Finite Group Databasebeta

← C3×C4×D5×D7 #13132641

Subgroups — C3×C4×D5×D7 #13132641

Order 1680.

216 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C3×D5×D7C211yesmaximal, characteristic
2C3×C4×D35C211yesmaximal, characteristic, complemented
3C3×C4×C7×D5C211yesmaximal, characteristic, complemented
4C3×C4×C5×D7C211yesmaximal, characteristic, complemented
5C3×#63304C211yesmaximal, characteristic, complemented
6C3×D5×Dic7C211yesmaximal, characteristic, complemented
7C3×D7×Dic5C211yesmaximal, characteristic, complemented
8C4×D5×D7C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×D35C2211yescharacteristic
10C2×C3×C7×D5C2211yescharacteristic
11C2×C3×C5×D7C2211yescharacteristic
12C3×D5×D7C414yesdirect product factor
13C3×C4×C5×C7C2211yescharacteristic, complemented
14C3×Dic35C2211yescharacteristic, complemented
15C3×C7×Dic5C2211yescharacteristic, complemented
16C3×C5×Dic7C2211yescharacteristic, complemented
17C2×C3×C4×D7—51nomaximal, Hall
18C2×D5×D7C2×C311yescharacteristic
19C4×D35C2×C311yescharacteristic, complemented
20C4×C7×D5C2×C311yescharacteristic, complemented
21C4×C5×D7C2×C311yescharacteristic, complemented
22#63304C2×C311yescharacteristic, complemented
23D5×Dic7C2×C311yescharacteristic, complemented
24D7×Dic5C2×C311yescharacteristic, complemented
25C2×C3×C4×D5—71nomaximal, Hall
26C2×C3×C5×C7C2311yescharacteristic
27C3×D35C2×C412yescomplemented
28C3×C7×D5C2×C412yescomplemented
29C3×C5×D7C2×C412yescomplemented
30C22×C3×D7—51no—
31C2×C3×C4×C7—51no—
32C2×C3×Dic7—51no—
33C3×C4×D7—51no—
34C3×C4×D7—51no—
35C3×C4×D7—51no—
36C3×C4×D7D511yescharacteristic, direct product factor
37C2×D35C22×C311yescharacteristic
38C2×C7×D5C22×C311yescharacteristic
39C2×C5×D7C22×C311yescharacteristic
40D5×D7C3×C414yesdirect product factor
41C4×C5×C7C22×C311yescharacteristic, complemented
42Dic35C22×C311yescharacteristic, complemented
43C5×Dic7C22×C311yescharacteristic, complemented
44C7×Dic5C22×C311yescharacteristic, complemented
45C22×C3×D5—71no—
46C2×C3×C4×C5—71no—
47C2×C3×Dic5—71no—
48C3×C4×D5—71no—
49C3×C4×D5—71no—
50C3×C4×D5—71no—
51C3×C4×D5D711yescharacteristic, direct product factor
52C2×C4×D7—51noHall
53C3×C5×C7C22×C411yescharacteristic, complemented, Hall
54C22×C3×C7—51no—
55C2×C3×D7—51no—
56C2×C3×D7—54no—
57C2×C3×D7C2×D511yescharacteristic
58C3×C4×C7—51no—
59C3×C4×C7C2×D511yescharacteristic, complemented
60C3×Dic7—51no—
61C3×Dic7C2×D511yescharacteristic, complemented
62C2×C4×D5—71noHall
63C2×C5×C7C23×C311yescharacteristic
64D35C2×C3×C412yescomplemented
65C5×D7C2×C3×C412yescomplemented
66C7×D5C2×C3×C412yescomplemented
67C22×C3×C5—71no—
68C2×C3×D5—71no—
69C2×C3×D5—74no—
70C2×C3×D5C2×D711yescharacteristic
71C3×C4×C5—71no—
72C3×C4×C5C2×D711yescharacteristic, complemented
73C3×Dic5—71no—
74C3×Dic5C2×D711yescharacteristic, complemented
75C22×D7—51no—
76C2×C4×C7—51no—
77C2×Dic7—51no—
78C4×D7—51no—
79C4×D7—51no—
80C4×D7—51no—
81C4×D7C3×D511yescharacteristic, direct product factor
82C22×C3×C4—351noHall
83C2×C3×C7—52no—
84C2×C3×C7C22×D511yescharacteristic
85C3×D7—52no—
86C3×D7C4×D512yesdirect product factor
87C22×D5—71no—
88C2×C4×C5—71no—
89C2×Dic5—71no—
90C4×D5—71no—
91C4×D5—71no—
92C4×D5—71no—
93C4×D5C3×D711yescharacteristic, direct product factor
94C5×C7C22×C3×C411yescharacteristic, complemented, Hall
95C2×C3×C5—72no—
96C2×C3×C5C22×D711yescharacteristic
97C3×D5—72no—
98C3×D5C4×D712yesdirect product factor
99C22×C7—51no—
100C2×D7—51no—
101C2×D7—54no—
102C2×D7C2×C3×D511yescharacteristic
103C4×C7—51no—
