Finite Group Databasebeta

← C3×D7×C17×D21 #44472941

Subgroups — C3×D7×C17×D21 #44472941

Order 29988.

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

normal subgroups only

This group has 230 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×C7×D7×C17C211yesmaximal, characteristic, complemented
2C3×C7×C17×D21C211yesmaximal, characteristic, complemented
3C3×C17×C3⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
4D7×C17×D21C311yesmaximal, characteristic, direct product factor
5C3×D72×C17—31nomaximal
6C32×C72×C17C2211yescharacteristic, complemented, Hall
7C17×C3⋊(C7⋊D7)C2×C311yescharacteristic, complemented
8C7×C17×D21C2×C311yescharacteristic, complemented
9C3×C17×C7⋊D7—31no—
10C3×C7×D7×C17—21no—
11C3×C7×D7×C17—31no—
12C3×C7×D7×C17D311yescharacteristic, complemented
13C3×C7×D7×C17C2×C311yescharacteristic, complemented
14C2×C3×C17×D21—71nomaximal
15C3×D3×D7×C17—71nomaximal
16D72×C17—31noHall
17C3×C72×C17—21no—
18C3×C72×C17C2×D311yescharacteristic, complemented
19C3×C72×C17C22×C311yescharacteristic, complemented
20C3×C17×D21—71no—
21C3×C17×D21—71no—
22C3×C17×D21—143no—
23C3×C17×D21D711yescharacteristic, direct product factor
24C3×D3×C7×C17—71no—
25C32×D7×C17D711yescharacteristic, complemented
26C2×C32×C7×C17—71no—
27C3×D7×D21C1711yesmaximal, characteristic, direct product factor, Hall
28C17×C7⋊D7—31no—
29C7×D7×C17—31no—
30C7×D7×C17C3×D311yescharacteristic, complemented
31C2×C17×D21—71no—
32C2×C3×D7×C17—211no—
33C2×C3×D7×C17—211no—
34D3×D7×C17—71no—
35C32×C7×C17—23no—
36C32×C7×C17C2×D711yescharacteristic, complemented
37C32×C7×C17C2×D711yescharacteristic, complemented
38C3×C3⋊(C7⋊D7)C2×C1711yescharacteristic, complemented
39C3×C7×D21C2×C1711yescharacteristic, complemented
40C32×C7×D7C2×C1711yescharacteristic, complemented
41C72×C17C2×C3×D311yescharacteristic, complemented, Hall
42C2×C3×C7×C17—71no—
43C2×C3×C7×C17—71no—
44C2×C3×C7×C17—141no—
45C2×C3×C7×C17—211no—
46C17×D21—71no—
47C17×D21—71no—
48C17×D21—143no—
49C17×D21C3×D711yescharacteristic, direct product factor
50D3×C7×C17—71no—
51C3×D7×C17—21no—
52C3×D7×C17—31no—
53C3×D7×C17—211no—
54C3×D7×C17—211no—
55C3×D7×C17—423no—
56C3×D7×C17C3×D711yescharacteristic, complemented
57C3×D7×C17D2111yescharacteristic, direct product factor
58C2×C3×D3×C17—491noHall
59D7×D21C3×C1711yescharacteristic, direct product factor
60C3×D72—31no—
61C2×D7×C17—211no—
62C2×D7×C17—211no—
63C32×C72C22×C1711yescharacteristic, complemented, Hall
64C3×C7×C17—21no—
65C3×C7×C17—21no—
66C3×C7×C17—23no—
67C3×C7×C17—23no—
68C3×C7×C17—43no—
69C3×C7×C17D3×D711yescharacteristic, complemented
70C3×C7×C17C2×C3×D711yescharacteristic, complemented
71C3×C7×C17C2×C3×D711yescharacteristic, complemented
72C3×C7×C17C2×D2111yescharacteristic, complemented
73C2×C32×C17—71no—
74C3×D3×C17—71no—
75C3×D3×C17—491no—
76C3⋊(C7⋊D7)C2×C3×C1711yescharacteristic, complemented
77C7×D21C2×C3×C1711yescharacteristic, complemented
78C3×C7⋊D7—31no—
79C3×C7×D7—21no—
80C3×C7×D7—31no—
81C3×C7×D7D3×C1711yescharacteristic, complemented
82C3×C7×D7C2×C3×C1711yescharacteristic, complemented
