Finite Group Databasebeta

← D3×C7×C17×D21 #44472957

Subgroups — D3×C7×C17×D21 #44472957

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
1C3×D3×C72×C17C211yesmaximal, characteristic, complemented
2C3×C7×C17×D21C211yesmaximal, characteristic, complemented
3C7×C17×C3⋊D21C211yesmaximal, characteristic, complemented
4C2×C7×C17×D21—31nomaximal
5D3×C7×D7×C17—31nomaximal
6C32×C72×C17C2211yescharacteristic, complemented, Hall
7C7×C17×D21—31no—
8C7×C17×D21—31no—
9C7×C17×D21—61no—
10C7×C17×D21D311yescharacteristic, direct product factor
11D3×C72×C17D311yescharacteristic, complemented
12C3×C7×D7×C17—31no—
13C2×C3×C72×C17—31no—
14D3×C17×D21C711yesmaximal, characteristic, direct product factor
15D32×C7×C17—71nomaximal
16C2×C7×D7×C17—91noHall
17C3×C72×C17—21no—
18C3×C72×C17C2×D311yescharacteristic, complemented
19C3×C72×C17C2×D311yescharacteristic, complemented
20C17×C3⋊D21C2×C711yescharacteristic, complemented
21C7×C17×C3⋊D3—71no—
22C3×C17×D21C2×C711yescharacteristic, complemented
23C3×D3×C7×C17—23no—
24C3×D3×C7×C17—71no—
25C3×D3×C7×C17D711yescharacteristic, complemented
26C3×D3×C7×C17C2×C711yescharacteristic, complemented
27D3×C7×D21C1711yesmaximal, characteristic, direct product factor, Hall
28C2×C72×C17—31no—
29C7×D7×C17—31no—
30C7×D7×C17—91no—
31C2×C17×D21—31no—
32C2×D3×C7×C17—211no—
33C2×D3×C7×C17—211no—
34D3×D7×C17—31no—
35C32×C7×C17—23no—
36C32×C7×C17C2×D711yescharacteristic, complemented
37C32×C7×C17C22×C711yescharacteristic, complemented
38C7×C3⋊D21C2×C1711yescharacteristic, complemented
39C3×C7×D21C2×C1711yescharacteristic, complemented
40C3×D3×C72C2×C1711yescharacteristic, complemented
41C72×C17D3211yescharacteristic, complemented, Hall
42C2×C3×C7×C17—31no—
43C2×C3×C7×C17—31no—
44C2×C3×C7×C17—63no—
45C2×C3×C7×C17—211no—
46C17×D21—31no—
47C17×D21—31no—
48C17×D21—61no—
49C17×D21D3×C711yescharacteristic, direct product factor
50D3×C7×C17—23no—
51D3×C7×C17—71no—
52D3×C7×C17—211no—
53D3×C7×C17—211no—
54D3×C7×C17—421no—
55D3×C7×C17D3×C711yescharacteristic, complemented
56D3×C7×C17D2111yescharacteristic, direct product factor
57C3×D7×C17—31no—
58D32×C17—71noHall
59C2×C7×D21—31no—
60D3×C7×D7—31no—
61C22×C7×C17—631no—
62C2×D7×C17—91no—
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×D3×C711yescharacteristic, complemented
71C3×C7×C17C2×D3×C711yescharacteristic, complemented
72C3×C7×C17C2×D2111yescharacteristic, complemented
73C17×C3⋊D3—71no—
74C3×D3×C17—71no—
75C3×D3×C17C7×D711yescharacteristic, complemented
76C2×C3×C72—31no—
77C7×D21—31no—
78C7×D21—31no—
79C7×D21—61no—
80C7×D21D3×C1711yescharacteristic, direct product factor
81D3×C72D3×C1711yescharacteristic, complemented
82C3×C7×D7—31no—
83D3×D21C7×C1711yescharacteristic, direct product factor
84D32×C7—71no—
85C2×C7×C17—31no—
86C2×C7×C17—31no—
87C2×C7×C17—63no—
88C2×C7×C17—211no—
89C2×C7×C17—631no—
90D7×C17—31no—
91D7×C17—91no—
92C2×D3×C17—211no—
93C2×D3×C17—211no—
