔽8)⋊C4C2×C5semidirect
25. C4×AGL(1,𝔽7)C2×C5semidirect
26. ΓL(1,𝔽8)⋊Q8C2×C5semidirect
37. #1159C3×C5semidirect
38. C5×ΓL(1,𝔽8)#41semidirect
53. C7×D4C2×C3×C5semidirect
54. C22⋊D7C2×C3×C5semidirect
55. C2×Dic7C2×C3×C5semidirect
56. C4×D7C2×C3×C5semidirect
57. Dic14C2×C3×C5semidirect
64. C5×D4AGL(1,𝔽7)semidirect
67. C5×C7C3×#41semidirect
81. ΓL(1,𝔽8)C5×#41semidirect
102. D4C5×AGL(1,𝔽7)semidirect
105. C7C3×C5×#41semidirect
109. C5#1159⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×D4×ΓL(1,𝔽8)2
C5×(C22⋊D7)⋊C32
C2×C5×ΓL(1,𝔽8)⋊C42
C4×C5×AGL(1,𝔽7)2
C5×ΓL(1,𝔽8)⋊Q82
C5×#11593
#1159⋊C35
C3×C5×#417