Finite Group Databasebeta

← ΓL(1,𝔽8)×Dic195 #30655253

Subgroups — ΓL(1,𝔽8)×Dic195 #30655253

Order 16380.

112 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 51 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×C3×C5×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic
2ΓL(1,𝔽8)×Dic65—31nomaximal
3C7×Dic195C311yesmaximal, characteristic, complemented
4C3×C5×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5ΓL(1,𝔽8)×Dic39—51nomaximal, Hall
6C2×C5×C13×ΓL(1,𝔽8)—21no—
7C2×C5×C13×ΓL(1,𝔽8)D311yescharacteristic
8C2×C3×C5×C7×C13C2×C311yescharacteristic
9C3×Dic195—71nomaximal, Hall
10C7×Dic65—31noHall
11C2×C3×C13×ΓL(1,𝔽8)D511yescharacteristic
12C3×C5×C7×C13C3×C411yescharacteristic, complemented
13C5×C13×ΓL(1,𝔽8)—21no—
14C5×C13×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
15ΓL(1,𝔽8)×Dic15—131nomaximal, Hall
16C2×C32×C5×C13—71no—
17C7×Dic39—51no—
18ΓL(1,𝔽8)×Dic13—151no—
19C2×C5×C7×C13C3×D311yescharacteristic
20C3×C13×ΓL(1,𝔽8)Dic511yescharacteristic, complemented, Hall
21Dic195ΓL(1,𝔽8)11yescharacteristic, direct product factor
22C3×Dic65—211no—
23C2×C3×C5×ΓL(1,𝔽8)D1311yescharacteristic
24C32×C5×C13—71noHall
25C2×C3×C7×C13C3×D511yescharacteristic
26C2×C13×ΓL(1,𝔽8)—21no—
27C2×C13×ΓL(1,𝔽8)D1511yescharacteristic
28C3×Dic39—351noHall
29C5×C7×C13C3×Dic311yescharacteristic, complemented, Hall
30C7×Dic15—131no—
31Dic5×ΓL(1,𝔽8)—391no—
32C2×C3×C5×C13—71no—
33C2×C3×C5×C13—141no—
34C2×C3×C5×C13C2×ΓL(1,𝔽8)11yescharacteristic
35C7×Dic13—151noHall
36C3×C5×ΓL(1,𝔽8)Dic1311yescharacteristic, complemented, Hall
37C3×C7×C13C3×Dic511yescharacteristic, complemented
38C13×ΓL(1,𝔽8)—21no—
39C13×ΓL(1,𝔽8)Dic1511yescharacteristic, complemented
40Dic65—31noHall
41Dic3×ΓL(1,𝔽8)—651noHall
42C2×C32×C13—71no—
43C2×C3×C5×C7C3×D1311yescharacteristic
44C2×C5×ΓL(1,𝔽8)—21no—
45C2×C5×ΓL(1,𝔽8)D3911yescharacteristic
46C3×C5×C13—71no—
47C3×C5×C13—141no—
48C3×C5×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
49C2×C7×C13C3×D1511yescharacteristic
50C3×Dic15—911noHall
51Dic39—51no—
52C3×Dic13—1051no—
53C7×Dic5—391noHall
54C2×C5×C13D3×ΓL(1,𝔽8)11yescharacteristic
55C2×C3×ΓL(1,𝔽8)D6511yescharacteristic
56C32×C13—71noHall
57C3×C5×C7C3×Dic1311yescharacteristic, complemented
58C5×ΓL(1,𝔽8)—21no—
59C5×ΓL(1,𝔽8)Dic3911yescharacteristic, complemented
60C7×C13C3×Dic1511yescharacteristic, complemented, Hall
61C2×C32×C5—71no—
62C7×Dic3—651no—
63C4×ΓL(1,𝔽8)—1951no—
64C2×C3×C13—71no—
65C2×C3×C13—141no—
66C2×C3×C13D5×ΓL(1,𝔽8)11yescharacteristic
67C2×C5×C7C3×D3911yescharacteristic
