Finite Group Databasebeta

← AGL(1,𝔽7)×#1042 #72243451

Subgroups — AGL(1,𝔽7)×#1042 #72243451

Order 4200.

136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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
1ΓL(1,𝔽8)×#1042C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)⋊#1042C211yesmaximal, characteristic, complemented
3C5×D5×AGL(1,𝔽7)C211yesmaximal, characteristic
4D7×#1042C311yesmaximal, characteristic, complemented, Hall
5C5×C5⋊AGL(1,𝔽7)C411yescharacteristic, complemented
6C5×D5×ΓL(1,𝔽8)C2211yescharacteristic
7C52×AGL(1,𝔽7)C411yescharacteristic, complemented
8AGL(1,𝔽5)×AGL(1,𝔽7)—51nomaximal
9Dic5×AGL(1,𝔽7)—51nomaximal
10C5×D5×D7C2×C311yescharacteristic
11C7⋊#1042C2×C311yescharacteristic, complemented
12C7×#1042C2×C311yescharacteristic, complemented
13C2×C3×#1042—71nomaximal, Hall
14C52×ΓL(1,𝔽8)C2×C411yescharacteristic, complemented, Hall
15C2×C5×AGL(1,𝔽7)—51no—
16D5×AGL(1,𝔽7)D511yescharacteristic
17ΓL(1,𝔽8)⋊AGL(1,𝔽5)—51no—
18AGL(1,𝔽5)×ΓL(1,𝔽8)—51no—
19ΓL(1,𝔽8)⋊Dic5—51no—
20Dic5×ΓL(1,𝔽8)—51no—
21C5×D35C3×C411yescharacteristic, complemented
22C5×C7×D5C22×C311yescharacteristic
23C52×D7C3×C411yescharacteristic, complemented
24C2×C3×C5×D5—71no—
25C3×#1042—71no—
26C3×#1042—141no—
27D7×AGL(1,𝔽5)—51no—
28D7×Dic5—51no—
29C2×C5×ΓL(1,𝔽8)—51no—
30C5⋊AGL(1,𝔽7)Dic511yescharacteristic, complemented
31D5×ΓL(1,𝔽8)C2×D511yescharacteristic
32C5×AGL(1,𝔽7)—41no—
33C5×AGL(1,𝔽7)—51no—
34C5×AGL(1,𝔽7)Dic511yescharacteristic, complemented
35C5×AGL(1,𝔽7)AGL(1,𝔽5)11yescharacteristic, complemented
36C2×#1042—71noHall
37C52×C7C2×C3×C411yescharacteristic, complemented, Hall
38C4×AGL(1,𝔽7)—251noHall
39C2×C3×C52—71no—
40C3×C5×D5—71no—
41C3×C5×D5—141no—
42C2×C5×D7—51no—
43D5×D7C3×D511yescharacteristic
44C7⋊AGL(1,𝔽5)—51no—
45C7×AGL(1,𝔽5)—51no—
46Dic35—51no—
47C7×Dic5—51no—
48C2×C3×AGL(1,𝔽5)—351no—
49C2×C3×Dic5—351no—
50C5×ΓL(1,𝔽8)—41no—
51C5×ΓL(1,𝔽8)C2×Dic511yescharacteristic, complemented
52C5×ΓL(1,𝔽8)C2×AGL(1,𝔽5)11yescharacteristic, complemented
53C2×C5×D5—71no—
54#1042—71no—
55#1042AGL(1,𝔽7)11yescharacteristic, direct product factor
56C2×AGL(1,𝔽7)—51no—
57C4×ΓL(1,𝔽8)—251no—
58ΓL(1,𝔽8)⋊C4—251no—
59C3×C52—141noHall
60C2×C5×C7—51no—
61D35C3×Dic511yescharacteristic, complemented
62C5×D7—41no—
63C5×D7—51no—
64C5×D7C3×Dic511yescharacteristic, complemented
65C5×D7C3×AGL(1,𝔽5)11yescharacteristic, complemented
66C7×D5C2×C3×D511yescharacteristic
67C22×C3×C5—351no—
68C2×C3×D5—71no—
69C3×AGL(1,𝔽5)—351no—
70C3×AGL(1,𝔽5)—701no—
71C3×Dic5—351no—
72C3×Dic5—701no—
73C4×D7—251noHall
74C2×C52—71no—
75C5×D5—71no—
