Finite Group Databasebeta

← ΓL(1,𝔽8)×Dic345 #44394451

Subgroups — ΓL(1,𝔽8)×Dic345 #44394451

Order 28980.

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×ΓL(1,𝔽8)×C23C211yesmaximal, characteristic
2ΓL(1,𝔽8)×Dic115—31nomaximal
3C7×Dic345C311yesmaximal, characteristic, complemented
4C3×C5×ΓL(1,𝔽8)×C23C411yescharacteristic, complemented, Hall
5ΓL(1,𝔽8)×Dic69—51nomaximal, Hall
6C2×C5×ΓL(1,𝔽8)×C23—21no—
7C2×C5×ΓL(1,𝔽8)×C23D311yescharacteristic
8C2×C3×C5×C7×C23C2×C311yescharacteristic
9C3×Dic345—71nomaximal, Hall
10C7×Dic115—31noHall
11C2×C3×ΓL(1,𝔽8)×C23D511yescharacteristic
12C5×ΓL(1,𝔽8)×C23—21no—
13C5×ΓL(1,𝔽8)×C23Dic311yescharacteristic, complemented
14C3×C5×C7×C23C3×C411yescharacteristic, complemented
15C2×C32×C5×C23—71no—
16C7×Dic69—51no—
17ΓL(1,𝔽8)×Dic23—151no—
18C2×C5×C7×C23C3×D311yescharacteristic
19C3×ΓL(1,𝔽8)×C23Dic511yescharacteristic, complemented, Hall
20Dic345ΓL(1,𝔽8)11yescharacteristic, direct product factor
21C3×Dic115—211no—
22ΓL(1,𝔽8)×Dic15—231nomaximal, Hall
23C32×C5×C23—71noHall
24C2×C3×C7×C23C3×D511yescharacteristic
25C2×ΓL(1,𝔽8)×C23—21no—
26C2×ΓL(1,𝔽8)×C23D1511yescharacteristic
27C3×Dic69—351noHall
28C5×C7×C23C3×Dic311yescharacteristic, complemented, Hall
29C2×C3×C5×C23—71no—
30C2×C3×C5×C23—141no—
31C2×C3×C5×C23C2×ΓL(1,𝔽8)11yescharacteristic
32C7×Dic23—151noHall
33C2×C3×C5×ΓL(1,𝔽8)D2311yescharacteristic
34C3×C7×C23C3×Dic511yescharacteristic, complemented
35ΓL(1,𝔽8)×C23—21no—
36ΓL(1,𝔽8)×C23Dic1511yescharacteristic, complemented
37Dic115—31noHall
38C7×Dic15—231no—
39Dic5×ΓL(1,𝔽8)—691no—
40C2×C32×C23—71no—
41C3×C5×C23—71no—
42C3×C5×C23—141no—
43C3×C5×C23C4×ΓL(1,𝔽8)11yescharacteristic, complemented
44C2×C7×C23C3×D1511yescharacteristic
45C3×C5×ΓL(1,𝔽8)Dic2311yescharacteristic, complemented, Hall
46Dic69—51no—
47C3×Dic23—1051no—
48Dic3×ΓL(1,𝔽8)—1151noHall
49C2×C5×C23D3×ΓL(1,𝔽8)11yescharacteristic
50C2×C3×C5×C7C3×D2311yescharacteristic
51C2×C5×ΓL(1,𝔽8)—21no—
52C2×C5×ΓL(1,𝔽8)D6911yescharacteristic
53C32×C23—71noHall
54C3×Dic15—1611noHall
55C7×C23C3×Dic1511yescharacteristic, complemented, Hall
56C7×Dic5—691noHall
57C2×C3×C23—71no—
58C2×C3×C23—141no—
59C2×C3×C23D5×ΓL(1,𝔽8)11yescharacteristic
60C2×C3×ΓL(1,𝔽8)D11511yescharacteristic
61C5×C23Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
62C3×C5×C7C3×Dic2311yescharacteristic, complemented
63C5×ΓL(1,𝔽8)—21no—
64C5×ΓL(1,𝔽8)Dic6911yescharacteristic, complemented
65Dic23—151noHall
66C2×C32×C5—71no—
67C7×Dic3—1151no—
68C4×ΓL(1,𝔽8)—3451no—
