Finite Group Databasebeta

← C7×ΓL(1,𝔽8)⋊Dic65 #50303614

Subgroups — C7×ΓL(1,𝔽8)⋊Dic65 #50303614

Order 38220.

104 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 52 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×C5×C7×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C7×Dic455C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C7×ΓL(1,𝔽8)⋊Dic13—51nomaximal, Hall
5C2×C5×C72×C13C2×C311yescharacteristic
6ΓL(1,𝔽8)⋊Dic65C711yesmaximal, characteristic, direct product factor
7C3×C7×Dic65—71nomaximal
8C2×C7×C13×ΓL(1,𝔽8)D511yescharacteristic
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10C7×ΓL(1,𝔽8)⋊Dic5—131nomaximal, Hall
11C2×C5×C13×ΓL(1,𝔽8)C2×C711yescharacteristic
12C2×C3×C5×C7×C13—71no—
13C7×Dic91—51noHall
14C7×C13×ΓL(1,𝔽8)Dic511yescharacteristic, complemented, Hall
15Dic455C3×C711yescharacteristic, complemented
16C7×Dic65—71no—
17C2×C5×C7×ΓL(1,𝔽8)D1311yescharacteristic
18C3×C5×C7×C13—71no—
19C5×C13×ΓL(1,𝔽8)C4×C711yescharacteristic, complemented
20C2×C72×C13C3×D511yescharacteristic
21C3×C7×Dic13—351no—
22ΓL(1,𝔽8)⋊Dic13—51no—
23C7×Dic35—131noHall
24C2×C5×C7×C13—61no—
25C2×C5×C7×C13AGL(1,𝔽7)11yescharacteristic
26C2×C5×C7×C13C2×C3×C711yescharacteristic
27C3×Dic65—71noHall
28C5×C7×ΓL(1,𝔽8)Dic1311yescharacteristic, complemented, Hall
29C72×C13C3×Dic511yescharacteristic, complemented, Hall
30C7×ΓL(1,𝔽8)⋊C4—651noHall
31C2×C3×C7×C13—71no—
32C2×C13×ΓL(1,𝔽8)C7×D511yescharacteristic
33C2×C5×C72C3×D1311yescharacteristic
34C5×C7×C13—61no—
35C5×C7×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
36C5×C7×C13C3×C4×C711yescharacteristic, complemented
37C3×C7×Dic5—911no—
38ΓL(1,𝔽8)⋊Dic5—131no—
39C2×C3×C5×C13—71no—
40Dic91—51no—
41C7×Dic13—351no—
42C2×C7×ΓL(1,𝔽8)D6511yescharacteristic
43C3×C7×C13—71no—
44C13×ΓL(1,𝔽8)C7×Dic511yescharacteristic, complemented
45Dic65—71noHall
46C5×C72C3×Dic1311yescharacteristic, complemented, Hall
47C2×C3×C5×C7—71no—
48C2×C5×ΓL(1,𝔽8)C7×D1311yescharacteristic
49C7×Dic7—651noHall
50C3×C5×C13—71noHall
51C2×C7×C13—61no—
52C2×C7×C13C5⋊AGL(1,𝔽7)11yescharacteristic
53C2×C7×C13C3×C7×D511yescharacteristic
54C3×Dic13—351noHall
55C7×ΓL(1,𝔽8)Dic6511yescharacteristic, complemented, Hall
56Dic35—131no—
57C7×Dic5—911no—
58C2×C5×C13C7×AGL(1,𝔽7)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)C7×Dic1311yescharacteristic, complemented
61C2×C72C3×D6511yescharacteristic
62C7×C13—61no—
63C7×C13ΓL(1,𝔽8)⋊Dic511yescharacteristic, complemented
64C7×C13C3×C7×Dic511yescharacteristic, complemented
65C3×C4×C7—4551no—
66ΓL(1,𝔽8)⋊C4—651no—
67C2×C3×C13—71no—
68C2×C5×C7—61no—
69C2×C5×C7ΓL(1,𝔽8)⋊D1311yescharacteristic
70C2×C5×C7C3×C7×D1311yescharacteristic
71C5×C13C7×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Hall
72C3×Dic5—911noHall
73Dic13—351noHall
74C72C3×Dic6511yescharacteristic, complemented, Sylow
75C2×C3×C7—71no—
76C2×ΓL(1,𝔽8)C7×D6511yescharacteristic
77C3×C13—71noHall
78C5×C7—61no—
79C5×C7ΓL(1,𝔽8)⋊Dic1311yescharacteristic, complemented
80C5×C7C3×C7×Dic1311yescharacteristic, complemented
81C2×C3×C5—71no—
82C4×C7—4551no—
83Dic7—651no—
84C2×C13C7×C5⋊AGL(1,𝔽7)11yescharacteristic
85C3×C7—71no—
86ΓL(1,𝔽8)C7×Dic6511yescharacteristic, complemented
87Dic5—911noHall
88C3×C5—71noHall
89C2×C7—61no—
90C2×C7C3×C7×D6511yescharacteristic
91C2×C7C5⋊(ΓL(1,𝔽8)⋊D13)11yescharacteristic
92C13C7×ΓL(1,𝔽8)⋊Dic511yescharacteristic, complemented, Sylow
93C3×C4—4551noHall
94C2×C5C7×ΓL(1,𝔽8)⋊D1311yescharacteristic
95C7—61no—
96C7C3×C7×Dic6511yescharacteristic, complemented
97C7ΓL(1,𝔽8)⋊Dic6511yescharacteristic, direct product factor
98C2×C3—71no—
99C5C7×ΓL(1,𝔽8)⋊Dic1311yescharacteristic, complemented, Sylow
100C4—4551noSylow
101C3—71noSylow
102C2C7×C5⋊(ΓL(1,𝔽8)⋊D13)11yescharacteristic
subgroupG/H
center, hypercenterC2×C7C5⋊(ΓL(1,𝔽8)⋊D13)
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC2C7×C5⋊(ΓL(1,𝔽8)⋊D13)
Fitting, socleC2×C5×C72×C13C2×C3
prime radicalC2×C5×C7×C13×ΓL(1,𝔽8)C2
cosocleC2×C5×C7×C13C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2×C7