83C2×C3×D21—71no—
84C3×D3×D7—71no—
85C2×C7×C17—71no—
86C2×C7×C17—211no—
87D7×C17—31no—
88D7×C17—211no—
89D7×C17—211no—
90D7×C17—423no—
91D7×C17C3×D2111yescharacteristic, direct product factor
92C22×C3×C17—1471no—
93C2×D3×C17—491no—
94D72—31noHall
95C32×C17D7211yescharacteristic, complemented, Hall
96C3×C72—21no—
97C3×C72C2×D3×C1711yescharacteristic, complemented
98C3×C72C22×C3×C1711yescharacteristic, complemented
99C2×C32×C7—71no—
100C3×D21—71no—
101C3×D21—71no—
102C3×D21—143no—
103C3×D21D7×C1711yescharacteristic, direct product factor
104C3×D3×C7—71no—
105C32×D7D7×C1711yescharacteristic, complemented
106C7×C17—23no—
107C7×C17C3×D3×D711yescharacteristic, complemented
108C7×C17C2×C3×D2111yescharacteristic, complemented
109C2×C3×C17—71no—
110C2×C3×C17—71no—
111C2×C3×C17—141no—
112C2×C3×C17—211no—
113C2×C3×C17—1471no—
114D3×C17—71no—
115D3×C17—491no—
116C7⋊D7—31no—
117C7×D7—31no—
118C7×D7C3×D3×C1711yescharacteristic, complemented
119C2×D21—71no—
120C2×C3×D7—211no—
121C2×C3×D7—211no—
122D3×D7—71no—
123C22×C17—1471noHall
124C32×C7—23no—
125C32×C7C2×D7×C1711yescharacteristic, complemented
126C32×C7C2×D7×C1711yescharacteristic, complemented
127C3×C17—21no—
128C3×C17C3×D7211yescharacteristic, complemented
129C3×C17D7×D2111yescharacteristic, direct product factor
130C72C2×C3×D3×C1711yescharacteristic, complemented, Sylow
131C2×C3×C7—71no—
132C2×C3×C7—71no—
133C2×C3×C7—141no—
134C2×C3×C7—211no—
135D21—71no—
136D21—71no—
137D21—143no—
138D21C3×D7×C1711yescharacteristic, direct product factor
139C3×D7—21no—
140C3×D7—31no—
141C3×D7—211no—
142C3×D7—211no—
143C3×D7—423no—
144C3×D7C3×D7×C1711yescharacteristic, complemented
145C3×D7C17×D2111yescharacteristic, direct product factor
146D3×C7—71no—
147C2×C3×D3—491noHall
148C2×C17—71no—
149C2×C17—211no—
150C2×C17—1471no—
151C2×D7—211no—
152C2×D7—211no—
153C3×C7—21no—
154C3×C7—21no—
155C3×C7—23no—
156C3×C7—23no—
157C3×C7—43no—
158C3×C7D3×D7×C1711yescharacteristic, complemented
159C3×C7C2×C3×D7×C1711yescharacteristic, complemented
160C3×C7C2×C3×D7×C1711yescharacteristic, complemented
161C3×C7C2×C17×D2111yescharacteristic, complemented
162C2×C32—71no—
163C3×D3—71no—
164C3×D3—491no—
165C17C3×D7×D2111yescharacteristic, direct product factor, Sylow
166C2×C7—71no—
167C2×C7—211no—
168D7—31no—
169D7—211no—
170D7—211no—
171D7—423no—
172D7C3×C17×D2111yescharacteristic, direct product factor
173C22×C3—1471no—
174C2×D3—491no—
175C32D72×C1711yescharacteristic, complemented, Sylow
176C7—23no—
177C7C3×D3×D7×C1711yescharacteristic, complemented
178C7C2×C3×C17×D2111yescharacteristic, complemented
179C2×C3—71no—
180C2×C3—71no—
181C2×C3—141no—
182C2×C3—211no—
183C2×C3—1471no—
184D3—71no—
185D3—491no—
186C22—1471noSylow
187C3—21no—
188C3C3×D72×C1711yescharacteristic, complemented
189C3D7×C17×D2111yescharacteristic, direct product factor
190C2—71no—
191C2—211no—
192C2—1471no—
subgroupG/H
center, hypercenterC3×C17D7×D21
commutator, cosocle, hypocenterC3×C72C22×C3×C17
FrattiniC1G
Fitting, socleC32×C72×C17C22
LCSG ▷ C3×C72
UCSC1 ◁ C3×C17

