Finite Group Databasebeta

← D5×C11×ΓL(1,𝔽8)⋊D7 #44568692

Subgroups — D5×C11×ΓL(1,𝔽8)⋊D7 #44568692

Order 32340.

200 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 52 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)⋊D7)C211yesmaximal, characteristic, complemented
2C7×D5×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C5×C11×ΓL(1,𝔽8)⋊D7C211yesmaximal, characteristic, complemented
4D5×C11×C7⋊D7C311yesmaximal, characteristic, complemented, Hall
5C5×C7×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C2×C11×ΓL(1,𝔽8)⋊D7—51nomaximal, Hall
7C11×C5⋊(C7⋊D7)C2×C311yescharacteristic, complemented
8C72×D5×C11C2×C311yescharacteristic, complemented
9C5×C11×C7⋊D7C2×C311yescharacteristic, complemented
10C3×D5×C11×D7—71nomaximal
11D5×C11×AGL(1,𝔽7)—71nomaximal
12C2×C7×C11×ΓL(1,𝔽8)—51no—
13C11×ΓL(1,𝔽8)⋊D7—51no—
14C11×ΓL(1,𝔽8)⋊D7D511yescharacteristic, direct product factor
15D5×ΓL(1,𝔽8)⋊D7C1111yesmaximal, characteristic, direct product factor, Hall
16C5×C72×C11C22×C311yescharacteristic, complemented, Hall
17C11×C5⋊AGL(1,𝔽7)—71no—
18D5×C11×ΓL(1,𝔽8)D711yescharacteristic, complemented
19C5×C11×AGL(1,𝔽7)—71no—
20C3×C11×D35—71no—
21C3×C7×D5×C11—71no—
22C3×C5×C11×D7—71no—
23C2×C11×C7⋊D7—51noHall
24C7×C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented, Hall
25D5×C11×D7—71no—
26D5×C11×D7—71no—
27D5×C11×D7—212no—
28C5⋊(ΓL(1,𝔽8)⋊D7)C2×C1111yescharacteristic, complemented
29C7×D5×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
30C5×ΓL(1,𝔽8)⋊D7C2×C1111yescharacteristic, complemented
31C3×C5×C7×C11—71no—
32C5×C11×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
33C2×C72×C11—51no—
34C11×C7⋊D7—51no—
35C11×C7⋊D7C3×D511yescharacteristic, complemented
36D5×C7⋊D7C3×C1111yescharacteristic, complemented, Hall
37C2×C3×C11×D7—351no—
38C2×C11×AGL(1,𝔽7)—351no—
39C11×D35—71no—
40C11×D35—71no—
41C11×D35—212no—
42C7×D5×C11—32no—
43C7×D5×C11AGL(1,𝔽7)11yescharacteristic, complemented
44C7×D5×C11C3×D711yescharacteristic, complemented
45C5×C11×D7—71no—
46C5×C11×D7—71no—
47C5×C11×D7—212no—
48C5×C7×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
49C2×C3×D5×C11—491noHall
50C2×ΓL(1,𝔽8)⋊D7—51noHall
51C72×C11C2×C3×D511yescharacteristic, complemented, Hall
52C5⋊(C7⋊D7)C2×C3×C1111yescharacteristic, complemented
53C72×D5C2×C3×C1111yescharacteristic, complemented
54C5×C7⋊D7C2×C3×C1111yescharacteristic, complemented
55C2×C3×C7×C11—351no—
56C3×C11×D7—71no—
57C3×C11×D7—351no—
58C2×C11×ΓL(1,𝔽8)—51no—
59C11×AGL(1,𝔽7)—71no—
60C11×AGL(1,𝔽7)—351no—
61C3×D5×D7—71no—
62D5×AGL(1,𝔽7)—71no—
63C5×C7×C11—32no—
64C5×C7×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
65C5×C7×C11C2×C3×D711yescharacteristic, complemented
66C2×C3×C5×C11—491no—
67C3×D5×C11—71no—
68C3×D5×C11—491no—
69C2×C11×D7—351no—
70C2×C11×D7—351no—
71C2×C11×D7—1052no—
72C2×C7×ΓL(1,𝔽8)—51no—
73ΓL(1,𝔽8)⋊D7—51no—