104C4×C7C2×C3×D511yescharacteristic, complemented
105Dic7—51no—
106Dic7C2×C3×D511yescharacteristic, complemented
107C23×C3—351no—
108C2×C3×C4—51no—
109C2×C3×C4—71no—
110C2×C3×C4—351no—
111C2×C3×C4—351no—
112C2×C3×C4—351no—
113C2×C3×C4—351no—
114C3×C7C2×C4×D511yescharacteristic, complemented, Hall
115C22×C5—71no—
116C2×D5—71no—
117C2×D5—74no—
118C2×D5C2×C3×D711yescharacteristic
119C4×C5—71no—
120C4×C5C2×C3×D711yescharacteristic, complemented
121Dic5—71no—
122Dic5C2×C3×D711yescharacteristic, complemented
123C22×C4—351noSylow
124C3×C5C2×C4×D711yescharacteristic, complemented, Hall
125C2×C7—52no—
126C2×C7C22×C3×D511yescharacteristic
127D7—52no—
128D7C3×C4×D512yesdirect product factor
129C22×C3—51no—
130C22×C3—71no—
131C22×C3—354no—
132C22×C3—351no—
133C3×C4—51no—
134C3×C4—71no—
135C3×C4—351no—
136C3×C4D5×D711yescharacteristic, direct product factor
137C2×C5—72no—
138C2×C5C22×C3×D711yescharacteristic
139D5—72no—
140D5C3×C4×D712yesdirect product factor
141C23—351no—
142C2×C4—51no—
143C2×C4—71no—
144C2×C4—351no—
145C2×C4—351no—
146C2×C4—351no—
147C2×C4—351no—
148C7C2×C3×C4×D511yescharacteristic, complemented, Sylow
149C2×C3—52no—
150C2×C3—72no—
151C2×C3—352no—
152C2×C3C2×D5×D711yescharacteristic
153C5C2×C3×C4×D711yescharacteristic, complemented, Sylow
154C22—51no—
155C22—71no—
156C22—354no—
157C22—351no—
158C4—51no—
159C4—71no—
160C4—351no—
161C4C3×D5×D711yescharacteristic, direct product factor
162C3C4×D5×D711yescharacteristic, direct product factor, Sylow
163C2—52no—
164C2—72no—
165C2—352no—
166C2C2×C3×D5×D711yescharacteristic
subgroupG/H
center, hypercenterC3×C4D5×D7
commutator, hypocenterC5×C7C22×C3×C4
FrattiniC2C2×C3×D5×D7
FittingC3×C4×C5×C7C22
socleC2×C3×C5×C7C23
prime radicalC2×C3×D5×D7C2
cosocleC2×C5×C7C23×C3
LCSG ▷ C5×C7
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×D35C2semidirect
3. C3×C4×C7×D5C2semidirect
4. C3×C4×C5×D7C2semidirect
5. C3×#63304C2semidirect
6. C3×D5×Dic7C2semidirect
7. C3×D7×Dic5C2semidirect
8. C4×D5×D7C3direct (N × H)
12. C3×D5×D7C4direct (N × H)
13. C3×C4×C5×C7C22semidirect
14. C3×Dic35C22semidirect
15. C3×C7×Dic5C22semidirect
16. C3×C5×Dic7C22semidirect
19. C4×D35C2×C3semidirect
20. C4×C7×D5C2×C3semidirect
21. C4×C5×D7C2×C3semidirect
22. #63304C2×C3semidirect
23. D5×Dic7C2×C3semidirect
24. D7×Dic5C2×C3semidirect
27. C3×D35C2×C4semidirect
28. C3×C7×D5C2×C4semidirect
29. C3×C5×D7C2×C4semidirect
36. C3×C4×D7D5direct (N × H)
40. D5×D7C3×C4direct (N × H)
41. C4×C5×C7C22×C3semidirect
42. Dic35C22×C3semidirect
43. C5×Dic7C22×C3semidirect
44. C7×Dic5C22×C3semidirect
51. C3×C4×D5D7direct (N × H)
53. C3×C5×C7C22×C4semidirect
59. C3×C4×C7C2×D5semidirect
61. C3×Dic7C2×D5semidirect
64. D35C2×C3×C4semidirect
65. C5×D7C2×C3×C4semidirect
66. C7×D5C2×C3×C4semidirect
72. C3×C4×C5C2×D7semidirect
74. C3×Dic5C2×D7semidirect
81. C4×D7C3×D5direct (N × H)
86. C3×D7C4×D5direct (N × H)
93. C4×D5C3×D7direct (N × H)
94. C5×C7C22×C3×C4semidirect
98. C3×D5C4×D7direct (N × H)
104. C4×C7C2×C3×D5semidirect
106. Dic7C2×C3×D5semidirect
114. C3×C7C2×C4×D5semidirect
120. C4×C5C2×C3×D7semidirect
122. Dic5C2×C3×D7semidirect
124. C3×C5C2×C4×D7semidirect
128. D7C3×C4×D5direct (N × H)
136. C3×C4D5×D7direct (N × H)
140. D5C3×C4×D7direct (N × H)
148. C7C2×C3×C4×D5semidirect
153. C5C2×C3×C4×D7semidirect
161. C4C3×D5×D7direct (N × H)
162. C3C4×D5×D7direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×D72
C3×C4×D352
C3×C4×C7×D52
C3×C4×C5×D72
C3×#633042
C3×D5×Dic72
C3×D7×Dic52
C4×D5×D73
C2×C3×C4×D75
C2×C3×C4×D57