Finite Group Databasebeta

← C11×AGL(1,𝔽7)×D35 #44568700

Subgroups — C11×AGL(1,𝔽7)×D35 #44568700

Order 32340.

168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 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×ΓL(1,𝔽8)×D35C211yesmaximal, characteristic, complemented
2C11×C5⋊(ΓL(1,𝔽8)⋊D7)C211yesmaximal, characteristic, complemented
3C5×C7×C11×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4C11×D7×D35C311yesmaximal, characteristic, complemented, Hall
5C5×C7×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C11×D7×AGL(1,𝔽7)—51nomaximal, Hall
7C11×C5⋊(C7⋊D7)C2×C311yescharacteristic, complemented
8C7×C11×D35C2×C311yescharacteristic, complemented
9C5×C7×C11×D7C2×C311yescharacteristic, complemented
10C2×C3×C11×D35—71nomaximal
11D5×C11×AGL(1,𝔽7)—71nomaximal
12C11×D7×ΓL(1,𝔽8)—51no—
13C11×ΓL(1,𝔽8)⋊D7—51no—
14C7×C11×AGL(1,𝔽7)D511yescharacteristic, complemented
15AGL(1,𝔽7)×D35C1111yesmaximal, characteristic, direct product factor, Hall
16C5×C72×C11C22×C311yescharacteristic, complemented, Hall
17C11×C5⋊AGL(1,𝔽7)—71no—
18D5×C11×ΓL(1,𝔽8)—71no—
19C5×C11×AGL(1,𝔽7)D711yescharacteristic, complemented
20C3×C11×D35—71no—
21C3×C11×D35—71no—
22C2×C3×C5×C7×C11—71no—
23C11×D72—51noHall
24C7×C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented, Hall
25C2×C11×D35—71no—
26D5×C11×D7—71no—
27ΓL(1,𝔽8)×D35C2×C1111yescharacteristic, complemented
28C5⋊(ΓL(1,𝔽8)⋊D7)C2×C1111yescharacteristic, complemented
29C5×C7×AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
30C3×C5×C7×C11—71no—
31C5×C11×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
32C11×C7⋊D7—51no—
33C7×C11×D7—51no—
34C7×C11×D7C3×D511yescharacteristic, complemented
35D7×D35C3×C1111yescharacteristic, complemented, Hall
36C2×C3×C11×D7—351no—
37C2×C11×AGL(1,𝔽7)—351no—
38C2×C5×C7×C11—71no—
39C11×D35—71no—
40C11×D35—71no—
41C11×D35—421no—
42C11×D35AGL(1,𝔽7)11yescharacteristic, direct product factor
43C7×D5×C11—71no—
44C5×C11×D7C3×D711yescharacteristic, complemented
45C5×C7×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
46C2×C3×D5×C11—491noHall
47D7×AGL(1,𝔽7)—51noHall
48C72×C11C2×C3×D511yescharacteristic, complemented, Hall
49C5⋊(C7⋊D7)C2×C3×C1111yescharacteristic, complemented
50C7×D35C2×C3×C1111yescharacteristic, complemented
51C5×C7×D7C2×C3×C1111yescharacteristic, complemented
52C2×C3×C7×C11—71no—
53C3×C11×D7—351no—
54C3×C11×D7—351no—
55C2×C11×ΓL(1,𝔽8)—351no—
56C11×AGL(1,𝔽7)—351no—
57C11×AGL(1,𝔽7)D3511yescharacteristic, direct product factor
