Finite Group Databasebeta

← C5×AGL(1,𝔽7)×D77 #44568630

Subgroups — C5×AGL(1,𝔽7)×D77 #44568630

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
1C5×ΓL(1,𝔽8)×D77C211yesmaximal, characteristic, complemented
2C5×C11⋊(ΓL(1,𝔽8)⋊D7)C211yesmaximal, characteristic, complemented
3C5×C7×C11×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4C5×D7×D77C311yesmaximal, characteristic, complemented, Hall
5C5×C7×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6AGL(1,𝔽7)×D77C511yesmaximal, characteristic, direct product factor, Hall
7C5×C7⋊D77C2×C311yescharacteristic, complemented
8C5×C7×D77C2×C311yescharacteristic, complemented
9C5×C7×C11×D7C2×C311yescharacteristic, complemented
10C2×C3×C5×D77—71nomaximal
11C5×D11×AGL(1,𝔽7)—71nomaximal
12ΓL(1,𝔽8)×D77C2×C511yescharacteristic, complemented
13C11⋊(ΓL(1,𝔽8)⋊D7)C2×C511yescharacteristic, complemented
14C7×C11×AGL(1,𝔽7)C2×C511yescharacteristic, complemented
15C5×D7×AGL(1,𝔽7)—111nomaximal, Hall
16C5×C72×C11C22×C311yescharacteristic, complemented, Hall
17C5×C11⋊AGL(1,𝔽7)—71no—
18C5×C11×AGL(1,𝔽7)D711yescharacteristic, complemented
19C5×ΓL(1,𝔽8)×D11—71no—
20C3×C5×D77—71no—
21C3×C5×D77—71no—
22C2×C3×C5×C7×C11—71no—
23D7×D77C3×C511yescharacteristic, complemented, Hall
24C7×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented, Hall
25C2×C5×D77—71no—
26C5×D7×D11—71no—
27C5×D7×ΓL(1,𝔽8)—111no—
28C5×ΓL(1,𝔽8)⋊D7—111no—
29C5×C7×AGL(1,𝔽7)D1111yescharacteristic, complemented
30C3×C5×C7×C11—71no—
31C5×C11×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
32C7⋊D77C2×C3×C511yescharacteristic, complemented
33C7×D77C2×C3×C511yescharacteristic, complemented
34C7×C11×D7C2×C3×C511yescharacteristic, complemented
35C5×D72—111noHall
36C2×C3×D77—71no—
37D11×AGL(1,𝔽7)—71no—
38C2×C5×C7×C11—71no—
39C5×D77—71no—
40C5×D77—71no—
41C5×D77—421no—
42C5×D77AGL(1,𝔽7)11yescharacteristic, direct product factor
43C5×C11×D7C3×D711yescharacteristic, complemented
44C5×C7×D11—71no—
45C5×C7×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
46C2×C3×C5×D11—491noHall
47D7×AGL(1,𝔽7)—111noHall
48C72×C11C22×C3×C511yescharacteristic, complemented, Hall
49C5×C7⋊D7—111no—
50C5×C7×D7—111no—
51C5×C7×D7C3×D1111yescharacteristic, complemented
52C2×C3×C7×C11—71no—
53C3×D77—71no—
54C3×D77—71no—
55C11⋊AGL(1,𝔽7)—71no—
56C11×AGL(1,𝔽7)C5×D711yescharacteristic, complemented
57ΓL(1,𝔽8)×D11—71no—
58C2×C3×C5×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—71no—
64C3×C5×D11—491no—
65C3×C5×D11—491no—
66C2×D77—71no—
67D7×D11—71no—
68ΓL(1,𝔽8)⋊D7—111no—
69D7×ΓL(1,𝔽8)—111no—
70C7×AGL(1,𝔽7)C5×D1111yescharacteristic, complemented
71C5×C72C2×C3×D1111yescharacteristic, complemented, Hall
72C3×C7×C11—71no—
73C11×ΓL(1,𝔽8)C2×C5×D711yescharacteristic, complemented
74C2×C5×D11—491noHall
75C2×C3×C5×C7—71no—
76C3×C5×D7—771no—
77C3×C5×D7—771no—
78C2×C5×ΓL(1,𝔽8)—771no—
79C5×AGL(1,𝔽7)—771no—
80C5×AGL(1,𝔽7)D7711yescharacteristic, direct product factor
81D72—111noHall
82C3×C5×C11—71noHall
83C2×C7×C11—71no—
84D77—71no—
85D77—71no—
86D77—421no—
87D77C5×AGL(1,𝔽7)11yescharacteristic, direct product factor
88C7×D11—71no—
89C11×D7C3×C5×D711yescharacteristic, complemented
90C7×ΓL(1,𝔽8)C2×C5×D1111yescharacteristic, complemented, Hall
91C2×C5×D7—771no—
92C2×C5×D7—771no—
93C2×C3×D11—491noHall
94C2×C5×C11—71no—
95C5×D11—71no—
96C5×D11—491no—
97C3×C5×C7—71no—
98C5×ΓL(1,𝔽8)C2×D7711yescharacteristic, complemented
99C7⋊D7—111no—
100C7×D7—111no—
101C7×D7C3×C5×D1111yescharacteristic, complemented
102C2×C3×D7—771no—
103C2×AGL(1,𝔽7)—771no—
104C7×C11—61no—
105C7×C11C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
106C7×C11C2×C3×C5×D711yescharacteristic, complemented
