Finite Group Databasebeta

← C13×ΓL(1,𝔽8)×Dic35 #50303661

Subgroups — C13×ΓL(1,𝔽8)×Dic35 #50303661

Order 38220.

104 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 54 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×C13×Dic35C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4C13×ΓL(1,𝔽8)×Dic7—51nomaximal, Hall
5C2×C5×C72×C13C2×C311yescharacteristic
6C13×Dic5×ΓL(1,𝔽8)—71nomaximal
7C3×C13×Dic35—71nomaximal
8C2×C7×C13×ΓL(1,𝔽8)D511yescharacteristic
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10ΓL(1,𝔽8)×Dic35C1311yesmaximal, characteristic, direct product factor, Hall
11C2×C5×C13×ΓL(1,𝔽8)D711yescharacteristic
12C2×C3×C5×C7×C13—71no—
13C7×C13×Dic7—51noHall
14C7×C13×ΓL(1,𝔽8)Dic511yescharacteristic, complemented, Hall
15C13×Dic35ΓL(1,𝔽8)11yescharacteristic, direct product factor
16C7×C13×Dic5—71no—
17C2×C5×C7×ΓL(1,𝔽8)C2×C1311yescharacteristic
18C3×C5×C7×C13—71no—
19C5×C13×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
20C2×C72×C13C3×D511yescharacteristic
21C3×C13×Dic7—351no—
22C4×C13×ΓL(1,𝔽8)—351no—
23C7×Dic35C3×C1311yescharacteristic, complemented, Hall
24C2×C5×C7×C13—61no—
25C2×C5×C7×C13C2×ΓL(1,𝔽8)11yescharacteristic
26C2×C5×C7×C13C3×D711yescharacteristic
27C3×C13×Dic5—491noHall
28C5×C7×ΓL(1,𝔽8)C4×C1311yescharacteristic, complemented, Hall
29C72×C13C3×Dic511yescharacteristic, complemented, Hall
30ΓL(1,𝔽8)×Dic7—51noHall
31C2×C3×C7×C13—71no—
32C2×C13×ΓL(1,𝔽8)D3511yescharacteristic
33C2×C5×C72C2×C3×C1311yescharacteristic
34C5×C7×C13—61no—
35C5×C7×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
36C5×C7×C13C3×Dic711yescharacteristic, complemented
37C3×Dic35—71no—
38Dic5×ΓL(1,𝔽8)—71no—
39C2×C3×C5×C13—71no—
40C4×C7×C13—351no—
41C13×Dic7—51no—
42C2×C7×ΓL(1,𝔽8)D5×C1311yescharacteristic
43C3×C7×C13—71no—
44C13×ΓL(1,𝔽8)Dic3511yescharacteristic, direct product factor
45C13×Dic5—71noHall
46C5×C72C3×C4×C1311yescharacteristic, complemented, Hall
47C2×C3×C5×C7—71no—
48C2×C5×ΓL(1,𝔽8)C13×D711yescharacteristic
49C7×Dic7—51noHall
50C3×C5×C13—71noHall
51C2×C7×C13—61no—
52C2×C7×C13D5×ΓL(1,𝔽8)11yescharacteristic
53C2×C7×C13C3×D3511yescharacteristic
54C3×C4×C13—2451noHall
55C7×ΓL(1,𝔽8)C13×Dic511yescharacteristic, complemented, Hall
56Dic35C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
57C7×Dic5—71no—
58C2×C5×C13D7×ΓL(1,𝔽8)11yescharacteristic
59C3×C5×C7—71no—
60C5×ΓL(1,𝔽8)C13×Dic711yescharacteristic, complemented
61C2×C72C3×D5×C1311yescharacteristic
62C7×C13—61no—
63C7×C13Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
64C7×C13C3×Dic3511yescharacteristic, complemented
65C3×Dic7—351no—
66C4×ΓL(1,𝔽8)—351no—
67C2×C3×C13—71no—
68C2×C5×C7—61no—
69C2×C5×C7C2×C13×ΓL(1,𝔽8)11yescharacteristic
70C2×C5×C7C3×C13×D711yescharacteristic
71C5×C13ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Hall
72C3×Dic5—491noHall
73C4×C13—351noHall
74C72C3×C13×Dic511yescharacteristic, complemented, Sylow
75C2×C3×C7—71no—
76C2×ΓL(1,𝔽8)C13×D3511yescharacteristic
77C3×C13—71noHall
78C5×C7—61no—
79C5×C7C4×C13×ΓL(1,𝔽8)11yescharacteristic, complemented
80C5×C7C3×C13×Dic711yescharacteristic, complemented
81C2×C3×C5—71no—
82C4×C7—351no—
83Dic7—51no—
84C2×C13ΓL(1,𝔽8)×D3511yescharacteristic
85C3×C7—71no—
86ΓL(1,𝔽8)C13×Dic3511yescharacteristic, direct product factor
87Dic5—71noHall
88C3×C5—71noHall
89C2×C7—61no—
90C2×C7C3×C13×D3511yescharacteristic
91C2×C7D5×C13×ΓL(1,𝔽8)11yescharacteristic
92C13ΓL(1,𝔽8)×Dic3511yescharacteristic, direct product factor, Sylow
93C3×C4—2451noHall
94C2×C5C13×D7×ΓL(1,𝔽8)11yescharacteristic
95C7—61no—
96C7C3×C13×Dic3511yescharacteristic, complemented
97C7C13×Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
98C2×C3—71no—
99C5C13×ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Sylow
100C4—351noSylow
101C3—71noSylow
102C2C13×ΓL(1,𝔽8)×D3511yescharacteristic
subgroupG/H
center, hypercenterC2×C13ΓL(1,𝔽8)×D35
commutator, hypocenterC5×C72C3×C4×C13
FrattiniC2C13×ΓL(1,𝔽8)×D35
Fitting, socleC2×C5×C72×C13C2×C3
prime radicalC2×C5×C7×C13×ΓL(1,𝔽8)C2
cosocleC2×C5×C72C2×C3×C13
LCSG ▷ C5×C72
UCSC1 ◁ C2×C13

