Finite Group Databasebeta

← C7×ΓL(1,𝔽8)⋊#63122 #50303855

Subgroups — C7×ΓL(1,𝔽8)⋊#63122 #50303855

Order 38220.

104 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 34 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×ΓL(1,𝔽8)×D65C211yesmaximal, characteristic
2C7×C7⋊#63122C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C7×ΓL(1,𝔽8)⋊#261—51nomaximal, Hall
5C72×D65C2×C311yescharacteristic
6C3×C7×#63122—71nomaximal
7ΓL(1,𝔽8)⋊#63122C711yesmaximal, characteristic, direct product factor
8C7×ΓL(1,𝔽8)×D13—51no—
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)—131nomaximal, Hall
11ΓL(1,𝔽8)×D65C2×C711yescharacteristic
12C3×C7×D65—71no—
13C7×C7⋊#261—51noHall
14C7×C13×ΓL(1,𝔽8)AGL(1,𝔽5)11yescharacteristic, complemented, Hall
15C7⋊#63122C3×C711yescharacteristic, complemented
16C7×#63122—71no—
17C7×D5×ΓL(1,𝔽8)—131no—
18C3×C5×C7×C13—71no—
19C5×C13×ΓL(1,𝔽8)C4×C711yescharacteristic, complemented
20C72×D13—51no—
21C3×C7×#261—351no—
22ΓL(1,𝔽8)⋊#261—51no—
23C7×C7⋊AGL(1,𝔽5)—131noHall
24C7×D65—61no—
25C7×D65AGL(1,𝔽7)11yescharacteristic
26C7×D65C2×C3×C711yescharacteristic
27C3×#63122—71noHall
28C5×C7×ΓL(1,𝔽8)#26111yescharacteristic, complemented, Hall
29C72×C13C3×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
30C7×ΓL(1,𝔽8)⋊C4—651noHall
31C3×C7×D13—351no—
32ΓL(1,𝔽8)×D13—51no—
33C72×D5—131no—
34C5×C7×C13—61no—
35C5×C7×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
36C5×C7×C13C3×C4×C711yescharacteristic, complemented
37C3×C7×AGL(1,𝔽5)—911no—
38ΓL(1,𝔽8)⋊AGL(1,𝔽5)—131no—
39C3×D65—71no—
40C7⋊#261—51no—
41C7×#261—351no—
42C2×C7×ΓL(1,𝔽8)—651no—
43C3×C7×C13—71no—
44C13×ΓL(1,𝔽8)C7×AGL(1,𝔽5)11yescharacteristic, complemented
45#63122—71noHall
46C5×C72C3×#26111yescharacteristic, complemented, Hall
47C3×C7×D5—911no—
48D5×ΓL(1,𝔽8)—131no—
49C7×Dic7—651noHall
50C3×C5×C13—71noHall
51C7×D13—51no—
52C7×D13—51no—
53C7×D13—301no—
54C3×#261—351noHall
55C7×ΓL(1,𝔽8)#6312211yescharacteristic, complemented, Hall
56C7⋊AGL(1,𝔽5)—131no—
57C7×AGL(1,𝔽5)—911no—
58D65C7×AGL(1,𝔽7)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)C7×#26111yescharacteristic, complemented
61C2×C72—651no—
62C7×C13—61no—
63C7×C13ΓL(1,𝔽8)⋊AGL(1,𝔽5)11yescharacteristic, complemented
64C7×C13C3×C7×AGL(1,𝔽5)11yescharacteristic, complemented
65C3×C4×C7—4551no—
66ΓL(1,𝔽8)⋊C4—651no—
67C3×D13—351no—
68C7×D5—131no—
69C7×D5—131no—
70C7×D5—781no—
71C5×C13C7×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Hall
72C3×AGL(1,𝔽5)—911noHall
73#261—351noHall
74C72C3×#6312211yescharacteristic, complemented, Sylow
75C2×C3×C7—4551no—
76C2×ΓL(1,𝔽8)—651no—
77C3×C13—71noHall
78C5×C7—61no—
79C5×C7ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
80C5×C7C3×C7×#26111yescharacteristic, complemented
81C3×D5—911no—
82C4×C7—4551no—
83Dic7—651no—
84D13—51no—
85C3×C7—71no—
86ΓL(1,𝔽8)C7×#6312211yescharacteristic, complemented
87AGL(1,𝔽5)—911noHall
88C3×C5—71noHall
89C2×C7—651no—
90C2×C7—651no—
91C2×C7—3901no—
92C13C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
93C3×C4—4551noHall
94D5—131no—
95C7—61no—
96C7ΓL(1,𝔽8)⋊#6312211yescharacteristic, direct product factor
97C7C3×C7×#6312211yescharacteristic, complemented
98C2×C3—4551no—
99C5C7×ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented, Sylow
100C4—4551noSylow
101C3—71noSylow
102C2—651no—
subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)⋊#63122
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C13C3×C4
prime radicalC7×ΓL(1,𝔽8)×D65C2
cosocleC7×D65C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
2. C7×C7⋊#63122C3semidirect
3. C5×C7×C13×ΓL(1,𝔽8)C4semidirect
7. ΓL(1,𝔽8)⋊#63122C7direct (N × H)
9. C5×C72×C13C3×C4semidirect
14. C7×C13×ΓL(1,𝔽8)AGL(1,𝔽5)semidirect
15. C7⋊#63122C3×C7semidirect
19. C5×C13×ΓL(1,𝔽8)C4×C7semidirect
28. C5×C7×ΓL(1,𝔽8)#261semidirect
29. C72×C13C3×AGL(1,𝔽5)semidirect
35. C5×C7×C13ΓL(1,𝔽8)⋊C4semidirect
36. C5×C7×C13C3×C4×C7semidirect
44. C13×ΓL(1,𝔽8)C7×AGL(1,𝔽5)semidirect
46. C5×C72C3×#261semidirect
55. C7×ΓL(1,𝔽8)#63122semidirect
60. C5×ΓL(1,𝔽8)C7×#261semidirect
63. C7×C13ΓL(1,𝔽8)⋊AGL(1,𝔽5)semidirect
64. C7×C13C3×C7×AGL(1,𝔽5)semidirect
71. C5×C13C7×ΓL(1,𝔽8)⋊C4semidirect
74. C72C3×#63122semidirect
79. C5×C7ΓL(1,𝔽8)⋊#261semidirect
80. C5×C7C3×C7×#261semidirect
86. ΓL(1,𝔽8)C7×#63122semidirect
92. C13C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)semidirect
96. C7ΓL(1,𝔽8)⋊#63122direct (N × H)
97. C7C3×C7×#63122semidirect
99. C5C7×ΓL(1,𝔽8)⋊#261semidirect

Maximal subgroups

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

subgroupindex
C7×ΓL(1,𝔽8)×D652
C7×C7⋊#631223
C7×ΓL(1,𝔽8)⋊#2615
C3×C7×#631227
ΓL(1,𝔽8)⋊#631227
C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)13