Finite Group Databasebeta

← C7×ΓL(1,𝔽8)×#63120 #50303847

Subgroups — C7×ΓL(1,𝔽8)×#63120 #50303847

Order 38220.

120 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C7×D5×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C72×#63120C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C7×ΓL(1,𝔽8)×Dic13—51nomaximal, Hall
5C72×D5×C13C2×C311yescharacteristic
6C3×C7×#63120—71nomaximal
7ΓL(1,𝔽8)×#63120C711yesmaximal, characteristic, direct product factor
8C2×C7×C13×ΓL(1,𝔽8)—51no—
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10C7×AGL(1,𝔽5)×ΓL(1,𝔽8)—131nomaximal, Hall
11D5×C13×ΓL(1,𝔽8)C2×C711yescharacteristic
12C3×C7×D5×C13—71no—
13C72×Dic13—51noHall
14C7×C13×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
15C7×#63120—32no—
16C7×#63120ΓL(1,𝔽8)11yescharacteristic, direct product factor
17C7×#63120C3×C711yescharacteristic, complemented
18C7×D5×ΓL(1,𝔽8)D1311yescharacteristic
19C3×C5×C7×C13—71no—
20C5×C13×ΓL(1,𝔽8)C4×C711yescharacteristic, complemented
21C2×C72×C13—51no—
22C3×C7×Dic13—351no—
23ΓL(1,𝔽8)×Dic13—51no—
24C72×AGL(1,𝔽5)—131noHall
25C7×D5×C13—32no—
26C7×D5×C13C2×ΓL(1,𝔽8)11yescharacteristic
27C7×D5×C13C2×C3×C711yescharacteristic
28C3×#63120—71noHall
29C5×C7×ΓL(1,𝔽8)Dic1311yescharacteristic, complemented, Hall
30C72×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
31C4×C7×ΓL(1,𝔽8)—651noHall
32C2×C3×C7×C13—351no—
33C2×C13×ΓL(1,𝔽8)—51no—
34C72×D5C3×D1311yescharacteristic
35C5×C7×C13—32no—
36C5×C7×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
37C5×C7×C13C3×C4×C711yescharacteristic, complemented
38C3×C7×AGL(1,𝔽5)—911no—
39AGL(1,𝔽5)×ΓL(1,𝔽8)—131no—
40C3×D5×C13—71no—
41C7×Dic13—51no—
42C7×Dic13—51no—
43C7×Dic13—152no—
44C2×C7×ΓL(1,𝔽8)—51no—
45C3×C7×C13—71no—
46C13×ΓL(1,𝔽8)C7×AGL(1,𝔽5)11yescharacteristic, complemented
47#63120C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
48C5×C72C3×Dic1311yescharacteristic, complemented, Hall
49C3×C7×D5—71no—
50D5×ΓL(1,𝔽8)C7×D1311yescharacteristic
51C4×C72—651noHall
52C3×C5×C13—71noHall
53C2×C7×C13—51no—
54C2×C7×C13—51no—
55C2×C7×C13—152no—
56C3×Dic13—351noHall
57C7×ΓL(1,𝔽8)#6312011yescharacteristic, direct product factor, Hall
58C7×AGL(1,𝔽5)—131no—
59C7×AGL(1,𝔽5)—131no—
60C7×AGL(1,𝔽5)—392no—
61D5×C13C2×C7×ΓL(1,𝔽8)11yescharacteristic
62C3×C5×C7—71no—
63C5×ΓL(1,𝔽8)C7×Dic1311yescharacteristic, complemented
64C2×C72—51no—
65C7×C13—32no—
66C7×C13AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented
67C7×C13C3×C7×AGL(1,𝔽5)11yescharacteristic, complemented
68C3×C4×C7—4551no—
69C4×ΓL(1,𝔽8)—651no—