74ΓL(1,𝔽8)⋊D7D5×C1111yescharacteristic, direct product factor
75C5×C72C22×C3×C1111yescharacteristic, complemented, Hall
76C3×C7×C11—71no—
77C11×ΓL(1,𝔽8)D5×D711yescharacteristic, complemented
78C2×D5×C11—491noHall
79C3×D35—71no—
80C3×C7×D5—71no—
81C3×C5×D7—71no—
82C5⋊AGL(1,𝔽7)—71no—
83D5×ΓL(1,𝔽8)C11×D711yescharacteristic, complemented
84C5×AGL(1,𝔽7)—71no—
85C2×C7⋊D7—51noHall
86C3×C5×C11—71noHall
87C2×C7×C11—51no—
88C2×C7×C11—51no—
89C2×C7×C11—152no—
90C11×D7—71no—
91C11×D7—71no—
92C11×D7—212no—
93C11×D7—351no—
94C11×D7—351no—
95C11×D7—1052no—
96C7×ΓL(1,𝔽8)C2×D5×C1111yescharacteristic, complemented, Hall
97D5×D7—71no—
98D5×D7—71no—
99D5×D7—212no—
100C22×C3×C11—2451noHall
101C2×C5×C11—491no—
102D5×C11—491no—
103D5×C11ΓL(1,𝔽8)⋊D711yescharacteristic, direct product factor
104C3×C5×C7—71no—
105C5×ΓL(1,𝔽8)C2×C11×D711yescharacteristic, complemented
106C2×C72—51no—
107C7⋊D7—51no—
108C7⋊D7C3×D5×C1111yescharacteristic, complemented
109C2×C3×D7—351no—
110C2×AGL(1,𝔽7)—351no—
111C7×C11—32no—
112C7×C11D5×AGL(1,𝔽7)11yescharacteristic, complemented
113C7×C11C3×D5×D711yescharacteristic, complemented
114D35—71no—
115D35—71no—
116D35—212no—
117C5×D7—71no—
118C5×D7—71no—
119C5×D7—212no—
120C7×D5—32no—
121C7×D5C11×AGL(1,𝔽7)11yescharacteristic, complemented
122C7×D5C3×C11×D711yescharacteristic, complemented
123C2×C3×C11—351no—
124C2×C3×C11—491no—
125C2×C3×C11—2451no—
126C2×C3×D5—491noHall
127C5×C11C2×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Hall
128C72C2×C3×D5×C1111yescharacteristic, complemented, Sylow
129C22×C11—2451noHall
130C2×C3×C7—351no—
131C3×D7—71no—
132C3×D7—351no—
133C2×ΓL(1,𝔽8)—51no—
134AGL(1,𝔽7)—71no—
135AGL(1,𝔽7)—351no—
136C5×C7—32no—
137C5×C7C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
138C5×C7C2×C3×C11×D711yescharacteristic, complemented
139C3×C11—71noHall
140C2×C3×C5—491no—
141C3×D5—71no—
142C3×D5—491no—
143C2×D7—351no—
144C2×D7—351no—
145C2×D7—1052no—
146C2×C11—51no—
147C2×C11—491no—
148C2×C11—2451no—
149C3×C7—71no—
150ΓL(1,𝔽8)D5×C11×D711yescharacteristic, complemented
151C2×D5—491noHall
152C3×C5—71noHall
153C2×C7—51no—
154C2×C7—51no—
155C2×C7—152no—
156D7—71no—
157D7—71no—
158D7—212no—
159D7—351no—
160D7—351no—
161D7—1052no—
162C22×C3—2451noHall
163C11D5×ΓL(1,𝔽8)⋊D711yescharacteristic, direct product factor, Sylow
164C2×C5—491no—
165D5—491no—
166D5C11×ΓL(1,𝔽8)⋊D711yescharacteristic, direct product factor
167C7—32no—
168C7D5×C11×AGL(1,𝔽7)11yescharacteristic, complemented
169C7C3×D5×C11×D711yescharacteristic, complemented
170C2×C3—351no—
171C2×C3—491no—
172C2×C3—2451no—
173C5C2×C11×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Sylow
174C22—2451noSylow
175C3—71noSylow
176C2—51no—
177C2—491no—
178C2—2451no—
subgroupG/H
center, hypercenterC11D5×ΓL(1,𝔽8)⋊D7
commutator, cosocle, hypocenterC5×C72C22×C3×C11
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C5×C72
UCSC1 ◁ C11