58C2×C3×D35—71no—
59D5×AGL(1,𝔽7)—71no—
60C5×C7×C11—61no—
61C5×C7×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
62C5×C7×C11C2×C3×D711yescharacteristic, complemented
63C2×C3×C5×C11—71no—
64C3×D5×C11—491no—
65C3×D5×C11—491no—
66C2×C11×D7—351no—
67C2×C11×D7—351no—
68ΓL(1,𝔽8)⋊D7—51no—
69D7×ΓL(1,𝔽8)—51no—
70C7×AGL(1,𝔽7)D5×C1111yescharacteristic, complemented
71C5×C72C22×C3×C1111yescharacteristic, complemented, Hall
72C3×C7×C11—71no—
73C11×ΓL(1,𝔽8)C2×D3511yescharacteristic, complemented
74C2×D5×C11—491noHall
75C2×C3×C5×C7—71no—
76C3×D35—71no—
77C3×D35—71no—
78C5⋊AGL(1,𝔽7)—71no—
79D5×ΓL(1,𝔽8)—71no—
80C5×AGL(1,𝔽7)C11×D711yescharacteristic, complemented
81D72—51noHall
82C3×C5×C11—71noHall
83C2×C7×C11—71no—
84C2×C7×C11—351no—
85C11×D7—51no—
86C11×D7—351no—
87C11×D7—351no—
88C11×D7—2101no—
89C11×D7C3×D3511yescharacteristic, complemented
90C7×ΓL(1,𝔽8)C2×D5×C1111yescharacteristic, complemented, Hall
91C2×D35—71no—
92D5×D7—71no—
93C22×C3×C11—2451noHall
94C2×C5×C11—71no—
95D5×C11—71no—
96D5×C11—491no—
97C3×C5×C7—71no—
98C5×ΓL(1,𝔽8)C2×C11×D711yescharacteristic, complemented
99C7⋊D7—51no—
100C7×D7—51no—
101C7×D7C3×D5×C1111yescharacteristic, complemented
102C2×C3×D7—351no—
103C2×AGL(1,𝔽7)—351no—
104C7×C11—61no—
105C7×C11D5×AGL(1,𝔽7)11yescharacteristic, complemented
106C7×C11C2×C3×D3511yescharacteristic, complemented
107C2×C5×C7—71no—
108D35—71no—
109D35—71no—
110D35—421no—
111D35C11×AGL(1,𝔽7)11yescharacteristic, direct product factor
112C5×D7C3×C11×D711yescharacteristic, complemented
113C7×D5—71no—
114C2×C3×C11—71no—
115C2×C3×C11—2451no—
116C2×C3×C11—2451no—
117C2×C3×D5—491noHall
118C5×C11D7×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
119C72C2×C3×D5×C1111yescharacteristic, complemented, Sylow
120C22×C11—2451noHall
121C2×C3×C7—71no—
122C3×D7—351no—
123C3×D7—351no—
124C2×ΓL(1,𝔽8)—351no—
125AGL(1,𝔽7)—351no—
126AGL(1,𝔽7)C11×D3511yescharacteristic, direct product factor
127C5×C7—61no—
128C5×C7C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
129C5×C7C2×C3×C11×D711yescharacteristic, complemented
130C3×C11—71noHall
131C2×C3×C5—71no—
132C3×D5—491no—
133C3×D5—491no—
134C2×D7—351no—
135C2×D7—351no—
136C2×C11—71no—
137C2×C11—351no—
138C2×C11—2451no—
139C3×C7—71no—
140ΓL(1,𝔽8)C2×C11×D3511yescharacteristic, complemented
141C2×D5—491noHall
142C3×C5—71noHall
143C2×C7—71no—
144C2×C7—351no—
145D7—51no—
146D7—351no—
147D7—351no—
148D7—2101no—
149D7C3×C11×D3511yescharacteristic, complemented
