Finite Group Databasebeta

← ΓL(1,𝔽8)×#6038 #72243166

Subgroups — ΓL(1,𝔽8)×#6038 #72243166

Order 4200.

168 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 42 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×ΓL(1,𝔽8)×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×C5×D5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C5×Dic5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4C7×#6038C311yesmaximal, characteristic, complemented, Hall
5C2×C52×ΓL(1,𝔽8)C2211yescharacteristic
6ΓL(1,𝔽8)×C5⋊D5—21no—
7C5×D5×ΓL(1,𝔽8)—21no—
8ΓL(1,𝔽8)×C22⋊D5—51nomaximal
9ΓL(1,𝔽8)×D20—51nomaximal
10C2×C7×C5⋊D5C2×C311yescharacteristic, complemented
11C2×C5×C7×D5C2×C311yescharacteristic, complemented
12C5×C7×Dic5C2×C311yescharacteristic, complemented
13C3×#6038—71nomaximal, Hall
14C52×ΓL(1,𝔽8)D411yescharacteristic, complemented, Hall
15C22×C5×ΓL(1,𝔽8)—51no—
16C2×D5×ΓL(1,𝔽8)—51no—
17C2×D5×ΓL(1,𝔽8)—51no—
18C2×D5×ΓL(1,𝔽8)—102no—
19C2×D5×ΓL(1,𝔽8)D511yescharacteristic, complemented
20C4×C5×ΓL(1,𝔽8)—51no—
21Dic5×ΓL(1,𝔽8)D511yescharacteristic, complemented
22C2×C52×C7C22×C311yescharacteristic
23C7×C5⋊D5—21no—
24C5×C7×D5—21no—
25C2×C3×C5⋊D5—71no—
26C2×C3×C5×D5—71no—
27C3×C5×Dic5—71no—
28C7×C22⋊D5—51no—
29C7×D20—51no—
30C2×C5×ΓL(1,𝔽8)—22no—
31C2×C5×ΓL(1,𝔽8)—101no—
32C2×C5×ΓL(1,𝔽8)C2×D511yescharacteristic
33C2×C5×ΓL(1,𝔽8)C2×D511yescharacteristic
34D5×ΓL(1,𝔽8)—21no—
35D5×ΓL(1,𝔽8)—102no—
36D5×ΓL(1,𝔽8)—102no—
37D5×ΓL(1,𝔽8)—101no—
38D5×ΓL(1,𝔽8)—101no—
39#6038ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
40C52×C7C3×D411yescharacteristic, complemented, Hall
41D4×ΓL(1,𝔽8)—251noHall
42C2×C3×C52—71no—
43C3×C5⋊D5—141no—
44C3×C5×D5—141no—
45C22×C5×C7—51no—
46C2×C7×D5—51no—
47C2×C7×D5—51no—
48C2×C7×D5—102no—
49C2×C7×D5C3×D511yescharacteristic, complemented
50C4×C5×C7—51no—
51C7×Dic5C3×D511yescharacteristic, complemented
52C3×C22⋊D5—351no—
53C3×D20—351no—
54C5×ΓL(1,𝔽8)—22no—
55C5×ΓL(1,𝔽8)D2011yescharacteristic, complemented
56C5×ΓL(1,𝔽8)C22⋊D511yescharacteristic, complemented
57C2×C5⋊D5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
58C2×C5×D5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
59C5×Dic5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
60C22×ΓL(1,𝔽8)—51no—
61C22×ΓL(1,𝔽8)—251no—
62C4×ΓL(1,𝔽8)—51no—
63C3×C52—71noHall
64C2×C5×C7—22no—
65C2×C5×C7—101no—
66C2×C5×C7C2×C3×D511yescharacteristic
67C2×C5×C7C2×C3×D511yescharacteristic
68C7×D5—21no—
69C7×D5—102no—
70C7×D5—102no—
71C7×D5—101no—
72C7×D5—101no—
73C22×C3×C5—351no—
74C2×C3×D5—71no—
75C2×C3×D5—351no—
76C2×C3×D5—351no—
77C2×C3×D5—702no—
78C3×C4×C5—351no—
79C3×Dic5—71no—
80C7×D4—251noHall
81C2×C52C22×ΓL(1,𝔽8)11yescharacteristic