69C2×C5×C7C3×D6911yescharacteristic
70C3×C23—71no—
71C3×C23—141no—
72C3×C23Dic5×ΓL(1,𝔽8)11yescharacteristic, complemented
73C3×ΓL(1,𝔽8)Dic11511yescharacteristic, complemented, Hall
74Dic15—231no—
75C3×Dic5—4831no—
76C2×C23ΓL(1,𝔽8)×D1511yescharacteristic
77C32×C5—71noHall
78C2×C3×C7C3×D11511yescharacteristic
79C2×ΓL(1,𝔽8)—21no—
80C2×ΓL(1,𝔽8)D34511yescharacteristic
81C3×Dic3—8051noHall
82C5×C7C3×Dic6911yescharacteristic, complemented, Hall
83C2×C3×C5—71no—
84C2×C3×C5—141no—
85C2×C3×C5ΓL(1,𝔽8)×D2311yescharacteristic
86C4×C7—3451noHall
87C23ΓL(1,𝔽8)×Dic1511yescharacteristic, complemented, Sylow
88C3×C7C3×Dic11511yescharacteristic, complemented
89ΓL(1,𝔽8)—21no—
90ΓL(1,𝔽8)Dic34511yescharacteristic, direct product factor
91Dic5—691noHall
92C2×C32—71no—
93C3×C5—71no—
94C3×C5—141no—
95C3×C5ΓL(1,𝔽8)×Dic2311yescharacteristic, complemented
96C2×C7C3×D34511yescharacteristic
97C3×C4—24151no—
98Dic3—1151no—
99C2×C5ΓL(1,𝔽8)×D6911yescharacteristic
100C32—71noSylow
101C7C3×Dic34511yescharacteristic, complemented, Sylow
102C2×C3—71no—
103C2×C3—141no—
104C2×C3ΓL(1,𝔽8)×D11511yescharacteristic
105C5ΓL(1,𝔽8)×Dic6911yescharacteristic, complemented, Sylow
106C4—3451noSylow
107C3—71no—
108C3—141no—
109C3ΓL(1,𝔽8)×Dic11511yescharacteristic, complemented
110C2ΓL(1,𝔽8)×D34511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D345
commutator, hypocenterC3×C5×C7×C23C3×C4
Fitting, socle, cosocleC2×C3×C5×C7×C23C2×C3
prime radicalC2×C3×C5×ΓL(1,𝔽8)×C23C2
LCSG ▷ C3×C5×C7×C23
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. C7×Dic345C3semidirect
4. C3×C5×ΓL(1,𝔽8)×C23C4semidirect
13. C5×ΓL(1,𝔽8)×C23Dic3semidirect
14. C3×C5×C7×C23C3×C4semidirect
19. C3×ΓL(1,𝔽8)×C23Dic5semidirect
20. Dic345ΓL(1,𝔽8)direct (N × H)
28. C5×C7×C23C3×Dic3semidirect
34. C3×C7×C23C3×Dic5semidirect
36. ΓL(1,𝔽8)×C23Dic15semidirect
43. C3×C5×C23C4×ΓL(1,𝔽8)semidirect
45. C3×C5×ΓL(1,𝔽8)Dic23semidirect
55. C7×C23C3×Dic15semidirect
61. C5×C23Dic3×ΓL(1,𝔽8)semidirect
62. C3×C5×C7C3×Dic23semidirect
64. C5×ΓL(1,𝔽8)Dic69semidirect
72. C3×C23Dic5×ΓL(1,𝔽8)semidirect
73. C3×ΓL(1,𝔽8)Dic115semidirect
82. C5×C7C3×Dic69semidirect
87. C23ΓL(1,𝔽8)×Dic15semidirect
88. C3×C7C3×Dic115semidirect
90. ΓL(1,𝔽8)Dic345direct (N × H)
95. C3×C5ΓL(1,𝔽8)×Dic23semidirect
101. C7C3×Dic345semidirect
105. C5ΓL(1,𝔽8)×Dic69semidirect
109. C3ΓL(1,𝔽8)×Dic115semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×ΓL(1,𝔽8)×C232
ΓL(1,𝔽8)×Dic1153
C7×Dic3453
ΓL(1,𝔽8)×Dic695
C3×Dic3457
ΓL(1,𝔽8)×Dic1523