76C5×D5C2×AGL(1,𝔽7)11yescharacteristic
77C2×ΓL(1,𝔽8)—51no—
78AGL(1,𝔽7)—51no—
79AGL(1,𝔽7)#104211yescharacteristic, direct product factor
80C2×AGL(1,𝔽5)—351no—
81C2×Dic5—351no—
82C5×C7—41no—
83C5×C7C2×C3×Dic511yescharacteristic, complemented
84C5×C7C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
85C2×C3×C5—71no—
86C2×C3×C5—71no—
87C2×C3×C5—281no—
88C2×C3×C5—351no—
89C2×C3×C5—701no—
90C3×D5—71no—
91C3×D5—141no—
92C2×D7—51no—
93C4×C7—251no—
94Dic7—251no—
95C52C4×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
96C2×C3×C4—1751noHall
97ΓL(1,𝔽8)C2×#104211yescharacteristic, complemented, Hall
98C22×C5—351no—
99C2×D5—71no—
100AGL(1,𝔽5)—51no—
101AGL(1,𝔽5)—351no—
102Dic5—51no—
103Dic5—351no—
104C3×C5—141no—
105C3×C5—141no—
106C3×C5—561no—
107C2×C7—51no—
108D7—51no—
109D7C3×#104211yescharacteristic, complemented
110C22×C3—351no—
111C3×C4—1751no—
112C3×C4—1751no—
113C2×C5—51no—
114C2×C5—71no—
115C2×C5—71no—
116C2×C5—281no—
117C2×C5—351no—
118D5—71no—
119D5D5×AGL(1,𝔽7)11yescharacteristic
120C2×C4—1751noSylow
121C7C2×C3×#104211yescharacteristic, complemented, Sylow
122C2×C3—71no—
123C2×C3—351no—
124C2×C3—351no—
125C5—41no—
126C5Dic5×AGL(1,𝔽7)11yescharacteristic, complemented
127C5AGL(1,𝔽5)×AGL(1,𝔽7)11yescharacteristic, complemented
128C22—351no—
129C4—251no—
130C4—1751no—
131C3—71noSylow
132C2—51no—
133C2—71no—
134C2—351no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52×C7C2×C3×C4
prime radicalC5×D5×AGL(1,𝔽7)C2
cosocleC5×C7×D5C22×C3
LCSG ▷ C52×C7
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#1042C2semidirect
2. ΓL(1,𝔽8)⋊#1042C2semidirect
4. D7×#1042C3semidirect
5. C5×C5⋊AGL(1,𝔽7)C4semidirect
7. C52×AGL(1,𝔽7)C4semidirect
11. C7⋊#1042C2×C3semidirect
12. C7×#1042C2×C3semidirect
14. C52×ΓL(1,𝔽8)C2×C4semidirect
21. C5×D35C3×C4semidirect
23. C52×D7C3×C4semidirect
30. C5⋊AGL(1,𝔽7)Dic5semidirect
34. C5×AGL(1,𝔽7)Dic5semidirect
35. C5×AGL(1,𝔽7)AGL(1,𝔽5)semidirect
37. C52×C7C2×C3×C4semidirect
51. C5×ΓL(1,𝔽8)C2×Dic5semidirect
52. C5×ΓL(1,𝔽8)C2×AGL(1,𝔽5)semidirect
55. #1042AGL(1,𝔽7)direct (N × H)
61. D35C3×Dic5semidirect
64. C5×D7C3×Dic5semidirect
65. C5×D7C3×AGL(1,𝔽5)semidirect
79. AGL(1,𝔽7)#1042direct (N × H)
83. C5×C7C2×C3×Dic5semidirect
84. C5×C7C2×C3×AGL(1,𝔽5)semidirect
95. C52C4×AGL(1,𝔽7)semidirect
97. ΓL(1,𝔽8)C2×#1042semidirect
109. D7C3×#1042semidirect
121. C7C2×C3×#1042semidirect
126. C5Dic5×AGL(1,𝔽7)semidirect
127. C5AGL(1,𝔽5)×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#10422
ΓL(1,𝔽8)⋊#10422
C5×D5×AGL(1,𝔽7)2
D7×#10423
AGL(1,𝔽5)×AGL(1,𝔽7)5
Dic5×AGL(1,𝔽7)5
C2×C3×#10427