Finite Group Databasebeta

← C3×C7×#6495 #72269339

Subgroups — C3×C7×#6495 #72269339

Order 4704.

434 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 116 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×C7×#1141C212yesmaximal, complemented
2C3×C7×#1142C212yesmaximal, complemented
3C3×C7×#1157C211yesmaximal, characteristic, complemented
4C3×C7×#1136C211yesmaximal, characteristic, complemented
5C2×C3×C72×D4C211yesmaximal, characteristic, complemented
6C7×#6495C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×C72—21no—
8C3×C72×D4—21no—
9C3×C72×D4C2212yes—
10C2×C3×C4×C72C2211yescharacteristic
11C3×C7×C22⋊D7—21no—
12C3×C7×D28C2211yescharacteristic, complemented
13C3×C4×C7×D7—21no—
14C3×C7×Dic14C2211yescharacteristic, complemented
15C3×C7×C7⋊C8C2212yescomplemented
16C2×C72×D4C2×C311yescharacteristic, complemented
17C7×#1157C2×C311yescharacteristic, complemented
18C7×#1142C2×C312yescomplemented
19C7×#1141C2×C312yescomplemented
20C7×#1136C2×C311yescharacteristic, complemented
21C3×C7×AGL(1,ℤ/8)—71nomaximal
22C3×#6495C711yesmaximal, characteristic, direct product factor
23C22×C3×C72—22no—
24C22×C3×C72—42no—
25C22×C3×C72D411yescharacteristic
26C2×C3×C7×D7—21no—
27C3×C4×C72D411yescharacteristic
28C3×C4×C72C2311yescharacteristic
29C3×C7×Dic7—21no—
30C23×C72—21no—
31C72×D4—21no—
32C72×D4C22×C312yes—
33C2×C4×C72C22×C311yescharacteristic
34C7×C22⋊D7—21no—
35C7×D28C22×C311yescharacteristic, complemented
36C4×C7×D7—21no—
37C7×Dic14C22×C311yescharacteristic, complemented
38C7×C7⋊C8C22×C312yescomplemented
39C3×C7×#41—71no—
40C2×C3×C7×D4—23no—
41C2×C3×C7×D4D711yescharacteristic, complemented
42C2×C3×C7×D4C2×C711yescharacteristic, complemented
43C3×#1157C2×C711yescharacteristic, complemented
44C3×C7×SD16—72no—
45C3×C7×D8—72no—
46C3×C7×M16—71no—
47C3×#1142C2×C712yescomplemented
48C3×#1141C2×C712yescomplemented
49C3×#1136C2×C711yescharacteristic, complemented
50C2×C3×C72—21no—
51C2×C3×C72—42no—
52C2×C3×C72C2×D411yescharacteristic
53C3×C7×D7—41no—
54C7×AGL(1,ℤ/8)—71no—
55#6495C3×C711yescharacteristic, direct product factor
56C22×C72—22no—
57C22×C72—42no—
58C22×C72C3×D411yescharacteristic
59C2×C7×D7—21no—
60C4×C72C3×D411yescharacteristic
61C4×C72C23×C311yescharacteristic
62C7×Dic7—21no—
63C23×C3×C7—26no—
64C23×C3×C7—21no—
65C23×C3×C7—21no—
66C3×C7×Q8—71no—
67C3×C7×D4—26no—
68C3×C7×D4—21no—
69C3×C7×D4—21no—
70C3×C7×D4—26no—
71C3×C7×D4—71no—
72C3×C7×D4—141no—
73C3×C7×D4C2×D712yes—
74C3×C7×D4C22×C712yes—
75C2×C3×C4×C7—23no—
76C2×C3×C4×C7—141no—
77C2×C3×C4×C7C2×D711yescharacteristic
78C2×C3×C4×C7C22×C711yescharacteristic
79C3×C22⋊D7—21no—
80C3×D28C22×C711yescharacteristic, complemented
81C3×C4×D7—21no—
82C3×Dic14C22×C711yescharacteristic, complemented
83C3×C7×C8—72no—
84C3×C7⋊C8C22×C712yescomplemented
85C3×C72AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
86C7×#41—71no—
87C2×C7×D4—23no—
88C2×C7×D4C3×D711yescharacteristic, complemented
89C2×C7×D4C2×C3×C711yescharacteristic, complemented
90#1157C2×C3×C711yescharacteristic, complemented
91C7×SD16—72no—
92C7×D8—72no—
93C7×M16—71no—
94#1142C2×C3×C712yescomplemented
95#1141C2×C3×C712yescomplemented
96#1136C2×C3×C711yescharacteristic, complemented
97C2×C72—21no—
98C2×C72—42no—
99C2×C72C2×C3×D411yescharacteristic