68C5×C13Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
69C3×ΓL(1,𝔽8)Dic6511yescharacteristic, complemented, Hall
70Dic15—131no—
71C3×Dic5—2731no—
72Dic13—151noHall
73C32×C5—71noHall
74C2×C3×C7C3×D6511yescharacteristic
75C2×ΓL(1,𝔽8)—21no—
76C2×ΓL(1,𝔽8)D19511yescharacteristic
77C3×C13—71no—
78C3×C13—141no—
79C3×C13Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
80C3×Dic3—4551noHall
81C5×C7C3×Dic3911yescharacteristic, complemented, Hall
82C2×C3×C5—71no—
83C2×C3×C5—141no—
84C2×C3×C5ΓL(1,𝔽8)×D1311yescharacteristic
85C4×C7—1951noHall
86C2×C13ΓL(1,𝔽8)×D1511yescharacteristic
87C3×C7C3×Dic6511yescharacteristic, complemented
88ΓL(1,𝔽8)—21no—
89ΓL(1,𝔽8)Dic19511yescharacteristic, direct product factor
90Dic5—391noHall
91C2×C32—71no—
92C3×C5—71no—
93C3×C5—141no—
94C3×C5ΓL(1,𝔽8)×Dic1311yescharacteristic, complemented
95C2×C7C3×D19511yescharacteristic
96C13ΓL(1,𝔽8)×Dic1511yescharacteristic, complemented, Sylow
97C3×C4—13651no—
98Dic3—651no—
99C2×C5ΓL(1,𝔽8)×D3911yescharacteristic
100C32—71noSylow
101C7C3×Dic19511yescharacteristic, complemented, Sylow
102C2×C3—71no—
103C2×C3—141no—
104C2×C3ΓL(1,𝔽8)×D6511yescharacteristic
105C5ΓL(1,𝔽8)×Dic3911yescharacteristic, complemented, Sylow
106C4—1951noSylow
107C3—71no—
108C3—141no—
109C3ΓL(1,𝔽8)×Dic6511yescharacteristic, complemented
110C2ΓL(1,𝔽8)×D19511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D195
commutator, hypocenterC3×C5×C7×C13C3×C4
Fitting, socle, cosocleC2×C3×C5×C7×C13C2×C3
prime radicalC2×C3×C5×C13×ΓL(1,𝔽8)C2
LCSG ▷ C3×C5×C7×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. C7×Dic195C3semidirect
4. C3×C5×C13×ΓL(1,𝔽8)C4semidirect
12. C3×C5×C7×C13C3×C4semidirect
14. C5×C13×ΓL(1,𝔽8)Dic3semidirect
20. C3×C13×ΓL(1,𝔽8)Dic5semidirect
21. Dic195ΓL(1,𝔽8)direct (N × H)
29. C5×C7×C13C3×Dic3semidirect
36. C3×C5×ΓL(1,𝔽8)Dic13semidirect
37. C3×C7×C13C3×Dic5semidirect
39. C13×ΓL(1,𝔽8)Dic15semidirect
48. C3×C5×C13C4×ΓL(1,𝔽8)semidirect
57. C3×C5×C7C3×Dic13semidirect
59. C5×ΓL(1,𝔽8)Dic39semidirect
60. C7×C13C3×Dic15semidirect
68. C5×C13Dic3×ΓL(1,𝔽8)semidirect
69. C3×ΓL(1,𝔽8)Dic65semidirect
79. C3×C13Dic5×ΓL(1,𝔽8)semidirect
81. C5×C7C3×Dic39semidirect
87. C3×C7C3×Dic65semidirect
89. ΓL(1,𝔽8)Dic195direct (N × H)
94. C3×C5ΓL(1,𝔽8)×Dic13semidirect
96. C13ΓL(1,𝔽8)×Dic15semidirect
101. C7C3×Dic195semidirect
105. C5ΓL(1,𝔽8)×Dic39semidirect
109. C3ΓL(1,𝔽8)×Dic65semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×C13×ΓL(1,𝔽8)2
ΓL(1,𝔽8)×Dic653
C7×Dic1953
ΓL(1,𝔽8)×Dic395
C3×Dic1957
ΓL(1,𝔽8)×Dic1513