Finite Group Databasebeta

← C7×D5×#10587720 #50304005

Subgroups — C7×D5×#10587720 #50304005

Order 38220.

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
1C7×D5×#63270C211yesmaximal, characteristic, complemented
2C7×C5⋊#10587720C211yesmaximal, characteristic, complemented
3C5×C7×#10587720C211yesmaximal, characteristic, complemented
4C7×D5×C13×D7C311yesmaximal, characteristic, complemented, Hall
5C5×C7×#63270C2211yescharacteristic, complemented, Hall
6C2×C7×#10587720—51nomaximal, Hall
7C7×C13×D35C2×C311yescharacteristic, complemented
8C72×D5×C13C2×C311yescharacteristic, complemented
9C5×C7×C13×D7C2×C311yescharacteristic, complemented
10C2×C7×D5×C13⋊C3—71nomaximal
11D5×#10587720C711yesmaximal, characteristic, direct product factor
12C2×C7×#63270—51no—
13C7×#10587720—51no—
14C7×#10587720D511yescharacteristic, direct product factor
15C5×C72×C13C22×C311yescharacteristic, complemented, Hall
16C7×D5×AGL(1,𝔽7)—131nomaximal, Hall
17C5⋊#10587720C2×C711yescharacteristic, complemented
18D5×#63270C2×C711yescharacteristic, complemented
19C7×D5×C13⋊C3—71no—
20C7×D5×C13⋊C3—71no—
21C5×#10587720C2×C711yescharacteristic, complemented
22C2×C5×C7×C13⋊C3—71no—
23C2×C7×C13×D7—51noHall
24C7×#63270C2×D511yescharacteristic, complemented, Hall
25C2×C7×D5×C13—71no—
26D5×C13×D7C3×C711yescharacteristic, complemented
27C7×D5×ΓL(1,𝔽8)—131no—
28C7×C5⋊AGL(1,𝔽7)—131no—
29C5×C7×AGL(1,𝔽7)—131no—
30C5×#63270C22×C711yescharacteristic, complemented
31C5×C7×C13⋊C3—71no—
32C2×C72×C13—51no—
33C7×C13×D7—51no—
34C7×C13×D7C3×D511yescharacteristic, complemented
35C22×C7×C13⋊C3—351no—
36C2×#10587720—51no—
37C7×D5×D7C13⋊C311yescharacteristic, complemented, Hall
38C2×C5×C7×C13—71no—
39C13×D35C2×C3×C711yescharacteristic, complemented
40C7×D5×C13—61no—
41C7×D5×C13—71no—
42C7×D5×C13AGL(1,𝔽7)11yescharacteristic, complemented
43C7×D5×C13C2×C3×C711yescharacteristic, complemented
44C5×C13×D7C2×C3×C711yescharacteristic, complemented
45C2×D5×C13⋊C3—71noHall
46C5×C7×ΓL(1,𝔽8)—131noHall
47C72×C13C2×C3×D511yescharacteristic, complemented, Hall
48C2×C7×AGL(1,𝔽7)—651noHall
49C2×#63270—51no—
50C2×C7×C13⋊C3—71no—
51C2×C7×C13⋊C3—351no—
52C2×C7×C13⋊C3—351no—
53#10587720—51no—
54#10587720C7×D511yescharacteristic, direct product factor
55C7×D35C2×C13⋊C311yescharacteristic, complemented
56C72×D5C2×C13⋊C311yescharacteristic, complemented
57C5×C7×D7C2×C13⋊C311yescharacteristic, complemented
58C5×C7×C13—61no—
59C5×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
60C5×C7×C13C22×C3×C711yescharacteristic, complemented
61C2×C3×C7×D5—911no—
62D5×AGL(1,𝔽7)—131no—
63C2×C5×C13⋊C3—71no—
64D5×C13⋊C3—71no—
65D5×C13⋊C3—71no—
66C22×C7×C13—351no—
67C2×C13×D7—51no—
68C2×C7×ΓL(1,𝔽8)—651no—
69C7×AGL(1,𝔽7)—131no—
70C7×AGL(1,𝔽7)—651no—
71#63270C2×C7×D511yescharacteristic, complemented
72C7×C13⋊C3—71no—
73C2×D5×C13—71noHall
74C5×C72C22×C13⋊C311yescharacteristic, complemented, Hall
75C2×C3×C5×C7—911no—
76C3×C7×D5—911no—
77C3×C7×D5—911no—
78C5⋊AGL(1,𝔽7)—131no—
79D5×ΓL(1,𝔽8)—131no—
80C5×AGL(1,𝔽7)—131no—
81C2×C7×D7—51noHall
82C5×C13⋊C3—71noHall
83C2×C7×C13—51no—
84C2×C7×C13—51no—
85C2×C7×C13—71no—
86C2×C7×C13—301no—
87C2×C7×C13—351no—
88C13×D7—51no—
89C13×D7C3×C7×D511yescharacteristic, complemented
90C22×C13⋊C3—351noHall
91C7×ΓL(1,𝔽8)—131noHall
92C2×C7×D5—71no—
93D5×D7C7×C13⋊C311yescharacteristic, complemented
94C2×C5×C13—71no—
95D5×C13—71no—
96D5×C13C7×AGL(1,𝔽7)11yescharacteristic, complemented
97C3×C5×C7—911no—
98C5×ΓL(1,𝔽8)—131no—
99C2×C72—51no—
100C7×D7—51no—
101C7×D7D5×C13⋊C311yescharacteristic, complemented
102C7×C13—61no—
103C7×C13D5×AGL(1,𝔽7)11yescharacteristic, complemented
104C7×C13C2×C3×C7×D511yescharacteristic, complemented
105C22×C3×C7—4551no—
106C2×AGL(1,𝔽7)—651no—