150C22×C3—2451noHall
151C11AGL(1,𝔽7)×D3511yescharacteristic, direct product factor, Sylow
152C2×C5—71no—
153D5—71no—
154D5—491no—
155C7—61no—
156C7D5×C11×AGL(1,𝔽7)11yescharacteristic, complemented
157C7C2×C3×C11×D3511yescharacteristic, complemented
158C2×C3—71no—
159C2×C3—2451no—
160C2×C3—2451no—
161C5C11×D7×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
162C22—2451noSylow
163C3—71noSylow
164C2—71no—
165C2—351no—
166C2—2451no—
subgroupG/H
center, hypercenterC11AGL(1,𝔽7)×D35
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×ΓL(1,𝔽8)×D35C2semidirect
2. C11×C5⋊(ΓL(1,𝔽8)⋊D7)C2semidirect
3. C5×C7×C11×AGL(1,𝔽7)C2semidirect
4. C11×D7×D35C3semidirect
5. C5×C7×C11×ΓL(1,𝔽8)C22semidirect
7. C11×C5⋊(C7⋊D7)C2×C3semidirect
8. C7×C11×D35C2×C3semidirect
9. C5×C7×C11×D7C2×C3semidirect
14. C7×C11×AGL(1,𝔽7)D5semidirect
15. AGL(1,𝔽7)×D35C11direct (N × H)
16. C5×C72×C11C22×C3semidirect
19. C5×C11×AGL(1,𝔽7)D7semidirect
24. C7×C11×ΓL(1,𝔽8)C2×D5semidirect
27. ΓL(1,𝔽8)×D35C2×C11semidirect
28. C5⋊(ΓL(1,𝔽8)⋊D7)C2×C11semidirect
29. C5×C7×AGL(1,𝔽7)C2×C11semidirect
31. C5×C11×ΓL(1,𝔽8)C2×D7semidirect
34. C7×C11×D7C3×D5semidirect
35. D7×D35C3×C11semidirect
42. C11×D35AGL(1,𝔽7)direct (N × H)
44. C5×C11×D7C3×D7semidirect
45. C5×C7×ΓL(1,𝔽8)C22×C11semidirect
48. C72×C11C2×C3×D5semidirect
49. C5⋊(C7⋊D7)C2×C3×C11semidirect
50. C7×D35C2×C3×C11semidirect
51. C5×C7×D7C2×C3×C11semidirect
57. C11×AGL(1,𝔽7)D35direct (N × H)
61. C5×C7×C11C2×AGL(1,𝔽7)semidirect
62. C5×C7×C11C2×C3×D7semidirect
70. C7×AGL(1,𝔽7)D5×C11semidirect
71. C5×C72C22×C3×C11semidirect
73. C11×ΓL(1,𝔽8)C2×D35semidirect
80. C5×AGL(1,𝔽7)C11×D7semidirect
89. C11×D7C3×D35semidirect
90. C7×ΓL(1,𝔽8)C2×D5×C11semidirect
98. C5×ΓL(1,𝔽8)C2×C11×D7semidirect
101. C7×D7C3×D5×C11semidirect
105. C7×C11D5×AGL(1,𝔽7)semidirect
106. C7×C11C2×C3×D35semidirect
111. D35C11×AGL(1,𝔽7)direct (N × H)
112. C5×D7C3×C11×D7semidirect
118. C5×C11D7×AGL(1,𝔽7)semidirect
119. C72C2×C3×D5×C11semidirect
126. AGL(1,𝔽7)C11×D35direct (N × H)
128. C5×C7C2×C11×AGL(1,𝔽7)semidirect
129. C5×C7C2×C3×C11×D7semidirect
140. ΓL(1,𝔽8)C2×C11×D35semidirect
149. D7C3×C11×D35semidirect
151. C11AGL(1,𝔽7)×D35direct (N × H)
156. C7D5×C11×AGL(1,𝔽7)semidirect
157. C7C2×C3×C11×D35semidirect
161. C5C11×D7×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C11×ΓL(1,𝔽8)×D352
C11×C5⋊(ΓL(1,𝔽8)⋊D7)2
C5×C7×C11×AGL(1,𝔽7)2
C11×D7×D353
C11×D7×AGL(1,𝔽7)5
C2×C3×C11×D357
D5×C11×AGL(1,𝔽7)7
AGL(1,𝔽7)×D3511