Finite Group Databasebeta

← C3×C5×D17×D21 #36339455

Subgroups — C3×C5×D17×D21 #36339455

Order 21420.

176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C32×C5×C7×D17C211yesmaximal, characteristic, complemented
2C3×C5×C17×D21C211yesmaximal, characteristic, complemented
3C3×C5×D357C211yesmaximal, characteristic, complemented
4C5×D17×D21C311yesmaximal, characteristic, direct product factor
5C3×C5×D7×D17—31nomaximal
6C32×C5×C7×C17C2211yescharacteristic, complemented, Hall
7C3×D17×D21C511yesmaximal, characteristic, direct product factor, Hall
8C5×D357C2×C311yescharacteristic, complemented
9C5×C17×D21C2×C311yescharacteristic, complemented
10C3×C5×D119—31no—
11C3×C5×D7×C17—31no—
12C3×C5×C7×D17—21no—
13C3×C5×C7×D17D311yescharacteristic, complemented
14C3×C5×C7×D17C2×C311yescharacteristic, complemented
15C3×C5×D3×D17—71nomaximal, Hall
16C5×D7×D17—31noHall
17C3×D357C2×C511yescharacteristic, complemented
18C3×C17×D21C2×C511yescharacteristic, complemented
19C32×C7×D17C2×C511yescharacteristic, complemented
20C3×C5×C7×C17—21no—
21C3×C5×C7×C17C2×D311yescharacteristic, complemented
22C3×C5×C7×C17C22×C311yescharacteristic, complemented
23C3×C5×D51—71no—
24C3×C5×D3×C17—71no—
25C32×C5×D17D711yescharacteristic, complemented
26D17×D21C3×C511yescharacteristic, direct product factor
27C3×D7×D17—31no—
28C2×C3×C5×D21—171nomaximal, Hall
29C5×D119—31no—
30C5×D7×C17—31no—
31C5×C7×D17C3×D311yescharacteristic, complemented
32C32×C7×C17C22×C511yescharacteristic, complemented, Hall
33C2×C3×C5×D17—211no—
34C5×D3×D17—71no—
35C32×C5×C17C2×D711yescharacteristic, complemented, Hall
36D357C2×C3×C511yescharacteristic, complemented
37C17×D21C2×C3×C511yescharacteristic, complemented
38C3×D119—31no—
39C3×D7×C17—31no—
40C3×C7×D17—21no—
41C3×C7×D17C5×D311yescharacteristic, complemented
42C3×C7×D17C2×C3×C511yescharacteristic, complemented
43C2×C32×C5×C7—171no—
44C3×C5×D21—171no—
45C3×C5×D21D1711yescharacteristic, direct product factor
46C3×D3×D17—71noHall
47C5×C7×C17C2×C3×D311yescharacteristic, complemented, Hall
48C2×C3×C5×C17—211no—
49C5×D51—71no—
50C5×D3×C17—71no—
51C3×C5×D17—21no—
52C3×C5×D17—211no—
53C3×C5×D17C3×D711yescharacteristic, complemented
54C3×C5×D17D2111yescharacteristic, direct product factor
55D7×D17—31noHall
56C2×C5×D21—171no—
57C2×C3×C5×D7—511no—
58C3×C7×C17—21no—
59C3×C7×C17C2×C5×D311yescharacteristic, complemented
60C3×C7×C17C22×C3×C511yescharacteristic, complemented
61C2×C5×D17—211noHall
62C32×C5×C7C2×D1711yescharacteristic, complemented, Hall
63C3×D51—71no—
64C3×D3×C17—71no—
65C32×D17C5×D711yescharacteristic, complemented
66C3×C5×C17—21no—
67C3×C5×C17C2×C3×D711yescharacteristic, complemented
