Finite Group Databasebeta

← C5×D7×#67 #13132682

Subgroups — C5×D7×#67 #13132682

Order 1680.

216 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C22×C3×C5×D7C211yesmaximal, characteristic, complemented
2C2×C5×D3×D7C211yesmaximal, characteristic, complemented
3C5×#4268C211yesmaximal, characteristic, complemented
4C5×C7×#67C211yesmaximal, characteristic, complemented
5C5×#4246C211yesmaximal, characteristic, complemented
6C5×#4245C211yesmaximal, characteristic, complemented
7C5×Dic3×D7C211yesmaximal, characteristic, complemented
8C5×D4×D7—31nomaximal, Hall
9C22×C3×C5×C7C2211yescharacteristic, complemented
10C2×C5×D21C2211yescharacteristic, complemented
11C2×C5×D3×C7C2211yescharacteristic, complemented
12C2×C3×C5×D7—22no—
13C2×C3×C5×D7C2211yescharacteristic
14C2×C3×C5×D7C2211yescharacteristic, complemented
15C5×D3×D7—22no—
16C5×Dic21C2211yescharacteristic, complemented
17C5×C7×Dic3C2211yescharacteristic, complemented
18D7×#67C511yesmaximal, characteristic, direct product factor, Hall
19C22×C5×D7—31no—
20C22×C5×D7D311yescharacteristic, complemented
21C5×C7×D4—31no—
22C5×C22⋊D7—31no—
23C5×C22⋊D7—31no—
24C5×D28—31no—
25C4×C5×D7—31no—
26C2×C5×#67—71nomaximal, Hall
27C2×C3×C5×C7—21no—
28C2×C3×C5×C7C2311yescharacteristic
29C5×D21—21no—
30C5×D3×C7—21no—
31C3×C5×D7—21no—
32C3×C5×D7D412yescomplemented
33C22×C3×D7C2×C511yescharacteristic, complemented
34C2×D3×D7C2×C511yescharacteristic, complemented
35#4268C2×C511yescharacteristic, complemented
36C7×#67C2×C511yescharacteristic, complemented
37#4246C2×C511yescharacteristic, complemented
38#4245C2×C511yescharacteristic, complemented
39Dic3×D7C2×C511yescharacteristic, complemented
40C22×C5×C7—31no—
41C22×C5×C7C2×D311yescharacteristic, complemented
42C2×C5×D7—22no—
43C2×C5×D7—31no—
44C2×C5×D7—62no—
45C2×C5×D7C2×D311yescharacteristic
46C2×C5×D7C2×D311yescharacteristic, complemented
47C4×C5×C7—31no—
48C5×Dic7—31no—
49C23×C3×C5—71no—
50C22×C5×D3—71no—
51C5×#67—71no—
52C5×#67—71no—
53C5×#67—71no—
54C5×#67D711yescharacteristic, direct product factor
55C2×C5×Dic3—71no—
56D4×D7—31noHall
57C3×C5×C7C2×D411yescharacteristic, complemented, Hall
58C22×C3×C7C22×C511yescharacteristic, complemented
59C2×D21C22×C511yescharacteristic, complemented
60C2×D3×C7C22×C511yescharacteristic, complemented
61C2×C3×D7—22no—
62C2×C3×D7C22×C511yescharacteristic
63C2×C3×D7C22×C511yescharacteristic, complemented
64D3×D7—22no—
65Dic21C22×C511yescharacteristic, complemented
66C7×Dic3C22×C511yescharacteristic, complemented
67C2×C5×D4—211noHall
68C2×C5×C7—21no—
69C2×C5×C7—61no—
70C2×C5×C7C22×D311yescharacteristic
71C5×D7—21no—
72C5×D7—61no—
73C5×D7#6712yesdirect product factor
74C22×C3×C5—71no—
75C22×C3×C5—71no—
76C22×C3×C5—142no—
77C22×C3×C5C2×D711yescharacteristic, complemented
78C2×C5×D3—71no—
79C2×C5×D3—142no—
80C2×C5×D3C2×D711yescharacteristic, complemented
81C5×Dic3—71no—
82C5×Dic3C2×D711yescharacteristic, complemented
83C22×D7—31no—
84C22×D7C5×D311yescharacteristic, complemented
85C7×D4—31no—
86C22⋊D7—31no—
87C22⋊D7—31no—
88D28—31no—
89C4×D7—31no—
90C2×#67—71noHall
91C2×C3×C7—21no—
92C2×C3×C7C23×C511yescharacteristic
93D21—21no—
94C3×D7—21no—
95C3×D7C5×D412yescomplemented
96D3×C7—21no—
97C23×C5—71no—
98C23×C5—211no—
99C5×D4—31no—
100C5×D4—211no—
101C5×D4—211no—
102C5×D4—211no—
103C2×C4×C5—211no—
104C5×C7C2×#6711yescharacteristic, complemented, Hall
105C2×C3×C5—21no—
106C2×C3×C5—72no—
107C2×C3×C5—141no—
108C2×C3×C5C22×D711yescharacteristic
109C5×D3—21no—
110C5×D3—141no—
111C22×C7—31no—
112C22×C7C2×C5×D311yescharacteristic, complemented
113C2×D7—22no—
114C2×D7—31no—
115C2×D7—62no—
