Finite Group Databasebeta

← C42×C5×AGL(1,𝔽7) #72219077

Subgroups — C42×C5×AGL(1,𝔽7) #72219077

Order 3360.

432 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 246 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×AGL(1,𝔽7)C213yesmaximal
2C42×C5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C4×C5×ΓL(1,𝔽8)⋊C4C213yesmaximal, complemented
4C42×C5×D7C311yesmaximal, characteristic, complemented, Hall
5C22×C5×AGL(1,𝔽7)C2211yescharacteristic
6C2×C4×C5×ΓL(1,𝔽8)C2213yes—
7C2×C5×ΓL(1,𝔽8)⋊C4C2213yes—
8C4×C5×AGL(1,𝔽7)C4112yesdirect product factor
9C42×AGL(1,𝔽7)C511yesmaximal, characteristic, direct product factor, Hall
10C2×C4×C5×D7C2×C313yes—
11C42×C5×C7C2×C311yescharacteristic, complemented
12C4×C5×Dic7C2×C313yescomplemented
13C2×C3×C42×C5—71nomaximal, Hall
14C22×C5×ΓL(1,𝔽8)C2311yescharacteristic
15C2×C5×AGL(1,𝔽7)C2×C416yes—
16C4×C5×ΓL(1,𝔽8)C2×C416yescomplemented
17C5×ΓL(1,𝔽8)⋊C4C2×C416yescomplemented
18C2×C4×AGL(1,𝔽7)C2×C513yes—
19C42×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
20C4×ΓL(1,𝔽8)⋊C4C2×C513yescomplemented
21C22×C5×D7C22×C311yescharacteristic
22C2×C4×C5×C7C22×C313yes—
23C2×C5×Dic7C22×C313yes—
24C4×C5×D7C3×C4112yescomplemented
25C22×C3×C4×C5—73no—
26C3×C42×C5—73no—
27C3×C42×C5—71no—
28C42×D7C3×C511yescharacteristic, complemented, Hall
29C2×C5×ΓL(1,𝔽8)C22×C413yes—
30C5×AGL(1,𝔽7)C4214yesdirect product factor
31C22×AGL(1,𝔽7)C22×C511yescharacteristic
32C2×C4×ΓL(1,𝔽8)C22×C513yes—
33C2×ΓL(1,𝔽8)⋊C4C22×C513yes—
34C4×AGL(1,𝔽7)C4×C5112yesdirect product factor
35C2×C42×C5—71noHall
36C22×C5×C7C23×C311yescharacteristic
37C2×C5×D7C2×C3×C416yes—
38C4×C5×C7C2×C3×C416yescomplemented
39C5×Dic7C2×C3×C416yescomplemented
40C23×C3×C5—71no—
41C2×C3×C4×C5—712no—
42C2×C3×C4×C5—73no—
43C2×C3×C4×C5—73no—
44C2×C4×D7C2×C3×C513yes—
45C42×C7C2×C3×C511yescharacteristic, complemented
46C4×Dic7C2×C3×C513yescomplemented
47C5×ΓL(1,𝔽8)C2×C4211yescharacteristic, complemented, Hall
48C2×C3×C42—71noHall
49C22×ΓL(1,𝔽8)C23×C511yescharacteristic
50C2×AGL(1,𝔽7)C2×C4×C516yes—
51C4×ΓL(1,𝔽8)C2×C4×C516yescomplemented
52ΓL(1,𝔽8)⋊C4C2×C4×C516yescomplemented
53C22×C4×C5—73no—
54C42×C5—73no—
55C42×C5AGL(1,𝔽7)11yescharacteristic, direct product factor
56C2×C5×C7C22×C3×C413yes—
57C5×D7C3×C4214yescomplemented
58C22×C3×C5—76no—
59C22×C3×C5—71no—
60C3×C4×C5—76no—
61C3×C4×C5—76no—
62C22×D7C22×C3×C511yescharacteristic
63C2×C4×C7C22×C3×C513yes—
64C2×Dic7C22×C3×C513yes—
65C4×D7C3×C4×C5112yescomplemented
66C22×C3×C4—73no—
67C3×C42—73no—
68C3×C42—71no—
69C2×ΓL(1,𝔽8)C22×C4×C513yes—
70AGL(1,𝔽7)C42×C514yesdirect product factor
71C23×C5—71no—
72C2×C4×C5—73no—
73C2×C4×C5—712no—
74C2×C4×C5C2×AGL(1,𝔽7)13yes—
75C5×C7C2×C3×C4211yescharacteristic, complemented, Hall
76C2×C42—71noSylow
77C2×C3×C5—74no—
78C2×C3×C5—73no—
79C22×C7C23×C3×C511yescharacteristic
80C2×D7C2×C3×C4×C516yes—
81C4×C7C2×C3×C4×C516yescomplemented
82Dic7C2×C3×C4×C516yescomplemented
83C23×C3—71no—
84C2×C3×C4—712no—
85C2×C3×C4—73no—
86C2×C3×C4—73no—
87ΓL(1,𝔽8)C2×C42×C511yescharacteristic, complemented, Hall
88C22×C5—76no—
89C22×C5C22×AGL(1,𝔽7)11yescharacteristic
90C4×C5—76no—
91C4×C5C4×AGL(1,𝔽7)16yesdirect product factor
92C22×C4—73no—
93C42—73no—
94C42C5×AGL(1,𝔽7)11yescharacteristic, direct product factor
95C3×C5—71noHall
96C2×C7C22×C3×C4×C513yes—
97D7C3×C42×C514yescomplemented
98C22×C3—76no—
99C22×C3—71no—
100C3×C4—76no—
101C3×C4—76no—
102C2×C5—74no—
103C2×C5C2×C4×AGL(1,𝔽7)13yes—
104C23—71no—
105C2×C4—73no—
106C2×C4—712no—
107C2×C4C2×C5×AGL(1,𝔽7)13yes—
108C7C2×C3×C42×C511yescharacteristic, complemented, Sylow
109C2×C3—74no—
110C2×C3—73no—
111C5C42×AGL(1,𝔽7)11yescharacteristic, direct product factor, Sylow
112C22—76no—
113C22C22×C5×AGL(1,𝔽7)11yescharacteristic
114C4—76no—
115C4C4×C5×AGL(1,𝔽7)16yesdirect product factor
116C3—71noSylow
117C2—74no—
118C2C2×C4×C5×AGL(1,𝔽7)13yes—
subgroupG/H
center, hypercenterC42×C5AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C42×C5
FrattiniC22C22×C5×AGL(1,𝔽7)
FittingC42×C5×C7C2×C3
socleC22×C5×C7C23×C3
prime radicalC22×C5×AGL(1,𝔽7)C22
cosocleC22×C7C23×C3×C5
LCSG ▷ C7
UCSC1 ◁ C42×C5

