Finite Group Databasebeta

← AGL(1,𝔽11)×ΓL(1,𝔽8)⋊D7 #44568549

Subgroups — AGL(1,𝔽11)×ΓL(1,𝔽8)⋊D7 #44568549

Order 32340.

200 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C11⋊C5×ΓL(1,𝔽8)⋊D7C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11))C211yesmaximal, characteristic, complemented
3C7×ΓL(1,𝔽8)×AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
4C7⋊D7×AGL(1,𝔽11)C311yesmaximal, characteristic, complemented, Hall
5C7×ΓL(1,𝔽8)×C11⋊C5C2211yescharacteristic, complemented, Hall
6D11×ΓL(1,𝔽8)⋊D7C511yesmaximal, characteristic, complemented, Hall
7C11⋊C5×C7⋊D7C2×C311yescharacteristic, complemented
8C7⋊(C7⋊AGL(1,𝔽11))C2×C311yescharacteristic, complemented
9C72×AGL(1,𝔽11)C2×C311yescharacteristic, complemented
10C3×D7×AGL(1,𝔽11)—71nomaximal
11AGL(1,𝔽7)×AGL(1,𝔽11)—71nomaximal
12C11⋊(ΓL(1,𝔽8)⋊D7)C2×C511yescharacteristic, complemented
13C11×ΓL(1,𝔽8)⋊D7C2×C511yescharacteristic, complemented
14C7×ΓL(1,𝔽8)×D11C2×C511yescharacteristic, complemented
15C2×C5×ΓL(1,𝔽8)⋊D7—111nomaximal, Hall
16C72×C11⋊C5C22×C311yescharacteristic, complemented, Hall
17ΓL(1,𝔽8)⋊AGL(1,𝔽11)—71no—
18AGL(1,𝔽7)×C11⋊C5—71no—
19ΓL(1,𝔽8)×AGL(1,𝔽11)D711yescharacteristic, complemented
20C3×C7⋊AGL(1,𝔽11)—71no—
21C3×D7×C11⋊C5—71no—
22C3×C7×AGL(1,𝔽11)—71no—
23D11×C7⋊D7C3×C511yescharacteristic, complemented, Hall
24C7×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented, Hall
25D7×AGL(1,𝔽11)—71no—
26D7×AGL(1,𝔽11)—71no—
27D7×AGL(1,𝔽11)—212no—
28C2×C5×C7×ΓL(1,𝔽8)—111no—
29C5×ΓL(1,𝔽8)⋊D7—111no—
30C5×ΓL(1,𝔽8)⋊D7—111no—
31C3×C7×C11⋊C5—71no—
32ΓL(1,𝔽8)×C11⋊C5C2×D711yescharacteristic, complemented
33C7⋊D77C2×C3×C511yescharacteristic, complemented
34C11×C7⋊D7C2×C3×C511yescharacteristic, complemented
35C72×D11C2×C3×C511yescharacteristic, complemented
36C2×C5×C7⋊D7—111noHall
37C3×D7×D11—71no—
38D11×AGL(1,𝔽7)—71no—
39C7⋊AGL(1,𝔽11)—71no—
40C7⋊AGL(1,𝔽11)—71no—
41C7⋊AGL(1,𝔽11)—212no—
42D7×C11⋊C5—71no—
43D7×C11⋊C5—71no—
44D7×C11⋊C5—212no—
45C7×AGL(1,𝔽11)—32no—
46C7×AGL(1,𝔽11)AGL(1,𝔽7)11yescharacteristic, complemented
47C7×AGL(1,𝔽11)C3×D711yescharacteristic, complemented
48C5×C7×ΓL(1,𝔽8)—111noHall
49C2×C3×AGL(1,𝔽11)—491noHall
50C2×ΓL(1,𝔽8)⋊D7—111noHall
51C72×C11C22×C3×C511yescharacteristic, complemented, Hall
52C2×C5×C72—111no—
53C5×C7⋊D7—111no—
54C5×C7⋊D7—111no—
55C3×D77—71no—
56C3×C11×D7—71no—
57C3×C7×D11—71no—
58C11⋊AGL(1,𝔽7)—71no—
59C11×AGL(1,𝔽7)—71no—
60ΓL(1,𝔽8)×D11C5×D711yescharacteristic, complemented
