Finite Group Databasebeta

← D7×C5⋊(C11⋊AGL(1,𝔽7)) #44568697

Subgroups — D7×C5⋊(C11⋊AGL(1,𝔽7)) #44568697

Order 32340.

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

normal subgroups only

This group has 168 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
1C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7))C211yesmaximal, characteristic, complemented
2C7×C5⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
3C5×C11×D7×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4D7×D385C311yesmaximal, characteristic, complemented, Hall
5C5×C7×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6D7×C11⋊AGL(1,𝔽7)—51nomaximal, Hall
7C5⋊(C7⋊D77)C2×C311yescharacteristic, complemented
8C7×D385C2×C311yescharacteristic, complemented
9C5×C7×C11×D7C2×C311yescharacteristic, complemented
10C3×D7×D55—71nomaximal
11C2×C5⋊(C11⋊AGL(1,𝔽7))—71nomaximal
12C11×D7×ΓL(1,𝔽8)D511yescharacteristic, complemented
13C11⋊(ΓL(1,𝔽8)⋊D7)—51no—
14C7×C11⋊AGL(1,𝔽7)—51no—
15D7×C5⋊AGL(1,𝔽7)—111nomaximal, Hall
16C5×C72×C11C22×C311yescharacteristic, complemented, Hall
17C5⋊(C11⋊AGL(1,𝔽7))—71no—
18C5⋊(C11⋊AGL(1,𝔽7))D711yescharacteristic, direct product factor
19C3×D385—71no—
20C3×C7×D55—71no—
21C3×C5×C11×D7—141no—
22C2×C5×C11×ΓL(1,𝔽8)—71no—
23D7×D77—51noHall
24C7×C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented, Hall
25C2×D385—71no—
26D7×D55—71no—
27C5⋊(ΓL(1,𝔽8)⋊D7)—111no—
28C7×C5⋊AGL(1,𝔽7)—111no—
29C5×D7×ΓL(1,𝔽8)D1111yescharacteristic, complemented
30C3×C5×C7×C11—141no—
31C5×C11×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
32C7⋊D77—51no—
33C7×D77—51no—
34C7×C11×D7C3×D511yescharacteristic, complemented
35D7×D35—111noHall
36C3×D7×D11—351no—
37C2×C11⋊AGL(1,𝔽7)—351no—
38C2×C5×C7×C11—71no—
39D385—71no—
40D385—71no—
41D385—421no—
42D385C3×D711yescharacteristic, complemented
43C7×D55—71no—
44C5×C11×D7AGL(1,𝔽7)11yescharacteristic, complemented
45C5×C7×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
46C2×C3×D55—491noHall
47D7×AGL(1,𝔽7)—551noHall
48C72×C11C2×C3×D511yescharacteristic, complemented, Hall
49C5⋊(C7⋊D7)—111no—
50C7×D35—111no—
51C5×C7×D7C3×D1111yescharacteristic, complemented
52C3×D77—351no—
53C3×C11×D7—141no—
54C3×C7×D11—351no—
55C2×C11×ΓL(1,𝔽8)—71no—
56C11⋊AGL(1,𝔽7)—51no—
57C11⋊AGL(1,𝔽7)—351no—
58C3×D5×D7—771no—
59C2×C5⋊AGL(1,𝔽7)—771no—
60C5×C7×C11—61no—
61C5×C7×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
62C5×C7×C11C2×C3×D711yescharacteristic, complemented
63C2×C3×C5×C11—981no—
64C3×D55—71no—
65C3×D55—491no—
66C2×D77—351no—
67D7×D11—351no—
68ΓL(1,𝔽8)⋊D7—551no—
69D7×ΓL(1,𝔽8)D5511yescharacteristic, complemented
70C7×AGL(1,𝔽7)—551no—
71C5×C72C2×C3×D1111yescharacteristic, complemented, Hall
72C3×C7×C11—141no—
73C11×ΓL(1,𝔽8)D5×D711yescharacteristic, complemented
74C2×D55—491noHall
75C3×D35—771no—
76C3×C7×D5—771no—
77C3×C5×D7—141no—
78C2×C5×ΓL(1,𝔽8)—71no—
79C5⋊AGL(1,𝔽7)—111no—
80C5⋊AGL(1,𝔽7)—771no—
81D72—551noHall
82C3×C5×C11—141noHall
83C2×C7×C11—71no—
84D77—51no—
85D77—351no—
86D77—351no—
87D77—2101no—
88C7×D11—351no—
89C11×D7C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
90C7×ΓL(1,𝔽8)C2×D5511yescharacteristic, complemented, Hall
91C2×D35—771no—
92D5×D7—771no—
93C2×C3×D11—2451noHall
94C2×C5×C11—71no—
95D55—71no—
96D55—491no—
97C3×C5×C7—141no—
98C5×ΓL(1,𝔽8)D7×D1111yescharacteristic, complemented
99C7⋊D7—551no—
100C7×D7—551no—