116C2×D7C2×C5×D311yescharacteristic
117C2×D7C2×C5×D311yescharacteristic, complemented
118C4×C7—31no—
119Dic7—31no—
120C23×C3—71no—
121C22×D3—71no—
122#67—71no—
123#67—71no—
124#67—71no—
125#67C5×D711yescharacteristic, direct product factor
126C2×Dic3—71no—
127C3×C7C2×C5×D411yescharacteristic, complemented, Hall
128C22×C5—31no—
129C22×C5—71no—
130C22×C5—71no—
131C22×C5—142no—
132C22×C5—211no—
133C22×C5—422no—
134C22×C5D3×D711yescharacteristic, complemented
135C4×C5—31no—
136C4×C5—211no—
137C2×D4—211noSylow
138C3×C5D4×D711yescharacteristic, complemented, Hall
139C2×C7—21no—
140C2×C7—61no—
141C2×C7C22×C5×D311yescharacteristic
142D7—21no—
143D7—61no—
144D7C5×#6712yesdirect product factor
145C22×C3—71no—
146C22×C3—71no—
147C22×C3—142no—
148C22×C3C2×C5×D711yescharacteristic, complemented
149C2×D3—71no—
150C2×D3—142no—
151C2×D3C2×C5×D711yescharacteristic, complemented
152Dic3—71no—
153Dic3C2×C5×D711yescharacteristic, complemented
154C2×C5—21no—
155C2×C5—61no—
156C2×C5—72no—
157C2×C5—141no—
158C2×C5—421no—
159C2×C5C2×D3×D711yescharacteristic
160C23—71no—
161C23—211no—
162D4—31no—
163D4—211no—
164D4—211no—
165D4—211no—
166C2×C4—211no—
167C7C2×C5×#6711yescharacteristic, complemented, Sylow
168C2×C3—21no—
169C2×C3—72no—
170C2×C3—141no—
171C2×C3C22×C5×D711yescharacteristic
172D3—21no—
173D3—141no—
174C5D7×#6711yescharacteristic, direct product factor, Sylow
175C22—31no—
176C22—71no—
177C22—71no—
178C22—142no—
179C22—211no—
180C22—422no—
181C22C5×D3×D711yescharacteristic, complemented
182C4—31no—
183C4—211no—
184C3C5×D4×D711yescharacteristic, complemented, Sylow
185C2—21no—
186C2—61no—
187C2—72no—
188C2—141no—
189C2—421no—
190C2C2×C5×D3×D711yescharacteristic
subgroupG/H
centerC2×C5C2×D3×D7
commutator, cosocleC2×C3×C7C23×C5
FrattiniC2C2×C5×D3×D7
FittingC22×C3×C5×C7C22
socleC2×C3×C5×C7C23
hypercenterC22×C5D3×D7
hypocenterC3×C7C2×C5×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C5 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×D7C2semidirect
2. C2×C5×D3×D7C2semidirect
3. C5×#4268C2semidirect
4. C5×C7×#67C2semidirect
5. C5×#4246C2semidirect
6. C5×#4245C2semidirect
7. C5×Dic3×D7C2semidirect
9. C22×C3×C5×C7C22semidirect
10. C2×C5×D21C22semidirect
11. C2×C5×D3×C7C22semidirect
14. C2×C3×C5×D7C22semidirect
16. C5×Dic21C22semidirect
17. C5×C7×Dic3C22semidirect
18. D7×#67C5direct (N × H)
20. C22×C5×D7D3semidirect
32. C3×C5×D7D4semidirect
33. C22×C3×D7C2×C5semidirect
34. C2×D3×D7C2×C5semidirect
35. #4268C2×C5semidirect
36. C7×#67C2×C5semidirect
37. #4246C2×C5semidirect
38. #4245C2×C5semidirect
39. Dic3×D7C2×C5semidirect
41. C22×C5×C7C2×D3semidirect
46. C2×C5×D7C2×D3semidirect
54. C5×#67D7direct (N × H)
57. C3×C5×C7C2×D4semidirect
58. C22×C3×C7C22×C5semidirect
59. C2×D21C22×C5semidirect
60. C2×D3×C7C22×C5semidirect
63. C2×C3×D7C22×C5semidirect
65. Dic21C22×C5semidirect
66. C7×Dic3C22×C5semidirect
73. C5×D7#67direct (N × H)
77. C22×C3×C5C2×D7semidirect
80. C2×C5×D3C2×D7semidirect
82. C5×Dic3C2×D7semidirect
84. C22×D7C5×D3semidirect
95. C3×D7C5×D4semidirect
104. C5×C7C2×#67semidirect
112. C22×C7C2×C5×D3semidirect
117. C2×D7C2×C5×D3semidirect
125. #67C5×D7direct (N × H)
127. C3×C7C2×C5×D4semidirect
134. C22×C5D3×D7semidirect
138. C3×C5D4×D7semidirect
144. D7C5×#67direct (N × H)
148. C22×C3C2×C5×D7semidirect
151. C2×D3C2×C5×D7semidirect
153. Dic3C2×C5×D7semidirect
167. C7C2×C5×#67semidirect
174. C5D7×#67direct (N × H)
181. C22C5×D3×D7semidirect
184. C3C5×D4×D7semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×D72
C2×C5×D3×D72
C5×#42682
C5×C7×#672
C5×#42462
C5×#42452
C5×Dic3×D72
C5×D4×D73
D7×#675
C2×C5×#677