68C3×C5×C17C2×D2111yescharacteristic, complemented
69C2×C3×D21—171noHall
70D119—31no—
71C7×D17C3×C5×D311yescharacteristic, complemented
72D7×C17—31no—
73C2×C3×C5×C7—171no—
74C2×C3×C5×C7—171no—
75C2×C3×C5×C7—341no—
76C5×D21—171no—
77C5×D21C3×D1711yescharacteristic, direct product factor
78C3×C5×D7—31no—
79C3×C5×D7—511no—
80C2×C3×D17—211no—
81D3×D17—71no—
82C2×C3×C5×D3—1191noHall
83C2×C5×C17—211no—
84C5×D17—211no—
85C5×D17C3×D2111yescharacteristic, direct product factor
86C32×C17C2×C5×D711yescharacteristic, complemented, Hall
87C2×C5×D7—511noHall
88C2×C32×C7—171no—
89C3×D21—171no—
90C3×D21C5×D1711yescharacteristic, direct product factor
91C7×C17C2×C3×C5×D311yescharacteristic, complemented, Hall
92C3×C5×C7—21no—
93C3×C5×C7D3×D1711yescharacteristic, complemented
94C3×C5×C7C2×C3×D1711yescharacteristic, complemented
95C2×C3×C17—211no—
96D51—71no—
97C3×D17—21no—
98C3×D17—211no—
99C3×D17C3×C5×D711yescharacteristic, complemented
100C3×D17C5×D2111yescharacteristic, direct product factor
101D3×C17—71no—
102C2×C32×C5—171no—
103C3×C5×D3—71no—
104C3×C5×D3—1191no—
105C5×C17C2×C3×D2111yescharacteristic, complemented, Hall
106C2×D21—171no—
107C2×C3×D7—511no—
108C2×C5×C7—171no—
109C5×D7—31no—
110C5×D7—511no—
111C2×D17—211noHall
112C32×C7C2×C5×D1711yescharacteristic, complemented, Hall
113C22×C3×C5—3571no—
114C2×C5×D3—1191no—
115C3×C17—21no—
116C3×C17C2×C3×C5×D711yescharacteristic, complemented
117C3×C17C2×C5×D2111yescharacteristic, complemented
118C32×C5D7×D1711yescharacteristic, complemented, Hall
119C2×C3×C7—171no—
120C2×C3×C7—171no—
121C2×C3×C7—341no—
122D21—171no—
123D21C3×C5×D1711yescharacteristic, direct product factor
124C3×D7—31no—
125C3×D7—511no—
126C2×C3×D3—1191noHall
127C5×C7C3×D3×D1711yescharacteristic, complemented, Hall
128C2×C17—211no—
129D17—211no—
130D17C3×C5×D2111yescharacteristic, direct product factor
131C2×C3×C5—171no—
132C2×C3×C5—171no—
133C2×C3×C5—211no—
134C2×C3×C5—341no—
135C2×C3×C5—3571no—
136C5×D3—71no—
137C5×D3—1191no—
138C2×D7—511noHall
139C3×C7—21no—
140C3×C7C5×D3×D1711yescharacteristic, complemented
141C3×C7C2×C3×C5×D1711yescharacteristic, complemented
142C22×C5—3571noHall
143C2×C32—171no—
144C3×D3—71no—
145C3×D3—1191no—
146C17C2×C3×C5×D2111yescharacteristic, complemented, Sylow
147C3×C5—21no—
148C3×C5C3×D7×D1711yescharacteristic, complemented
149C3×C5D17×D2111yescharacteristic, direct product factor
150C2×C7—171no—
151D7—31no—
152D7—511no—
153C22×C3—3571no—
154C2×D3—1191no—
155C2×C5—171no—
156C2×C5—211no—
157C2×C5—3571no—
158C32C5×D7×D1711yescharacteristic, complemented, Sylow
159C7C3×C5×D3×D1711yescharacteristic, complemented, Sylow