Complements

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

NH ≅ G/Ntype
2. C42×C5×ΓL(1,𝔽8)C2semidirect
3. C4×C5×ΓL(1,𝔽8)⋊C4C2semidirect
4. C42×C5×D7C3semidirect
8. C4×C5×AGL(1,𝔽7)C4direct (N × H)
9. C42×AGL(1,𝔽7)C5direct (N × H)
11. C42×C5×C7C2×C3semidirect
12. C4×C5×Dic7C2×C3semidirect
16. C4×C5×ΓL(1,𝔽8)C2×C4semidirect
17. C5×ΓL(1,𝔽8)⋊C4C2×C4semidirect
19. C42×ΓL(1,𝔽8)C2×C5semidirect
20. C4×ΓL(1,𝔽8)⋊C4C2×C5semidirect
24. C4×C5×D7C3×C4semidirect
28. C42×D7C3×C5semidirect
30. C5×AGL(1,𝔽7)C42direct (N × H)
34. C4×AGL(1,𝔽7)C4×C5direct (N × H)
38. C4×C5×C7C2×C3×C4semidirect
39. C5×Dic7C2×C3×C4semidirect
45. C42×C7C2×C3×C5semidirect
46. C4×Dic7C2×C3×C5semidirect
47. C5×ΓL(1,𝔽8)C2×C42semidirect
51. C4×ΓL(1,𝔽8)C2×C4×C5semidirect
52. ΓL(1,𝔽8)⋊C4C2×C4×C5semidirect
55. C42×C5AGL(1,𝔽7)direct (N × H)
57. C5×D7C3×C42semidirect
65. C4×D7C3×C4×C5semidirect
70. AGL(1,𝔽7)C42×C5direct (N × H)
75. C5×C7C2×C3×C42semidirect
81. C4×C7C2×C3×C4×C5semidirect
82. Dic7C2×C3×C4×C5semidirect
87. ΓL(1,𝔽8)C2×C42×C5semidirect
91. C4×C5C4×AGL(1,𝔽7)direct (N × H)
94. C42C5×AGL(1,𝔽7)direct (N × H)
97. D7C3×C42×C5semidirect
108. C7C2×C3×C42×C5semidirect
111. C5C42×AGL(1,𝔽7)direct (N × H)
115. C4C4×C5×AGL(1,𝔽7)direct (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×AGL(1,𝔽7)2
C42×C5×ΓL(1,𝔽8)2
C4×C5×ΓL(1,𝔽8)⋊C42
C42×C5×D73
C42×AGL(1,𝔽7)5
C2×C3×C42×C57