82C5⋊D5—21no—
83C5×D5—21no—
84C2×ΓL(1,𝔽8)—101no—
85C2×ΓL(1,𝔽8)—501no—
86C2×ΓL(1,𝔽8)D5211yescharacteristic
87C22⋊D5—51no—
88D20—51no—
89C5×C7—22no—
90C5×C7C3×D2011yescharacteristic, complemented
91C5×C7C3×C22⋊D511yescharacteristic, complemented
92C2×C3×C5—71no—
93C2×C3×C5—71no—
94C2×C3×C5—142no—
95C2×C3×C5—701no—
96C3×D5—141no—
97C3×D5—701no—
98C3×D5—701no—
99C3×D5—702no—
100C3×D5—702no—
101C22×C7—51no—
102C22×C7—251no—
103C4×C7—51no—
104C52D4×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
105C3×D4—1751noHall
106ΓL(1,𝔽8)#603811yescharacteristic, direct product factor, Hall
107C22×C5—51no—
108C2×D5—51no—
109C2×D5—51no—
110C2×D5—102no—
111C2×D5D5×ΓL(1,𝔽8)11yescharacteristic, complemented
112C4×C5—51no—
113Dic5D5×ΓL(1,𝔽8)11yescharacteristic, complemented
114C3×C5—71no—
115C3×C5—71no—
116C3×C5—142no—
117C2×C7—101no—
118C2×C7—501no—
119C2×C7C3×D5211yescharacteristic
120C22×C3—351no—
121C22×C3—1751no—
122C3×C4—351no—
123C2×C5—22no—
124C2×C5—101no—
125C2×C5C2×D5×ΓL(1,𝔽8)11yescharacteristic
126C2×C5C2×D5×ΓL(1,𝔽8)11yescharacteristic
127D5—21no—
128D5—102no—
129D5—102no—
130D5—101no—
131D5—101no—
132D4—251noSylow
133C7C3×#603811yescharacteristic, complemented, Sylow
134C2×C3—71no—
135C2×C3—701no—
136C2×C3—3501no—
137C5—22no—
138C5ΓL(1,𝔽8)×D2011yescharacteristic, complemented
139C5ΓL(1,𝔽8)×C22⋊D511yescharacteristic, complemented
140C22—51no—
141C22—251no—
142C4—51no—
143C3—71noSylow
144C2—101no—
145C2—501no—
146C2D52×ΓL(1,𝔽8)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D52×ΓL(1,𝔽8)
commutator, Fitting, socle, cosocleC2×C52×C7C22×C3
hypocenterC52×C7C3×D4
LCSG ▷ C2×C52×C7 ▷ C52×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×ΓL(1,𝔽8)×C5⋊D5C2semidirect
2. C2×C5×D5×ΓL(1,𝔽8)C2semidirect
3. C5×Dic5×ΓL(1,𝔽8)C2semidirect
4. C7×#6038C3semidirect
10. C2×C7×C5⋊D5C2×C3semidirect
11. C2×C5×C7×D5C2×C3semidirect
12. C5×C7×Dic5C2×C3semidirect
14. C52×ΓL(1,𝔽8)D4semidirect
19. C2×D5×ΓL(1,𝔽8)D5semidirect
21. Dic5×ΓL(1,𝔽8)D5semidirect
39. #6038ΓL(1,𝔽8)direct (N × H)
40. C52×C7C3×D4semidirect
49. C2×C7×D5C3×D5semidirect
51. C7×Dic5C3×D5semidirect
55. C5×ΓL(1,𝔽8)D20semidirect
56. C5×ΓL(1,𝔽8)C22⋊D5semidirect
57. C2×C5⋊D5C2×ΓL(1,𝔽8)semidirect
58. C2×C5×D5C2×ΓL(1,𝔽8)semidirect
59. C5×Dic5C2×ΓL(1,𝔽8)semidirect
90. C5×C7C3×D20semidirect
91. C5×C7C3×C22⋊D5semidirect
104. C52D4×ΓL(1,𝔽8)semidirect
106. ΓL(1,𝔽8)#6038direct (N × H)
111. C2×D5D5×ΓL(1,𝔽8)semidirect
113. Dic5D5×ΓL(1,𝔽8)semidirect
133. C7C3×#6038semidirect
138. C5ΓL(1,𝔽8)×D20semidirect
139. C5ΓL(1,𝔽8)×C22⋊D5semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×C5⋊D52
C2×C5×D5×ΓL(1,𝔽8)2
C5×Dic5×ΓL(1,𝔽8)2
C7×#60383
ΓL(1,𝔽8)×C22⋊D55
ΓL(1,𝔽8)×D205
C3×#60387