107C2×C5×C7—71no—
108C2×C5×C7—771no—
109C5×D7—111no—
110C5×D7—771no—
111C5×D7—771no—
112C5×D7—4621no—
113C5×D7C3×D7711yescharacteristic, complemented
114C2×C3×C11—71no—
115C3×D11—491no—
116C3×D11—491no—
117C22×C3×C5—5391noHall
118C5×C11D7×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
119C72C2×C3×C5×D1111yescharacteristic, complemented, Sylow
120C2×D11—491noHall
121C2×C3×C7—71no—
122C3×D7—771no—
123C3×D7—771no—
124C2×ΓL(1,𝔽8)—771no—
125AGL(1,𝔽7)—771no—
126AGL(1,𝔽7)C5×D7711yescharacteristic, direct product factor
127C5×C7—61no—
128C5×C7D11×AGL(1,𝔽7)11yescharacteristic, complemented
129C5×C7C2×C3×D7711yescharacteristic, complemented
130C3×C11—71noHall
131C2×C3×C5—71no—
132C2×C3×C5—5391no—
133C2×C3×C5—5391no—
134C2×D7—771no—
135C2×D7—771no—
136C2×C11—71no—
137D11—71no—
138D11—491no—
139C3×C7—71no—
140ΓL(1,𝔽8)C2×C5×D7711yescharacteristic, complemented
141C22×C5—5391noHall
142C3×C5—71noHall
143C2×C7—71no—
144C2×C7—771no—
145D7—111no—
146D7—771no—
147D7—771no—
148D7—4621no—
149D7C3×C5×D7711yescharacteristic, complemented
150C22×C3—5391noHall
151C11C5×D7×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
152C2×C5—71no—
153C2×C5—5391no—
154C2×C5—771no—
155C7—61no—
156C7C5×D11×AGL(1,𝔽7)11yescharacteristic, complemented
157C7C2×C3×C5×D7711yescharacteristic, complemented
158C2×C3—71no—
159C2×C3—5391no—
160C2×C3—5391no—
161C5AGL(1,𝔽7)×D7711yescharacteristic, direct product factor, Sylow
162C22—5391noSylow
163C3—71noSylow
164C2—71no—
165C2—5391no—
166C2—771no—
subgroupG/H
center, hypercenterC5AGL(1,𝔽7)×D77
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C72×C11
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C5×ΓL(1,𝔽8)×D77C2semidirect
2. C5×C11⋊(ΓL(1,𝔽8)⋊D7)C2semidirect
3. C5×C7×C11×AGL(1,𝔽7)C2semidirect
4. C5×D7×D77C3semidirect
5. C5×C7×C11×ΓL(1,𝔽8)C22semidirect
6. AGL(1,𝔽7)×D77C5direct (N × H)
7. C5×C7⋊D77C2×C3semidirect
8. C5×C7×D77C2×C3semidirect
9. C5×C7×C11×D7C2×C3semidirect
12. ΓL(1,𝔽8)×D77C2×C5semidirect
13. C11⋊(ΓL(1,𝔽8)⋊D7)C2×C5semidirect
14. C7×C11×AGL(1,𝔽7)C2×C5semidirect
16. C5×C72×C11C22×C3semidirect
18. C5×C11×AGL(1,𝔽7)D7semidirect
23. D7×D77C3×C5semidirect
24. C7×C11×ΓL(1,𝔽8)C22×C5semidirect
29. C5×C7×AGL(1,𝔽7)D11semidirect
31. C5×C11×ΓL(1,𝔽8)C2×D7semidirect
32. C7⋊D77C2×C3×C5semidirect
33. C7×D77C2×C3×C5semidirect
34. C7×C11×D7C2×C3×C5semidirect
42. C5×D77AGL(1,𝔽7)direct (N × H)
43. C5×C11×D7C3×D7semidirect
45. C5×C7×ΓL(1,𝔽8)C2×D11semidirect
48. C72×C11C22×C3×C5semidirect
51. C5×C7×D7C3×D11semidirect
56. C11×AGL(1,𝔽7)C5×D7semidirect
61. C5×C7×C11C2×AGL(1,𝔽7)semidirect
62. C5×C7×C11C2×C3×D7semidirect
70. C7×AGL(1,𝔽7)C5×D11semidirect
71. C5×C72C2×C3×D11semidirect
73. C11×ΓL(1,𝔽8)C2×C5×D7semidirect
80. C5×AGL(1,𝔽7)D77direct (N × H)
87. D77C5×AGL(1,𝔽7)direct (N × H)
89. C11×D7C3×C5×D7semidirect
90. C7×ΓL(1,𝔽8)C2×C5×D11semidirect
98. C5×ΓL(1,𝔽8)C2×D77semidirect
101. C7×D7C3×C5×D11semidirect
105. C7×C11C2×C5×AGL(1,𝔽7)semidirect
106. C7×C11C2×C3×C5×D7semidirect
113. C5×D7C3×D77semidirect
118. C5×C11D7×AGL(1,𝔽7)semidirect
119. C72C2×C3×C5×D11semidirect
126. AGL(1,𝔽7)C5×D77direct (N × H)
128. C5×C7D11×AGL(1,𝔽7)semidirect
129. C5×C7C2×C3×D77semidirect
140. ΓL(1,𝔽8)C2×C5×D77semidirect
149. D7C3×C5×D77semidirect
151. C11C5×D7×AGL(1,𝔽7)semidirect
156. C7C5×D11×AGL(1,𝔽7)semidirect
157. C7C2×C3×C5×D77semidirect
161. C5AGL(1,𝔽7)×D77direct (N × H)

Maximal subgroups

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

subgroupindex
C5×ΓL(1,𝔽8)×D772
C5×C11⋊(ΓL(1,𝔽8)⋊D7)2
C5×C7×C11×AGL(1,𝔽7)2
C5×D7×D773
AGL(1,𝔽7)×D775
C2×C3×C5×D777
C5×D11×AGL(1,𝔽7)7
C5×D7×AGL(1,𝔽7)11