Complements

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

NH ≅ G/Ntype
2. C7×Dic455C3semidirect
3. C5×C7×C13×ΓL(1,𝔽8)C4semidirect
6. ΓL(1,𝔽8)⋊Dic65C7direct (N × H)
9. C5×C72×C13C3×C4semidirect
14. C7×C13×ΓL(1,𝔽8)Dic5semidirect
15. Dic455C3×C7semidirect
19. C5×C13×ΓL(1,𝔽8)C4×C7semidirect
28. C5×C7×ΓL(1,𝔽8)Dic13semidirect
29. C72×C13C3×Dic5semidirect
35. C5×C7×C13ΓL(1,𝔽8)⋊C4semidirect
36. C5×C7×C13C3×C4×C7semidirect
44. C13×ΓL(1,𝔽8)C7×Dic5semidirect
46. C5×C72C3×Dic13semidirect
55. C7×ΓL(1,𝔽8)Dic65semidirect
60. C5×ΓL(1,𝔽8)C7×Dic13semidirect
63. C7×C13ΓL(1,𝔽8)⋊Dic5semidirect
64. C7×C13C3×C7×Dic5semidirect
71. C5×C13C7×ΓL(1,𝔽8)⋊C4semidirect
74. C72C3×Dic65semidirect
79. C5×C7ΓL(1,𝔽8)⋊Dic13semidirect
80. C5×C7C3×C7×Dic13semidirect
86. ΓL(1,𝔽8)C7×Dic65semidirect
92. C13C7×ΓL(1,𝔽8)⋊Dic5semidirect
96. C7C3×C7×Dic65semidirect
97. C7ΓL(1,𝔽8)⋊Dic65direct (N × H)
99. 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
C2×C5×C7×C13×ΓL(1,𝔽8)2
C7×Dic4553
C7×ΓL(1,𝔽8)⋊Dic135
ΓL(1,𝔽8)⋊Dic657
C3×C7×Dic657
C7×ΓL(1,𝔽8)⋊Dic513