61C2×C3×C5×D7—771no—
62C2×C5×AGL(1,𝔽7)—771no—
63C7×C11⋊C5—32no—
64C7×C11⋊C5C2×AGL(1,𝔽7)11yescharacteristic, complemented
65C7×C11⋊C5C2×C3×D711yescharacteristic, complemented
66C2×C3×C11⋊C5—491no—
67C3×AGL(1,𝔽11)—71no—
68C3×AGL(1,𝔽11)—491no—
69D7×D11—71no—
70D7×D11—71no—
71D7×D11—212no—
72C2×C7×ΓL(1,𝔽8)—111no—
73ΓL(1,𝔽8)⋊D7—111no—
74ΓL(1,𝔽8)⋊D7AGL(1,𝔽11)11yescharacteristic, direct product factor
75C5×C72—111noHall
76C3×C7×C11—71no—
77C11×ΓL(1,𝔽8)C2×C5×D711yescharacteristic, complemented
78C2×AGL(1,𝔽11)—491noHall
79C2×C3×C5×C7—771no—
80C3×C5×D7—771no—
81C3×C5×D7—771no—
82C2×C5×ΓL(1,𝔽8)—111no—
83C5×AGL(1,𝔽7)—771no—
84C5×AGL(1,𝔽7)—771no—
85C2×C7⋊D7—111noHall
86C3×C11⋊C5—71noHall
87D77—71no—
88D77—71no—
89D77—212no—
90C7×D11—32no—
91C7×D11C5×AGL(1,𝔽7)11yescharacteristic, complemented
92C7×D11C3×C5×D711yescharacteristic, complemented
93C11×D7—71no—
94C11×D7—71no—
95C11×D7—212no—
96C7×ΓL(1,𝔽8)C2×AGL(1,𝔽11)11yescharacteristic, complemented, Hall
97C2×C5×D7—771no—
98C2×C5×D7—771no—
99C2×C5×D7—2312no—
100C2×C3×D11—491noHall
101C2×C11⋊C5—491no—
102AGL(1,𝔽11)—491no—
103AGL(1,𝔽11)ΓL(1,𝔽8)⋊D711yescharacteristic, direct product factor
104C3×C5×C7—771no—
105C5×ΓL(1,𝔽8)—111no—
106C2×C72—111no—
107C7⋊D7—111no—
108C7⋊D7C3×AGL(1,𝔽11)11yescharacteristic, complemented
109C2×C3×D7—771no—
110C2×AGL(1,𝔽7)—771no—
111C7×C11—32no—
112C7×C11C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
113C7×C11C2×C3×C5×D711yescharacteristic, complemented
114C2×C5×C7—111no—
115C2×C5×C7—111no—
116C2×C5×C7—332no—
117C5×D7—771no—
118C5×D7—771no—
119C5×D7—771no—
120C5×D7—771no—
121C5×D7—2312no—
122C5×D7—2312no—
123C2×C3×C11—491no—
124C3×D11—71no—
125C3×D11—491no—
126C22×C3×C5—5391noHall
127C11⋊C5C2×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Hall
128C72C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented, Sylow
129C2×D11—491noHall
130C2×C3×C7—771no—
131C3×D7—71no—
132C3×D7—771no—
133C2×ΓL(1,𝔽8)—111no—
134AGL(1,𝔽7)—71no—
135AGL(1,𝔽7)—771no—
136C5×C7—111no—
137C5×C7—111no—
138C5×C7—332no—
139C3×C11—71noHall
140C2×C3×C5—5391no—
141C2×C3×C5—5391no—
142C2×C3×C5—771no—
143C2×D7—771no—
144C2×D7—771no—
145C2×D7—2312no—
146C2×C11—491no—
147D11—491no—
148D11C5×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
149C3×C7—71no—
150ΓL(1,𝔽8)D7×AGL(1,𝔽11)11yescharacteristic, complemented
151C22×C5—5391noHall
152C3×C5—771noHall
153C2×C7—111no—
154C2×C7—111no—