70C2×C3×C13—351no—
71C7×D5—32no—
72C7×D5ΓL(1,𝔽8)×D1311yescharacteristic
73C7×D5C3×C7×D1311yescharacteristic
74C5×C13C4×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
75C3×AGL(1,𝔽5)—911noHall
76Dic13—51noHall
77C72C3×#6312011yescharacteristic, complemented, Sylow
78C2×C3×C7—351no—
79C2×ΓL(1,𝔽8)—51no—
80C3×C13—71noHall
81C5×C7—32no—
82C5×C7ΓL(1,𝔽8)×Dic1311yescharacteristic, complemented
83C5×C7C3×C7×Dic1311yescharacteristic, complemented
84C3×D5—71no—
85C4×C7—651no—
86C4×C7—651no—
87C4×C7—1952no—
88C2×C13—51no—
89C3×C7—71no—
90ΓL(1,𝔽8)C7×#6312011yescharacteristic, direct product factor
91AGL(1,𝔽5)—131noHall
92C3×C5—71noHall
93C2×C7—51no—
94C2×C7—51no—
95C2×C7—152no—
96C13C7×AGL(1,𝔽5)×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
97C3×C4—4551noHall
98D5C7×ΓL(1,𝔽8)×D1311yescharacteristic
99C7—32no—
100C7ΓL(1,𝔽8)×#6312011yescharacteristic, direct product factor
101C7C3×C7×#6312011yescharacteristic, complemented
102C2×C3—351no—
103C5C7×ΓL(1,𝔽8)×Dic1311yescharacteristic, complemented, Sylow
104C4—651noSylow
105C3—71noSylow
106C2—51no—
subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)×#63120
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C13C3×C4
prime radicalC7×D5×C13×ΓL(1,𝔽8)C2
cosocleC7×D5×C13C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
2. C72×#63120C3semidirect
3. C5×C7×C13×ΓL(1,𝔽8)C4semidirect
7. ΓL(1,𝔽8)×#63120C7direct (N × H)
9. C5×C72×C13C3×C4semidirect
14. C7×C13×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
16. C7×#63120ΓL(1,𝔽8)direct (N × H)
17. C7×#63120C3×C7semidirect
20. C5×C13×ΓL(1,𝔽8)C4×C7semidirect
29. C5×C7×ΓL(1,𝔽8)Dic13semidirect
30. C72×C13C3×AGL(1,𝔽5)semidirect
36. C5×C7×C13C4×ΓL(1,𝔽8)semidirect
37. C5×C7×C13C3×C4×C7semidirect
46. C13×ΓL(1,𝔽8)C7×AGL(1,𝔽5)semidirect
47. #63120C7×ΓL(1,𝔽8)direct (N × H)
48. C5×C72C3×Dic13semidirect
57. C7×ΓL(1,𝔽8)#63120direct (N × H)
63. C5×ΓL(1,𝔽8)C7×Dic13semidirect
66. C7×C13AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
67. C7×C13C3×C7×AGL(1,𝔽5)semidirect
74. C5×C13C4×C7×ΓL(1,𝔽8)semidirect
77. C72C3×#63120semidirect
82. C5×C7ΓL(1,𝔽8)×Dic13semidirect
83. C5×C7C3×C7×Dic13semidirect
90. ΓL(1,𝔽8)C7×#63120direct (N × H)
96. C13C7×AGL(1,𝔽5)×ΓL(1,𝔽8)semidirect
100. C7ΓL(1,𝔽8)×#63120direct (N × H)
101. C7C3×C7×#63120semidirect
103. C5C7×ΓL(1,𝔽8)×Dic13semidirect

Maximal subgroups

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

subgroupindex
C7×D5×C13×ΓL(1,𝔽8)2
C72×#631203
C7×ΓL(1,𝔽8)×Dic135
C3×C7×#631207
ΓL(1,𝔽8)×#631207
C7×AGL(1,𝔽5)×ΓL(1,𝔽8)13