94C2×C7×D7—91noHall
95C32×C17C2×C7×D711yescharacteristic, complemented, Hall
96C3×C72—21no—
97C3×C72C2×D3×C1711yescharacteristic, complemented
98C3×C72C2×D3×C1711yescharacteristic, complemented
99C3⋊D21C2×C7×C1711yescharacteristic, complemented
100C7×C3⋊D3—71no—
101C3×D21C2×C7×C1711yescharacteristic, complemented
102C3×D3×C7—23no—
103C3×D3×C7—71no—
104C3×D3×C7D7×C1711yescharacteristic, complemented
105C3×D3×C7C2×C7×C1711yescharacteristic, complemented
106C7×C17—23no—
107C7×C17D32×C711yescharacteristic, complemented
108C7×C17D3×D2111yescharacteristic, direct product factor
109C2×C3×C17—31no—
110C2×C3×C17—211no—
111D3×C17—71no—
112D3×C17—211no—
113D3×C17—211no—
114D3×C17—421no—
115D3×C17C7×D2111yescharacteristic, direct product factor
116C2×C72—31no—
117C7×D7—31no—
118C7×D7—91no—
119C2×D21—31no—
120C2×D3×C7—211no—
121C2×D3×C7—211no—
122D3×D7—31no—
123C22×C17—631noHall
124C32×C7—23no—
125C32×C7C2×D7×C1711yescharacteristic, complemented
126C32×C7C22×C7×C1711yescharacteristic, complemented
127C3×C17—21no—
128C3×C17D3×C7×D711yescharacteristic, complemented
129C3×C17C2×C7×D2111yescharacteristic, complemented
130C72D32×C1711yescharacteristic, complemented, Sylow
131C2×C3×C7—31no—
132C2×C3×C7—31no—
133C2×C3×C7—63no—
134C2×C3×C7—211no—
135D21—31no—
136D21—31no—
137D21—61no—
138D21D3×C7×C1711yescharacteristic, direct product factor
139C3×D7—31no—
140D3×C7—23no—
141D3×C7—71no—
142D3×C7—211no—
143D3×C7—211no—
144D3×C7—421no—
145D3×C7D3×C7×C1711yescharacteristic, complemented
146D3×C7C17×D2111yescharacteristic, direct product factor
147D32—71noHall
148C2×C17—31no—
149C2×C17—211no—
150C2×C17—631no—
151C22×C7—631no—
152C2×D7—91no—
153C3×C7—21no—
154C3×C7—21no—
155C3×C7—23no—
156C3×C7—23no—
157C3×C7—43no—
158C3×C7D3×D7×C1711yescharacteristic, complemented
159C3×C7C2×D3×C7×C1711yescharacteristic, complemented
160C3×C7C2×D3×C7×C1711yescharacteristic, complemented
161C3×C7C2×C17×D2111yescharacteristic, complemented
162C3⋊D3—71no—
163C3×D3—71no—
164C3×D3C7×D7×C1711yescharacteristic, complemented
165C17D3×C7×D2111yescharacteristic, direct product factor, Sylow
166C2×C7—31no—
167C2×C7—31no—
168C2×C7—63no—
169C2×C7—211no—
170C2×C7—631no—
171D7—31no—
172D7—91no—
173C2×D3—211no—
174C2×D3—211no—
175C32C2×C7×D7×C1711yescharacteristic, complemented, Sylow
176C7—23no—
177C7D32×C7×C1711yescharacteristic, complemented
178C7D3×C17×D2111yescharacteristic, direct product factor
179C2×C3—31no—
180C2×C3—211no—
181D3—71no—
182D3—211no—
183D3—211no—
184D3—421no—
185D3C7×C17×D2111yescharacteristic, direct product factor
186C22—631noSylow
187C3—21no—
188C3D3×C7×D7×C1711yescharacteristic, complemented
189C3C2×C7×C17×D2111yescharacteristic, complemented
190C2—31no—
191C2—211no—
192C2—631no—
subgroupG/H
center, hypercenterC7×C17D3×D21
commutator, cosocle, hypocenterC32×C7C22×C7×C17
FrattiniC1G
Fitting, socleC32×C72×C17C22
LCSG ▷ C32×C7
UCSC1 ◁ C7×C17

