Finite Group Databasebeta

← ΓL(1,𝔽8)×#4099 #72218297

Subgroups — ΓL(1,𝔽8)×#4099 #72218297

Order 3360.

520 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C24×C5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×ΓL(1,𝔽8)×C22⋊D5C213yesmaximal, complemented
3ΓL(1,𝔽8)×C22⋊Dic5C213yesmaximal, complemented
4C7×#4099C311yesmaximal, characteristic, complemented, Hall
5C23×C5×ΓL(1,𝔽8)—26no—
6C23×C5×ΓL(1,𝔽8)C2213yes—
7C22×D5×ΓL(1,𝔽8)C2211yescharacteristic, complemented
8ΓL(1,𝔽8)×C22⋊D5—26no—
9C2×Dic5×ΓL(1,𝔽8)C2213yescomplemented
10ΓL(1,𝔽8)×#120—51nomaximal, Hall
11C24×C5×C7C2×C311yescharacteristic, complemented
12C2×C7×C22⋊D5C2×C313yescomplemented
13C7×C22⋊Dic5C2×C313yescomplemented
14C3×#4099—71nomaximal, Hall
15C22×C5×ΓL(1,𝔽8)—28no—
16C22×C5×ΓL(1,𝔽8)—26no—
17C22×C5×ΓL(1,𝔽8)D416yescomplemented
18C22×C5×ΓL(1,𝔽8)C2311yescharacteristic
19C2×D5×ΓL(1,𝔽8)—23no—
20Dic5×ΓL(1,𝔽8)—23no—
21C24×ΓL(1,𝔽8)D511yescharacteristic, complemented
22C2×D4×ΓL(1,𝔽8)—53no—
23C22⋊C4×ΓL(1,𝔽8)—53no—
24C23×C5×C7—26no—
25C23×C5×C7C22×C313yes—
26C22×C7×D5C22×C311yescharacteristic, complemented
27C7×C22⋊D5—26no—
28C2×C7×Dic5C22×C313yescomplemented
29C24×C3×C5—71no—
30C2×C3×C22⋊D5—73no—
31C3×C22⋊Dic5—73no—
32C7×#120—51noHall
33C2×C5×ΓL(1,𝔽8)—26no—
34C2×C5×ΓL(1,𝔽8)C2×D413yes—
35D5×ΓL(1,𝔽8)—41no—
36C23×ΓL(1,𝔽8)—26no—
37C23×ΓL(1,𝔽8)—51no—
38C23×ΓL(1,𝔽8)C2×D513yes—
39D4×ΓL(1,𝔽8)—106no—
40C2×C4×ΓL(1,𝔽8)—53no—
41#4099ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
42C22×C5×C7—28no—
43C22×C5×C7—26no—
44C22×C5×C7C3×D416yescomplemented
45C22×C5×C7C23×C311yescharacteristic
46C2×C7×D5—23no—
47C7×Dic5—23no—
48C23×C3×C5—73no—
49C23×C3×C5—146no—
50C22×C3×D5—71no—
51C3×C22⋊D5—146no—
52C2×C3×Dic5—73no—
53C24×C7C3×D511yescharacteristic, complemented
54C2×C7×D4—53no—
55C7×C22⋊C4—53no—
56C5×ΓL(1,𝔽8)#12011yescharacteristic, complemented, Hall
57C3×#120—351noHall
58C22×ΓL(1,𝔽8)—28no—
59C22×ΓL(1,𝔽8)—26no—
60C22×ΓL(1,𝔽8)—103no—
61C22×ΓL(1,𝔽8)C22⋊D516yescomplemented
62C22×ΓL(1,𝔽8)C22×D511yescharacteristic
63C4×ΓL(1,𝔽8)—103no—
64C24×C5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
65C2×C22⋊D5C2×ΓL(1,𝔽8)13yescomplemented
66C22⋊Dic5C2×ΓL(1,𝔽8)13yescomplemented
67C2×C5×C7—26no—
68C2×C5×C7C2×C3×D413yes—
69C7×D5—41no—
70C22×C3×C5—76no—
71C22×C3×C5—71no—
72C22×C3×C5—146no—
73C22×C3×C5—148no—
74C2×C3×D5—143no—
75C3×Dic5—143no—
76C23×C7—26no—
77C23×C7—51no—
78C23×C7C2×C3×D513yes—
79C7×D4—106no—
80C2×C4×C7—53no—
81C24×C3—71no—
82C2×C3×D4—353no—
83C3×C22⋊C4—353no—
84C2×ΓL(1,𝔽8)—26no—
85C2×ΓL(1,𝔽8)—201no—
86C2×ΓL(1,𝔽8)C2×C22⋊D513yes—
87C23×C5—26no—
88C23×C5C22×ΓL(1,𝔽8)13yes—
89C22×D5C22×ΓL(1,𝔽8)11yescharacteristic, complemented
90C22⋊D5—26no—
91C2×Dic5C22×ΓL(1,𝔽8)13yescomplemented