155C2×C7—332no—
156D7—71no—
157D7—71no—
158D7—212no—
159D7—771no—
160D7—771no—
161D7—2312no—
162C22×C3—5391noHall
163C11C2×C5×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Sylow
164C2×C5—111no—
165C2×C5—5391no—
166C2×C5—5391no—
167C7—32no—
168C7AGL(1,𝔽7)×AGL(1,𝔽11)11yescharacteristic, complemented
169C7C3×D7×AGL(1,𝔽11)11yescharacteristic, complemented
170C2×C3—5391no—
171C2×C3—491no—
172C2×C3—771no—
173C5—111noSylow
174C22—5391noSylow
175C3—71noSylow
176C2—111no—
177C2—5391no—
178C2—491no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C11C22×C3×C5
LCSG ▷ C72×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. C11⋊C5×ΓL(1,𝔽8)⋊D7C2semidirect
2. ΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11))C2semidirect
3. C7×ΓL(1,𝔽8)×AGL(1,𝔽11)C2semidirect
4. C7⋊D7×AGL(1,𝔽11)C3semidirect
5. C7×ΓL(1,𝔽8)×C11⋊C5C22semidirect
6. D11×ΓL(1,𝔽8)⋊D7C5semidirect
7. C11⋊C5×C7⋊D7C2×C3semidirect
8. C7⋊(C7⋊AGL(1,𝔽11))C2×C3semidirect
9. C72×AGL(1,𝔽11)C2×C3semidirect
12. C11⋊(ΓL(1,𝔽8)⋊D7)C2×C5semidirect
13. C11×ΓL(1,𝔽8)⋊D7C2×C5semidirect
14. C7×ΓL(1,𝔽8)×D11C2×C5semidirect
16. C72×C11⋊C5C22×C3semidirect
19. ΓL(1,𝔽8)×AGL(1,𝔽11)D7semidirect
23. D11×C7⋊D7C3×C5semidirect
24. C7×C11×ΓL(1,𝔽8)C22×C5semidirect
32. ΓL(1,𝔽8)×C11⋊C5C2×D7semidirect
33. C7⋊D77C2×C3×C5semidirect
34. C11×C7⋊D7C2×C3×C5semidirect
35. C72×D11C2×C3×C5semidirect
46. C7×AGL(1,𝔽11)AGL(1,𝔽7)semidirect
47. C7×AGL(1,𝔽11)C3×D7semidirect
51. C72×C11C22×C3×C5semidirect
60. ΓL(1,𝔽8)×D11C5×D7semidirect
64. C7×C11⋊C5C2×AGL(1,𝔽7)semidirect
65. C7×C11⋊C5C2×C3×D7semidirect
74. ΓL(1,𝔽8)⋊D7AGL(1,𝔽11)direct (N × H)
77. C11×ΓL(1,𝔽8)C2×C5×D7semidirect
91. C7×D11C5×AGL(1,𝔽7)semidirect
92. C7×D11C3×C5×D7semidirect
96. C7×ΓL(1,𝔽8)C2×AGL(1,𝔽11)semidirect
103. AGL(1,𝔽11)ΓL(1,𝔽8)⋊D7direct (N × H)
108. C7⋊D7C3×AGL(1,𝔽11)semidirect
112. C7×C11C2×C5×AGL(1,𝔽7)semidirect
113. C7×C11C2×C3×C5×D7semidirect
127. C11⋊C5C2×ΓL(1,𝔽8)⋊D7semidirect
128. C72C2×C3×AGL(1,𝔽11)semidirect
148. D11C5×ΓL(1,𝔽8)⋊D7semidirect
150. ΓL(1,𝔽8)D7×AGL(1,𝔽11)semidirect
163. C11C2×C5×ΓL(1,𝔽8)⋊D7semidirect
168. C7AGL(1,𝔽7)×AGL(1,𝔽11)semidirect
169. C7C3×D7×AGL(1,𝔽11)semidirect

Maximal subgroups

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

subgroupindex
C11⋊C5×ΓL(1,𝔽8)⋊D72
ΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11))2
C7×ΓL(1,𝔽8)×AGL(1,𝔽11)2
C7⋊D7×AGL(1,𝔽11)3
D11×ΓL(1,𝔽8)⋊D75
C3×D7×AGL(1,𝔽11)7
AGL(1,𝔽7)×AGL(1,𝔽11)7
C2×C5×ΓL(1,𝔽8)⋊D711