Complements

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

NH ≅ G/Ntype
1. C3×D3×C72×C17C2semidirect
2. C3×C7×C17×D21C2semidirect
3. C7×C17×C3⋊D21C2semidirect
6. C32×C72×C17C22semidirect
10. C7×C17×D21D3direct (N × H)
11. D3×C72×C17D3semidirect
14. D3×C17×D21C7direct (N × H)
18. C3×C72×C17C2×D3semidirect
19. C3×C72×C17C2×D3semidirect
20. C17×C3⋊D21C2×C7semidirect
22. C3×C17×D21C2×C7semidirect
25. C3×D3×C7×C17D7semidirect
26. C3×D3×C7×C17C2×C7semidirect
27. D3×C7×D21C17direct (N × H)
36. C32×C7×C17C2×D7semidirect
37. C32×C7×C17C22×C7semidirect
38. C7×C3⋊D21C2×C17semidirect
39. C3×C7×D21C2×C17semidirect
40. C3×D3×C72C2×C17semidirect
41. C72×C17D32semidirect
49. C17×D21D3×C7direct (N × H)
55. D3×C7×C17D3×C7semidirect
56. D3×C7×C17D21direct (N × H)
63. C32×C72C22×C17semidirect
69. C3×C7×C17D3×D7semidirect
70. C3×C7×C17C2×D3×C7semidirect
71. C3×C7×C17C2×D3×C7semidirect
72. C3×C7×C17C2×D21semidirect
75. C3×D3×C17C7×D7semidirect
80. C7×D21D3×C17direct (N × H)
81. D3×C72D3×C17semidirect
83. D3×D21C7×C17direct (N × H)
95. C32×C17C2×C7×D7semidirect
97. C3×C72C2×D3×C17semidirect
98. C3×C72C2×D3×C17semidirect
99. C3⋊D21C2×C7×C17semidirect
101. C3×D21C2×C7×C17semidirect
104. C3×D3×C7D7×C17semidirect
105. C3×D3×C7C2×C7×C17semidirect
107. C7×C17D32×C7semidirect
108. C7×C17D3×D21direct (N × H)
115. D3×C17C7×D21direct (N × H)
125. C32×C7C2×D7×C17semidirect
126. C32×C7C22×C7×C17semidirect
128. C3×C17D3×C7×D7semidirect
129. C3×C17C2×C7×D21semidirect
130. C72D32×C17semidirect
138. D21D3×C7×C17direct (N × H)
145. D3×C7D3×C7×C17semidirect
146. D3×C7C17×D21direct (N × H)
158. C3×C7D3×D7×C17semidirect
159. C3×C7C2×D3×C7×C17semidirect
160. C3×C7C2×D3×C7×C17semidirect
161. C3×C7C2×C17×D21semidirect
164. C3×D3C7×D7×C17semidirect
165. C17D3×C7×D21direct (N × H)
175. C32C2×C7×D7×C17semidirect
177. C7D32×C7×C17semidirect
178. C7D3×C17×D21direct (N × H)
185. D3C7×C17×D21direct (N × H)
188. C3D3×C7×D7×C17semidirect
189. C3C2×C7×C17×D21semidirect

Maximal subgroups

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

subgroupindex
C3×D3×C72×C172
C3×C7×C17×D212
C7×C17×C3⋊D212
C2×C7×C17×D213
D3×C7×D7×C173
D3×C17×D217
D32×C7×C177
D3×C7×D2117