Complements

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

NH ≅ G/Ntype
1. C11×C5⋊(ΓL(1,𝔽8)⋊D7)C2semidirect
2. C7×D5×C11×ΓL(1,𝔽8)C2semidirect
3. C5×C11×ΓL(1,𝔽8)⋊D7C2semidirect
4. D5×C11×C7⋊D7C3semidirect
5. C5×C7×C11×ΓL(1,𝔽8)C22semidirect
7. C11×C5⋊(C7⋊D7)C2×C3semidirect
8. C72×D5×C11C2×C3semidirect
9. C5×C11×C7⋊D7C2×C3semidirect
14. C11×ΓL(1,𝔽8)⋊D7D5direct (N × H)
15. D5×ΓL(1,𝔽8)⋊D7C11direct (N × H)
16. C5×C72×C11C22×C3semidirect
18. D5×C11×ΓL(1,𝔽8)D7semidirect
24. C7×C11×ΓL(1,𝔽8)C2×D5semidirect
28. C5⋊(ΓL(1,𝔽8)⋊D7)C2×C11semidirect
29. C7×D5×ΓL(1,𝔽8)C2×C11semidirect
30. C5×ΓL(1,𝔽8)⋊D7C2×C11semidirect
32. C5×C11×ΓL(1,𝔽8)C2×D7semidirect
35. C11×C7⋊D7C3×D5semidirect
36. D5×C7⋊D7C3×C11semidirect
43. C7×D5×C11AGL(1,𝔽7)semidirect
44. C7×D5×C11C3×D7semidirect
48. C5×C7×ΓL(1,𝔽8)C22×C11semidirect
51. C72×C11C2×C3×D5semidirect
52. C5⋊(C7⋊D7)C2×C3×C11semidirect
53. C72×D5C2×C3×C11semidirect
54. C5×C7⋊D7C2×C3×C11semidirect
64. C5×C7×C11C2×AGL(1,𝔽7)semidirect
65. C5×C7×C11C2×C3×D7semidirect
74. ΓL(1,𝔽8)⋊D7D5×C11direct (N × H)
75. C5×C72C22×C3×C11semidirect
77. C11×ΓL(1,𝔽8)D5×D7semidirect
83. D5×ΓL(1,𝔽8)C11×D7semidirect
96. C7×ΓL(1,𝔽8)C2×D5×C11semidirect
103. D5×C11ΓL(1,𝔽8)⋊D7direct (N × H)
105. C5×ΓL(1,𝔽8)C2×C11×D7semidirect
108. C7⋊D7C3×D5×C11semidirect
112. C7×C11D5×AGL(1,𝔽7)semidirect
113. C7×C11C3×D5×D7semidirect
121. C7×D5C11×AGL(1,𝔽7)semidirect
122. C7×D5C3×C11×D7semidirect
127. C5×C11C2×ΓL(1,𝔽8)⋊D7semidirect
128. C72C2×C3×D5×C11semidirect
137. C5×C7C2×C11×AGL(1,𝔽7)semidirect
138. C5×C7C2×C3×C11×D7semidirect
150. ΓL(1,𝔽8)D5×C11×D7semidirect
163. C11D5×ΓL(1,𝔽8)⋊D7direct (N × H)
166. D5C11×ΓL(1,𝔽8)⋊D7direct (N × H)
168. C7D5×C11×AGL(1,𝔽7)semidirect
169. C7C3×D5×C11×D7semidirect
173. C5C2×C11×ΓL(1,𝔽8)⋊D7semidirect

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)⋊D7)2
C7×D5×C11×ΓL(1,𝔽8)2
C5×C11×ΓL(1,𝔽8)⋊D72
D5×C11×C7⋊D73
C2×C11×ΓL(1,𝔽8)⋊D75
C3×D5×C11×D77
D5×C11×AGL(1,𝔽7)7
D5×ΓL(1,𝔽8)⋊D711