160C2×C3—171no—
161C2×C3—171no—
162C2×C3—211no—
163C2×C3—341no—
164C2×C3—3571no—
165D3—71no—
166D3—1191no—
167C5C3×D17×D2111yescharacteristic, direct product factor, Sylow
168C22—3571noSylow
169C3—21no—
170C3C3×C5×D7×D1711yescharacteristic, complemented
171C3C5×D17×D2111yescharacteristic, direct product factor
172C2—171no—
173C2—211no—
174C2—3571no—
subgroupG/H
center, hypercenterC3×C5D17×D21
commutator, cosocle, hypocenterC3×C7×C17C22×C3×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C17C22
LCSG ▷ C3×C7×C17
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C32×C5×C7×D17C2semidirect
2. C3×C5×C17×D21C2semidirect
3. C3×C5×D357C2semidirect
4. C5×D17×D21C3direct (N × H)
6. C32×C5×C7×C17C22semidirect
7. C3×D17×D21C5direct (N × H)
8. C5×D357C2×C3semidirect
9. C5×C17×D21C2×C3semidirect
13. C3×C5×C7×D17D3semidirect
14. C3×C5×C7×D17C2×C3semidirect
17. C3×D357C2×C5semidirect
18. C3×C17×D21C2×C5semidirect
19. C32×C7×D17C2×C5semidirect
21. C3×C5×C7×C17C2×D3semidirect
22. C3×C5×C7×C17C22×C3semidirect
25. C32×C5×D17D7semidirect
26. D17×D21C3×C5direct (N × H)
31. C5×C7×D17C3×D3semidirect
32. C32×C7×C17C22×C5semidirect
35. C32×C5×C17C2×D7semidirect
36. D357C2×C3×C5semidirect
37. C17×D21C2×C3×C5semidirect
41. C3×C7×D17C5×D3semidirect
42. C3×C7×D17C2×C3×C5semidirect
45. C3×C5×D21D17direct (N × H)
47. C5×C7×C17C2×C3×D3semidirect
53. C3×C5×D17C3×D7semidirect
54. C3×C5×D17D21direct (N × H)
59. C3×C7×C17C2×C5×D3semidirect
60. C3×C7×C17C22×C3×C5semidirect
62. C32×C5×C7C2×D17semidirect
65. C32×D17C5×D7semidirect
67. C3×C5×C17C2×C3×D7semidirect
68. C3×C5×C17C2×D21semidirect
71. C7×D17C3×C5×D3semidirect
77. C5×D21C3×D17direct (N × H)
85. C5×D17C3×D21direct (N × H)
86. C32×C17C2×C5×D7semidirect
90. C3×D21C5×D17direct (N × H)
91. C7×C17C2×C3×C5×D3semidirect
93. C3×C5×C7D3×D17semidirect
94. C3×C5×C7C2×C3×D17semidirect
99. C3×D17C3×C5×D7semidirect
100. C3×D17C5×D21direct (N × H)
105. C5×C17C2×C3×D21semidirect
112. C32×C7C2×C5×D17semidirect
116. C3×C17C2×C3×C5×D7semidirect
117. C3×C17C2×C5×D21semidirect
118. C32×C5D7×D17semidirect
123. D21C3×C5×D17direct (N × H)
127. C5×C7C3×D3×D17semidirect
130. D17C3×C5×D21direct (N × H)
140. C3×C7C5×D3×D17semidirect
141. C3×C7C2×C3×C5×D17semidirect
146. C17C2×C3×C5×D21semidirect
148. C3×C5C3×D7×D17semidirect
149. C3×C5D17×D21direct (N × H)
158. C32C5×D7×D17semidirect
159. C7C3×C5×D3×D17semidirect
167. C5C3×D17×D21direct (N × H)
170. C3C3×C5×D7×D17semidirect
171. C3C5×D17×D21direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C5×C7×D172
C3×C5×C17×D212
C3×C5×D3572
C5×D17×D213
C3×C5×D7×D173
C3×D17×D215
C3×C5×D3×D177
C2×C3×C5×D2117