Finite Group Databasebeta

← C5×#1157⋊C3 #13132558

Subgroups — C5×#1157⋊C3 #13132558

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
1C2×C4×C5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C5×(C22⋊D7)⋊C3C212yesmaximal, complemented
3C5×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4C4×C5×AGL(1,𝔽7)C212yesmaximal, complemented
5C5×ΓL(1,𝔽8)⋊Q8C211yesmaximal, characteristic, complemented
6C5×#1157C311yesmaximal, characteristic, complemented, Hall
7C22×C5×ΓL(1,𝔽8)C2211yescharacteristic
8C2×C5×AGL(1,𝔽7)C2212yes—
9C4×C5×ΓL(1,𝔽8)C2211yescharacteristic
10C4×C5×ΓL(1,𝔽8)C2211yescharacteristic
11C5×ΓL(1,𝔽8)⋊C4C2212yes—
12#1157⋊C3C511yesmaximal, characteristic, direct product factor, Hall
13C2×C4×C5×C7C2×C311yescharacteristic, complemented
14C5×C22⋊D7C2×C312yescomplemented
15C5×D28C2×C311yescharacteristic, complemented
16C4×C5×D7C2×C312yescomplemented
17C5×Dic14C2×C311yescharacteristic, complemented
18C3×C5×#41—71nomaximal, Hall
19C2×C5×ΓL(1,𝔽8)—21no—
20C2×C5×ΓL(1,𝔽8)C2311yescharacteristic
21C5×AGL(1,𝔽7)—22no—
22C2×C4×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
23(C22⋊D7)⋊C3C2×C512yescomplemented
24C4⋊AGL(1,𝔽7)C2×C511yescharacteristic, complemented
25C4×AGL(1,𝔽7)C2×C512yescomplemented
26ΓL(1,𝔽8)⋊Q8C2×C511yescharacteristic, complemented
27C22×C5×C7C22×C311yescharacteristic
28C2×C5×D7C22×C312yes—
29C4×C5×C7C22×C311yescharacteristic
30C4×C5×C7C22×C311yescharacteristic
31C5×Dic7C22×C312yes—
32C3×C5×Q8—71no—
33C3×C5×D4—71no—
34C3×C5×D4—72no—
35C2×C3×C4×C5—72no—
36C2×C3×C4×C5—71no—
37#1157C3×C511yescharacteristic, complemented, Hall
38C5×ΓL(1,𝔽8)#4111yescharacteristic, complemented, Hall
39C22×ΓL(1,𝔽8)C22×C511yescharacteristic
40C2×AGL(1,𝔽7)C22×C512yes—
41C4×ΓL(1,𝔽8)C22×C511yescharacteristic
42C4×ΓL(1,𝔽8)C22×C511yescharacteristic
43ΓL(1,𝔽8)⋊C4C22×C512yes—
44C5×#41—71noHall
45C2×C5×C7—21no—
46C2×C5×C7C23×C311yescharacteristic
47C5×D7—22no—
48C22×C3×C5—72no—
49C22×C3×C5—71no—
50C3×C4×C5—72no—
51C3×C4×C5—71no—
52C3×C4×C5—71no—
53C2×C4×C7C2×C3×C511yescharacteristic, complemented
54C22⋊D7C2×C3×C512yescomplemented
55D28C2×C3×C511yescharacteristic, complemented
56C4×D7C2×C3×C512yescomplemented
57Dic14C2×C3×C511yescharacteristic, complemented
58C3×#41—71noHall
59C2×ΓL(1,𝔽8)—21no—
60C2×ΓL(1,𝔽8)C23×C511yescharacteristic
61AGL(1,𝔽7)—22no—
62C5×Q8—71no—
63C5×D4—72no—
64C5×D4—71no—
65C2×C4×C5—72no—
66C2×C4×C5AGL(1,𝔽7)11yescharacteristic, complemented
67C5×C7C3×#4111yescharacteristic, complemented, Hall