Complements

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

NH ≅ G/Ntype
2. C7×C13×Dic35C3semidirect
3. C5×C7×C13×ΓL(1,𝔽8)C4semidirect
9. C5×C72×C13C3×C4semidirect
10. ΓL(1,𝔽8)×Dic35C13direct (N × H)
14. C7×C13×ΓL(1,𝔽8)Dic5semidirect
15. C13×Dic35ΓL(1,𝔽8)direct (N × H)
19. C5×C13×ΓL(1,𝔽8)Dic7semidirect
23. C7×Dic35C3×C13semidirect
28. C5×C7×ΓL(1,𝔽8)C4×C13semidirect
29. C72×C13C3×Dic5semidirect
35. C5×C7×C13C4×ΓL(1,𝔽8)semidirect
36. C5×C7×C13C3×Dic7semidirect
44. C13×ΓL(1,𝔽8)Dic35direct (N × H)
46. C5×C72C3×C4×C13semidirect
55. C7×ΓL(1,𝔽8)C13×Dic5semidirect
56. Dic35C13×ΓL(1,𝔽8)direct (N × H)
60. C5×ΓL(1,𝔽8)C13×Dic7semidirect
63. C7×C13Dic5×ΓL(1,𝔽8)semidirect
64. C7×C13C3×Dic35semidirect
71. C5×C13ΓL(1,𝔽8)×Dic7semidirect
74. C72C3×C13×Dic5semidirect
79. C5×C7C4×C13×ΓL(1,𝔽8)semidirect
80. C5×C7C3×C13×Dic7semidirect
86. ΓL(1,𝔽8)C13×Dic35direct (N × H)
92. C13ΓL(1,𝔽8)×Dic35direct (N × H)
96. C7C3×C13×Dic35semidirect
97. C7C13×Dic5×ΓL(1,𝔽8)semidirect
99. C5C13×ΓL(1,𝔽8)×Dic7semidirect

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×C13×Dic353
C13×ΓL(1,𝔽8)×Dic75
C13×Dic5×ΓL(1,𝔽8)7
C3×C13×Dic357
ΓL(1,𝔽8)×Dic3513