Finite Group Databasebeta

← #72243196

Subgroups — #72243196

Order 4200.

168 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 37 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×C5⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C5×ΓL(1,𝔽8)⋊Dic5C211yesmaximal, characteristic, complemented
4#13063794C311yesmaximal, characteristic, complemented, Hall
5C2×C52×ΓL(1,𝔽8)C2211yescharacteristic
6ΓL(1,𝔽8)×C5⋊D5—21no—
7C5×C5⋊AGL(1,𝔽7)—21no—
8#63307⋊C3—51nomaximal
9#63305⋊C3—51nomaximal
10C2×C7×C5⋊D5C2×C311yescharacteristic, complemented
11C2×C5×D35C2×C311yescharacteristic, complemented
12C5×Dic35C2×C311yescharacteristic, complemented
13C3×#6038—71nomaximal, Hall
14C52×ΓL(1,𝔽8)D411yescharacteristic, complemented, Hall
15C2×C5⋊AGL(1,𝔽7)D511yescharacteristic, complemented
16C2×D5×ΓL(1,𝔽8)—51no—
17C2×D5×ΓL(1,𝔽8)—51no—
18C2×D5×ΓL(1,𝔽8)—102no—
19C2×C5×AGL(1,𝔽7)—51no—
20ΓL(1,𝔽8)⋊Dic5D511yescharacteristic, complemented
21C5×ΓL(1,𝔽8)⋊C4—51no—
22C2×C52×C7C22×C311yescharacteristic
23C7×C5⋊D5—21no—
24C5×D35—21no—
25C2×C3×C5⋊D5—141no—
26C2×C3×C5×D5—71no—
27C3×C5×Dic5—71no—
28#63307—51no—
29#63305—51no—
30C2×C5×ΓL(1,𝔽8)—22no—
31C2×C5×ΓL(1,𝔽8)C2×D511yescharacteristic
32C2×C5×ΓL(1,𝔽8)C2×D511yescharacteristic
33C5⋊AGL(1,𝔽7)—21no—
34D5×ΓL(1,𝔽8)—102no—
35D5×ΓL(1,𝔽8)—102no—
36D5×ΓL(1,𝔽8)—101no—
37D5×ΓL(1,𝔽8)—101no—
38C5×AGL(1,𝔽7)—101no—
39#6038—71noHall
40C52×C7C3×D411yescharacteristic, complemented, Hall
41(C22⋊D7)⋊C3—251noHall
42C2×C3×C52—141no—
43C3×C5⋊D5—141no—
44C3×C5×D5—141no—
45C2×D35C3×D511yescharacteristic, complemented
46C2×C7×D5—51no—
47C2×C7×D5—51no—
48C2×C7×D5—102no—
49C2×C5×D7—51no—
50Dic35C3×D511yescharacteristic, complemented
51C5×Dic7—51no—
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⋊D5AGL(1,𝔽7)11yescharacteristic, complemented
58C2×C5×D5—71no—
59C5×Dic5—71no—
60C22×ΓL(1,𝔽8)—251no—
61C2×AGL(1,𝔽7)—51no—
62ΓL(1,𝔽8)⋊C4—51no—
63C3×C52—141noHall
64C2×C5×C7—22no—
65C2×C5×C7C2×C3×D511yescharacteristic
66C2×C5×C7C2×C3×D511yescharacteristic
67D35—21no—
68C5×D7—101no—
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—701no—
77C2×C3×D5—702no—
78C3×C4×C5—351no—
79C3×Dic5—71no—
80C22⋊D7—251noHall
81C2×C52C2×AGL(1,𝔽7)11yescharacteristic
82C5⋊D5—21no—
83C5×D5—141no—
84C2×ΓL(1,𝔽8)—501no—
85C2×ΓL(1,𝔽8)D5211yescharacteristic
86AGL(1,𝔽7)—101no—
87C22⋊D5—351no—
88D20—351no—
89C5×C7—22no—
90C5×C7C3×D2011yescharacteristic, complemented
91C5×C7C3×C22⋊D511yescharacteristic, complemented
92C2×C3×C5—71no—
93C2×C3×C5—142no—
94C2×C3×C5—141no—
95C2×C3×C5—701no—
96C3×D5—141no—
97C3×D5—701no—
98C3×D5—702no—
99C3×D5—701no—
100C3×D5—702no—
101C22×C7—251no—
102C2×D7—51no—
103Dic7—51no—
104C52(C22⋊D7)⋊C311yescharacteristic, complemented, Sylow
105C3×D4—1751noHall
106ΓL(1,𝔽8)#603811yescharacteristic, complemented, Hall
107C22×C5—351no—
108C2×D5—51no—
109C2×D5—51no—
110C2×D5—71no—
111C2×D5—102no—
112C4×C5—351no—
113Dic5—71no—
114C3×C5—71no—
115C3×C5—142no—
116C3×C5—141no—
117C2×C7—501no—
118C2×C7C3×D5211yescharacteristic
119D7—101no—
120C22×C3—351no—
121C22×C3—3501no—
122C3×C4—351no—
123C2×C5—22no—
124C2×C5—701no—
125C2×C5D5×AGL(1,𝔽7)11yescharacteristic
126C2×C5D5×AGL(1,𝔽7)11yescharacteristic
127D5—102no—
128D5—102no—
129D5—101no—
130D5—101no—
131D5—141no—
132D4—1751noSylow
133C7C3×#603811yescharacteristic, complemented, Sylow
134C2×C3—141no—
135C2×C3—701no—
136C2×C3—3501no—
137C5—22no—
138C5#63305⋊C311yescharacteristic, complemented
139C5#63307⋊C311yescharacteristic, complemented
140C22—251no—
141C22—351no—
142C4—351no—
143C3—141noSylow
144C2—501no—
145C2—701no—
146C2(C5⋊AGL(1,𝔽7))⋊D511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2(C5⋊AGL(1,𝔽7))⋊D5
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×C5⋊AGL(1,𝔽7)C2semidirect
3. C5×ΓL(1,𝔽8)⋊Dic5C2semidirect
4. #13063794C3semidirect
10. C2×C7×C5⋊D5C2×C3semidirect
11. C2×C5×D35C2×C3semidirect
12. C5×Dic35C2×C3semidirect
14. C52×ΓL(1,𝔽8)D4semidirect
15. C2×C5⋊AGL(1,𝔽7)D5semidirect
20. ΓL(1,𝔽8)⋊Dic5D5semidirect
40. C52×C7C3×D4semidirect
45. C2×D35C3×D5semidirect
50. Dic35C3×D5semidirect
55. C5×ΓL(1,𝔽8)D20semidirect
56. C5×ΓL(1,𝔽8)C22⋊D5semidirect
57. C2×C5⋊D5AGL(1,𝔽7)semidirect
90. C5×C7C3×D20semidirect
91. C5×C7C3×C22⋊D5semidirect
104. C52(C22⋊D7)⋊C3semidirect
106. ΓL(1,𝔽8)#6038semidirect
133. C7C3×#6038semidirect
138. C5#63305⋊C3semidirect
139. C5#63307⋊C3semidirect

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×C5⋊AGL(1,𝔽7)2
C5×ΓL(1,𝔽8)⋊Dic52
#130637943
#63307⋊C35
#63305⋊C35
C3×#60387