68C2×C3×C5—71no—
69C2×C3×C5—141no—
70C2×C3×C5—142no—
71C22×C7C22×C3×C511yescharacteristic
72C2×D7C22×C3×C512yes—
73C4×C7C22×C3×C511yescharacteristic
74C4×C7C22×C3×C511yescharacteristic
75Dic7C22×C3×C512yes—
76C3×Q8—71no—
77C3×D4—71no—
78C3×D4—72no—
79C2×C3×C4—72no—
80C2×C3×C4—71no—
81ΓL(1,𝔽8)C5×#4111yescharacteristic, complemented, Hall
82C22×C5—72no—
83C22×C5C2×AGL(1,𝔽7)11yescharacteristic
84C4×C5—72no—
85C4×C5C2×AGL(1,𝔽7)11yescharacteristic
86C4×C5C2×AGL(1,𝔽7)11yescharacteristic
87#41—71noSylow
88C3×C5—71noHall
89C2×C7—21no—
90C2×C7C23×C3×C511yescharacteristic
91D7—22no—
92C22×C3—72no—
93C22×C3—71no—
94C3×C4—72no—
95C3×C4—71no—
96C3×C4—71no—
97C2×C5—21no—
98C2×C5—142no—
99C2×C5C22×AGL(1,𝔽7)11yescharacteristic
100Q8—71no—
101D4—72no—
102D4—71no—
103C2×C4—72no—
104C2×C4C5×AGL(1,𝔽7)11yescharacteristic, complemented
105C7C3×C5×#4111yescharacteristic, complemented, Sylow
106C2×C3—71no—
107C2×C3—141no—
108C2×C3—142no—
109C5#1157⋊C311yescharacteristic, direct product factor, Sylow
110C22—72no—
111C22C2×C5×AGL(1,𝔽7)11yescharacteristic
112C4—72no—
113C4C2×C5×AGL(1,𝔽7)11yescharacteristic
114C4C2×C5×AGL(1,𝔽7)11yescharacteristic
115C3—71noSylow
116C2—21no—
117C2—142no—
118C2C22×C5×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC4×C5C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
FittingC2×C4×C5×C7C2×C3
socleC2×C5×C7C23×C3
hypercenterC2×C4×C5AGL(1,𝔽7)
hypocenterC7C3×C5×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×ΓL(1,𝔽8)C2semidirect
2. C5×(C22⋊D7)⋊C3C2semidirect
3. C5×C4⋊AGL(1,𝔽7)C2semidirect
4. C4×C5×AGL(1,𝔽7)C2semidirect
5. C5×ΓL(1,𝔽8)⋊Q8C2semidirect
6. C5×#1157C3semidirect
12. #1157⋊C3C5direct (N × H)
13. C2×C4×C5×C7C2×C3semidirect
14. C5×C22⋊D7C2×C3semidirect
15. C5×D28C2×C3semidirect
16. C4×C5×D7C2×C3semidirect
17. C5×Dic14C2×C3semidirect
22. C2×C4×ΓL(1,𝔽8)C2×C5semidirect
23. (C22⋊D7)⋊C3C2×C5semidirect
24. C4⋊AGL(1,𝔽7)C2×C5semidirect
25. C4×AGL(1,𝔽7)C2×C5semidirect
26. ΓL(1,𝔽8)⋊Q8C2×C5semidirect
37. #1157C3×C5semidirect
38. C5×ΓL(1,𝔽8)#41semidirect
53. C2×C4×C7C2×C3×C5semidirect
54. C22⋊D7C2×C3×C5semidirect
55. D28C2×C3×C5semidirect
56. C4×D7C2×C3×C5semidirect
57. Dic14C2×C3×C5semidirect
66. C2×C4×C5AGL(1,𝔽7)semidirect
67. C5×C7C3×#41semidirect
81. ΓL(1,𝔽8)C5×#41semidirect
104. C2×C4C5×AGL(1,𝔽7)semidirect
105. C7C3×C5×#41semidirect
109. C5#1157⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×ΓL(1,𝔽8)2
C5×(C22⋊D7)⋊C32
C5×C4⋊AGL(1,𝔽7)2
C4×C5×AGL(1,𝔽7)2
C5×ΓL(1,𝔽8)⋊Q82
C5×#11573
#1157⋊C35
C3×C5×#417