Complements

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

NH ≅ G/Ntype
1. C32×C7×D7×C17C2semidirect
2. C3×C7×C17×D21C2semidirect
3. C3×C17×C3⋊(C7⋊D7)C2semidirect
4. D7×C17×D21C3direct (N × H)
6. C32×C72×C17C22semidirect
7. C17×C3⋊(C7⋊D7)C2×C3semidirect
8. C7×C17×D21C2×C3semidirect
12. C3×C7×D7×C17D3semidirect
13. C3×C7×D7×C17C2×C3semidirect
18. C3×C72×C17C2×D3semidirect
19. C3×C72×C17C22×C3semidirect
23. C3×C17×D21D7direct (N × H)
25. C32×D7×C17D7semidirect
27. C3×D7×D21C17direct (N × H)
30. C7×D7×C17C3×D3semidirect
36. C32×C7×C17C2×D7semidirect
37. C32×C7×C17C2×D7semidirect
38. C3×C3⋊(C7⋊D7)C2×C17semidirect
39. C3×C7×D21C2×C17semidirect
40. C32×C7×D7C2×C17semidirect
41. C72×C17C2×C3×D3semidirect
49. C17×D21C3×D7direct (N × H)
56. C3×D7×C17C3×D7semidirect
57. C3×D7×C17D21direct (N × H)
59. D7×D21C3×C17direct (N × H)
63. C32×C72C22×C17semidirect
69. C3×C7×C17D3×D7semidirect
70. C3×C7×C17C2×C3×D7semidirect
71. C3×C7×C17C2×C3×D7semidirect
72. C3×C7×C17C2×D21semidirect
76. C3⋊(C7⋊D7)C2×C3×C17semidirect
77. C7×D21C2×C3×C17semidirect
81. C3×C7×D7D3×C17semidirect
82. C3×C7×D7C2×C3×C17semidirect
91. D7×C17C3×D21direct (N × H)
95. C32×C17D72semidirect
97. C3×C72C2×D3×C17semidirect
98. C3×C72C22×C3×C17semidirect
103. C3×D21D7×C17direct (N × H)
105. C32×D7D7×C17semidirect
107. C7×C17C3×D3×D7semidirect
108. C7×C17C2×C3×D21semidirect
118. C7×D7C3×D3×C17semidirect
125. C32×C7C2×D7×C17semidirect
126. C32×C7C2×D7×C17semidirect
128. C3×C17C3×D72semidirect
129. C3×C17D7×D21direct (N × H)
130. C72C2×C3×D3×C17semidirect
138. D21C3×D7×C17direct (N × H)
144. C3×D7C3×D7×C17semidirect
145. C3×D7C17×D21direct (N × H)
158. C3×C7D3×D7×C17semidirect
159. C3×C7C2×C3×D7×C17semidirect
160. C3×C7C2×C3×D7×C17semidirect
161. C3×C7C2×C17×D21semidirect
165. C17C3×D7×D21direct (N × H)
172. D7C3×C17×D21direct (N × H)
175. C32D72×C17semidirect
177. C7C3×D3×D7×C17semidirect
178. C7C2×C3×C17×D21semidirect
188. C3C3×D72×C17semidirect
189. C3D7×C17×D21direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C7×D7×C172
C3×C7×C17×D212
C3×C17×C3⋊(C7⋊D7)2
D7×C17×D213
C3×D72×C173
C2×C3×C17×D217
C3×D3×D7×C177
C3×D7×D2117