101C7×D7C3×D5511yescharacteristic, complemented
102C2×C3×D7—3851no—
103C2×AGL(1,𝔽7)—3851no—
104C7×C11—61no—
105C7×C11C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
106C7×C11C3×D5×D711yescharacteristic, complemented
107C2×C5×C7—71no—
108D35—111no—
109D35—771no—
110D35—771no—
111D35—4621no—
112C5×D7C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
113C7×D5—771no—
114C2×C3×C11—981no—
115C3×D11—351no—
116C3×D11—2451no—
117C2×C3×D5—5391noHall
118C5×C11D7×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
119C72C2×C3×D5511yescharacteristic, complemented, Sylow
120C2×D11—2451noHall
121C2×C3×C7—3851no—
122C3×D7—141no—
123C3×D7—3851no—
124C2×ΓL(1,𝔽8)—71no—
125AGL(1,𝔽7)—551no—
126AGL(1,𝔽7)—3851no—
127C5×C7—61no—
128C5×C7C2×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
129C5×C7C3×D7×D1111yescharacteristic, complemented
130C3×C11—141noHall
131C2×C3×C5—981no—
132C3×D5—5391no—
133C3×D5—771no—
134C2×D7—3851no—
135C2×D7—3851no—
136C2×C11—71no—
137D11—351no—
138D11—2451no—
139C3×C7—141no—
140ΓL(1,𝔽8)D7×D5511yescharacteristic, complemented
141C2×D5—5391noHall
142C3×C5—141noHall
143C2×C7—71no—
144C2×C7—3851no—
145D7—23101no—
146D7—551no—
147D7—3851no—
148D7—3851no—
149D7C5⋊(C11⋊AGL(1,𝔽7))11yescharacteristic, direct product factor
150C22×C3—26951noHall
151C11D7×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
152C2×C5—71no—
153D5—5391no—
154D5—771no—
155C7—61no—
156C7C2×C5⋊(C11⋊AGL(1,𝔽7))11yescharacteristic, complemented
157C7C3×D7×D5511yescharacteristic, complemented
158C2×C3—981no—
159C2×C3—3851no—
160C2×C3—26951no—
161C5D7×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
162C22—26951noSylow
163C3—141noSylow
164C2—71no—
165C2—3851no—
166C2—26951no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C11C22×C3
LCSG ▷ C5×C72×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7))C2semidirect
2. C7×C5⋊(C11⋊AGL(1,𝔽7))C2semidirect
3. C5×C11×D7×ΓL(1,𝔽8)C2semidirect
4. D7×D385C3semidirect
5. C5×C7×C11×ΓL(1,𝔽8)C22semidirect
7. C5⋊(C7⋊D77)C2×C3semidirect
8. C7×D385C2×C3semidirect
9. C5×C7×C11×D7C2×C3semidirect
12. C11×D7×ΓL(1,𝔽8)D5semidirect
16. C5×C72×C11C22×C3semidirect
18. C5⋊(C11⋊AGL(1,𝔽7))D7direct (N × H)
24. C7×C11×ΓL(1,𝔽8)C2×D5semidirect
29. C5×D7×ΓL(1,𝔽8)D11semidirect
31. C5×C11×ΓL(1,𝔽8)C2×D7semidirect
34. C7×C11×D7C3×D5semidirect
42. D385C3×D7semidirect
44. C5×C11×D7AGL(1,𝔽7)semidirect
45. C5×C7×ΓL(1,𝔽8)C2×D11semidirect
48. C72×C11C2×C3×D5semidirect
51. C5×C7×D7C3×D11semidirect
61. C5×C7×C11C2×AGL(1,𝔽7)semidirect
62. C5×C7×C11C2×C3×D7semidirect
69. D7×ΓL(1,𝔽8)D55semidirect
71. C5×C72C2×C3×D11semidirect
73. C11×ΓL(1,𝔽8)D5×D7semidirect
89. C11×D7C5⋊AGL(1,𝔽7)semidirect
90. C7×ΓL(1,𝔽8)C2×D55semidirect
98. C5×ΓL(1,𝔽8)D7×D11semidirect
101. C7×D7C3×D55semidirect
105. C7×C11C2×C5⋊AGL(1,𝔽7)semidirect
106. C7×C11C3×D5×D7semidirect
112. C5×D7C11⋊AGL(1,𝔽7)semidirect
118. C5×C11D7×AGL(1,𝔽7)semidirect
119. C72C2×C3×D55semidirect
128. C5×C7C2×C11⋊AGL(1,𝔽7)semidirect
129. C5×C7C3×D7×D11semidirect
140. ΓL(1,𝔽8)D7×D55semidirect
149. D7C5⋊(C11⋊AGL(1,𝔽7))direct (N × H)
151. C11D7×C5⋊AGL(1,𝔽7)semidirect
156. C7C2×C5⋊(C11⋊AGL(1,𝔽7))semidirect
157. C7C3×D7×D55semidirect
161. C5D7×C11⋊AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7))2
C7×C5⋊(C11⋊AGL(1,𝔽7))2
C5×C11×D7×ΓL(1,𝔽8)2
D7×D3853
D7×C11⋊AGL(1,𝔽7)5
C3×D7×D557
C2×C5⋊(C11⋊AGL(1,𝔽7))7
D7×C5⋊AGL(1,𝔽7)11