107C2×C13⋊C3—71no—
108C2×C13⋊C3—351no—
109C2×C13⋊C3—351no—
110C2×C5×C7—71no—
111D35C2×C7×C13⋊C311yescharacteristic, complemented
112C5×D7C2×C7×C13⋊C311yescharacteristic, complemented
113C7×D5—61no—
114C7×D5—71no—
115C7×D5#1058772011yescharacteristic, direct product factor
116C7×D5C2×C7×C13⋊C311yescharacteristic, complemented
117C5×C13C2×C7×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
118C2×C3×D5—911noHall
119C22×C13—351noHall
120C72C2×D5×C13⋊C311yescharacteristic, complemented, Sylow
121C2×C3×C7—911no—
122C2×C3×C7—4551no—
123C2×C3×C7—4551no—
124C2×ΓL(1,𝔽8)—651no—
125AGL(1,𝔽7)—131no—
126AGL(1,𝔽7)—651no—
127C13⋊C3—71noHall
128C5×C7—61no—
129C5×C7C2×#1058772011yescharacteristic, complemented
130C5×C7C22×C7×C13⋊C311yescharacteristic, complemented
131C2×C3×C5—911no—
132C3×D5—911no—
133C3×D5—911no—
134C22×C7—351no—
135C2×D7—51no—
136C2×C13—51no—
137C2×C13—71no—
138C2×C13—351no—
139C3×C7—911no—
140ΓL(1,𝔽8)—131no—
141C2×D5—71noHall
142C3×C5—911noHall
143C2×C7—51no—
144C2×C7—51no—
145C2×C7—71no—
146C2×C7—301no—
147C2×C7—351no—
148D7—51no—
149D7C7×D5×C13⋊C311yescharacteristic, complemented
150C13C7×D5×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
151C22×C3—4551noHall
152C2×C5—71no—
153D5—71no—
154D5C7×#1058772011yescharacteristic, direct product factor
155C7—61no—
156C7D5×#1058772011yescharacteristic, direct product factor
157C7C2×C7×D5×C13⋊C311yescharacteristic, complemented
158C2×C3—911no—
159C2×C3—4551no—
160C2×C3—4551no—
161C5C2×C7×#1058772011yescharacteristic, complemented, Sylow
162C22—351noSylow
163C3—911noSylow
164C2—51no—
165C2—71no—
166C2—351no—
subgroupG/H
center, hypercenterC7D5×#10587720
commutator, cosocle, hypocenterC5×C7×C13C22×C3×C7
FrattiniC1G
Fitting, socleC5×C72×C13C22×C3
LCSG ▷ C5×C7×C13
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
1. C7×D5×#63270C2semidirect
2. C7×C5⋊#10587720C2semidirect
3. C5×C7×#10587720C2semidirect
4. C7×D5×C13×D7C3semidirect
5. C5×C7×#63270C22semidirect
7. C7×C13×D35C2×C3semidirect
8. C72×D5×C13C2×C3semidirect
9. C5×C7×C13×D7C2×C3semidirect
11. D5×#10587720C7direct (N × H)
14. C7×#10587720D5direct (N × H)
15. C5×C72×C13C22×C3semidirect
17. C5⋊#10587720C2×C7semidirect
18. D5×#63270C2×C7semidirect
21. C5×#10587720C2×C7semidirect
24. C7×#63270C2×D5semidirect
26. D5×C13×D7C3×C7semidirect
30. C5×#63270C22×C7semidirect
34. C7×C13×D7C3×D5semidirect
37. C7×D5×D7C13⋊C3semidirect
39. C13×D35C2×C3×C7semidirect
42. C7×D5×C13AGL(1,𝔽7)semidirect
43. C7×D5×C13C2×C3×C7semidirect
44. C5×C13×D7C2×C3×C7semidirect
47. C72×C13C2×C3×D5semidirect
54. #10587720C7×D5direct (N × H)
55. C7×D35C2×C13⋊C3semidirect
56. C72×D5C2×C13⋊C3semidirect
57. C5×C7×D7C2×C13⋊C3semidirect
59. C5×C7×C13C2×AGL(1,𝔽7)semidirect
60. C5×C7×C13C22×C3×C7semidirect
71. #63270C2×C7×D5semidirect
74. C5×C72C22×C13⋊C3semidirect
89. C13×D7C3×C7×D5semidirect
93. D5×D7C7×C13⋊C3semidirect
96. D5×C13C7×AGL(1,𝔽7)semidirect
101. C7×D7D5×C13⋊C3semidirect
103. C7×C13D5×AGL(1,𝔽7)semidirect
104. C7×C13C2×C3×C7×D5semidirect
111. D35C2×C7×C13⋊C3semidirect
112. C5×D7C2×C7×C13⋊C3semidirect
115. C7×D5#10587720direct (N × H)
116. C7×D5C2×C7×C13⋊C3semidirect
117. C5×C13C2×C7×AGL(1,𝔽7)semidirect
120. C72C2×D5×C13⋊C3semidirect
129. C5×C7C2×#10587720semidirect
130. C5×C7C22×C7×C13⋊C3semidirect
149. D7C7×D5×C13⋊C3semidirect
150. C13C7×D5×AGL(1,𝔽7)semidirect
154. D5C7×#10587720direct (N × H)
156. C7D5×#10587720direct (N × H)
157. C7C2×C7×D5×C13⋊C3semidirect
161. C5C2×C7×#10587720semidirect

Maximal subgroups

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

subgroupindex
C7×D5×#632702
C7×C5⋊#105877202
C5×C7×#105877202
C7×D5×C13×D73
C2×C7×#105877205
C2×C7×D5×C13⋊C37
D5×#105877207
C7×D5×AGL(1,𝔽7)13