92C5×C7C3×#12011yescharacteristic, complemented, Hall
93#120—51noSylow
94C2×C3×C5—73no—
95C2×C3×C5—146no—
96C3×D5—281no—
97C22×C7—28no—
98C22×C7—26no—
99C22×C7—103no—
100C22×C7C3×C22⋊D516yescomplemented
101C22×C7C22×C3×D511yescharacteristic
102C4×C7—103no—
103C23×C3—73no—
104C23×C3—146no—
105C23×C3—351no—
106C3×D4—706no—
107C2×C3×C4—353no—
108ΓL(1,𝔽8)#409911yescharacteristic, direct product factor, Hall
109C22×C5—28no—
110C22×C5—26no—
111C22×C5D4×ΓL(1,𝔽8)16yescomplemented
112C22×C5C23×ΓL(1,𝔽8)11yescharacteristic
113C2×D5—23no—
114Dic5—23no—
115C24D5×ΓL(1,𝔽8)11yescharacteristic, complemented
116C2×D4—53no—
117C22⋊C4—53no—
118C3×C5—71noHall
119C2×C7—26no—
120C2×C7—201no—
121C2×C7C2×C3×C22⋊D513yes—
122C22×C3—76no—
123C22×C3—71no—
124C22×C3—146no—
125C22×C3—148no—
126C22×C3—703no—
127C3×C4—703no—
128C2×C5—26no—
129C2×C5C2×D4×ΓL(1,𝔽8)13yes—
130D5—41no—
131C23—26no—
132C23—51no—
133C23C2×D5×ΓL(1,𝔽8)13yes—
134D4—106no—
135C2×C4—53no—
136C7C3×#409911yescharacteristic, complemented, Sylow
137C2×C3—73no—
138C2×C3—146no—
139C2×C3—1401no—
140C5ΓL(1,𝔽8)×#12011yescharacteristic, complemented, Sylow
141C22—28no—
142C22—26no—
143C22—103no—
144C22ΓL(1,𝔽8)×C22⋊D516yescomplemented
145C22C22×D5×ΓL(1,𝔽8)11yescharacteristic
146C4—103no—
147C3—71noSylow
148C2—26no—
149C2—201no—
150C2C2×ΓL(1,𝔽8)×C22⋊D513yes—
subgroupG/H
center, FrattiniC22C22×D5×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C5×C7C23×C3
FittingC24×C5×C7C2×C3
hypercenterC24D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×#120
LCSG ▷ C22×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C5×ΓL(1,𝔽8)C2semidirect
2. C2×ΓL(1,𝔽8)×C22⋊D5C2semidirect
3. ΓL(1,𝔽8)×C22⋊Dic5C2semidirect
4. C7×#4099C3semidirect
7. C22×D5×ΓL(1,𝔽8)C22semidirect
9. C2×Dic5×ΓL(1,𝔽8)C22semidirect
11. C24×C5×C7C2×C3semidirect
12. C2×C7×C22⋊D5C2×C3semidirect
13. C7×C22⋊Dic5C2×C3semidirect
17. C22×C5×ΓL(1,𝔽8)D4semidirect
21. C24×ΓL(1,𝔽8)D5semidirect
26. C22×C7×D5C22×C3semidirect
28. C2×C7×Dic5C22×C3semidirect
41. #4099ΓL(1,𝔽8)direct (N × H)
44. C22×C5×C7C3×D4semidirect
53. C24×C7C3×D5semidirect
56. C5×ΓL(1,𝔽8)#120semidirect
61. C22×ΓL(1,𝔽8)C22⋊D5semidirect
64. C24×C5C2×ΓL(1,𝔽8)semidirect
65. C2×C22⋊D5C2×ΓL(1,𝔽8)semidirect
66. C22⋊Dic5C2×ΓL(1,𝔽8)semidirect
89. C22×D5C22×ΓL(1,𝔽8)semidirect
91. C2×Dic5C22×ΓL(1,𝔽8)semidirect
92. C5×C7C3×#120semidirect
100. C22×C7C3×C22⋊D5semidirect
108. ΓL(1,𝔽8)#4099direct (N × H)
111. C22×C5D4×ΓL(1,𝔽8)semidirect
115. C24D5×ΓL(1,𝔽8)semidirect
136. C7C3×#4099semidirect
140. C5ΓL(1,𝔽8)×#120semidirect
144. C22ΓL(1,𝔽8)×C22⋊D5semidirect

Maximal subgroups

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

subgroupindex
C24×C5×ΓL(1,𝔽8)2
C2×ΓL(1,𝔽8)×C22⋊D52
ΓL(1,𝔽8)×C22⋊Dic52
C7×#40993
ΓL(1,𝔽8)×#1205
C3×#40997