100C7×D7—41no—
101C3×AGL(1,ℤ/8)—71noHall
102C22×C3×C7—212no—
103C22×C3×C7—22no—
104C22×C3×C7—22no—
105C22×C3×C7—23no—
106C22×C3×C7—412no—
107C22×C3×C7—42no—
108C22×C3×C7—42no—
109C22×C3×C7—141no—
110C22×C3×C7C22⋊D711yescharacteristic
111C22×C3×C7C7×D411yescharacteristic
112C2×C3×D7—21no—
113C3×C4×C7—23no—
114C3×C4×C7—23no—
115C3×C4×C7—141no—
116C3×C4×C7C22⋊D711yescharacteristic
117C3×C4×C7C7×D411yescharacteristic
118C3×C4×C7C22×D711yescharacteristic
119C3×C4×C7C23×C711yescharacteristic
120C3×Dic7—21no—
121C23×C7—26no—
122C23×C7—21no—
123C23×C7—21no—
124C7×Q8—71no—
125C7×D4—26no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—26no—
129C7×D4—71no—
130C7×D4—141no—
131C7×D4C2×C3×D712yes—
132C7×D4C22×C3×C712yes—
133C2×C4×C7—23no—
134C2×C4×C7—141no—
135C2×C4×C7C2×C3×D711yescharacteristic
136C2×C4×C7C22×C3×C711yescharacteristic
137C22⋊D7—21no—
138D28C22×C3×C711yescharacteristic, complemented
139C4×D7—21no—
140Dic14C22×C3×C711yescharacteristic, complemented
141C7×C8—72no—
142C7⋊C8C22×C3×C712yescomplemented
143C72C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
144C3×#41—71no—
145C2×C3×D4C7×D711yescharacteristic, complemented
146C3×SD16—72no—
147C3×D8—72no—
148C3×M16—71no—
149C2×C3×C7—26no—
150C2×C3×C7—21no—
151C2×C3×C7—21no—
152C2×C3×C7—23no—
153C2×C3×C7—412no—
154C2×C3×C7—42no—
155C2×C3×C7—42no—
156C2×C3×C7—281no—
157C2×C3×C7C2×C22⋊D711yescharacteristic
158C2×C3×C7C2×C7×D411yescharacteristic
159C3×D7—41no—
160AGL(1,ℤ/8)—71noSylow
161C22×C7—212no—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—23no—
165C22×C7—412no—
166C22×C7—42no—
167C22×C7—42no—
168C22×C7—141no—
169C22×C7C3×C22⋊D711yescharacteristic
170C22×C7C3×C7×D411yescharacteristic
171C2×D7—21no—
172C4×C7—23no—
173C4×C7—23no—
174C4×C7—141no—
175C4×C7C3×C22⋊D711yescharacteristic
176C4×C7C3×C7×D411yescharacteristic
177C4×C7C22×C3×D711yescharacteristic
178C4×C7C23×C3×C711yescharacteristic
179Dic7—21no—
180C23×C3—21no—
181C3×Q8—71no—
182C3×D4—21no—
183C3×D4—71no—
184C3×D4—141no—
185C3×D4C2×C7×D712yes—
186C2×C3×C4—141no—
187C2×C3×C4C2×C7×D711yescharacteristic
188C3×C8—72no—
189C3×C7—23no—
190C3×C7#649511yescharacteristic, direct product factor
191C3×C7C7×AGL(1,ℤ/8)11yescharacteristic, complemented
192#41—71no—
193C2×D4C3×C7×D711yescharacteristic, complemented
194SD16—72no—
195D8—72no—
196M16—71no—
197C2×C7—26no—
198C2×C7—21no—
199C2×C7—21no—
200C2×C7—23no—
201C2×C7—412no—
202C2×C7—42no—
203C2×C7—42no—
204C2×C7—281no—
205C2×C7C2×C3×C22⋊D711yescharacteristic
206C2×C7C2×C3×C7×D411yescharacteristic
207D7—41no—
208C22×C3—22no—
209C22×C3—42no—
210C22×C3—141no—
211C22×C3C7×C22⋊D711yescharacteristic
212C3×C4—141no—
213C3×C4C7×C22⋊D711yescharacteristic
214C3×C4C22×C7×D711yescharacteristic
215C23—21no—
216Q8—71no—
217D4—21no—
218D4—71no—
219D4—141no—
220D4C2×C3×C7×D712yes—
221C2×C4—141no—
222C2×C4C2×C3×C7×D711yescharacteristic
223C8—72no—
224C7—23no—
225C7C3×#649511yescharacteristic, direct product factor
226C7C3×C7×AGL(1,ℤ/8)11yescharacteristic, complemented
227C2×C3—21no—
228C2×C3—42no—
229C2×C3—281no—
230C2×C3C2×C7×C22⋊D711yescharacteristic
231C22—22no—
232C22—42no—
233C22—141no—
234C22C3×C7×C22⋊D711yescharacteristic
235C4—141no—
236C4C3×C7×C22⋊D711yescharacteristic
237C4C22×C3×C7×D711yescharacteristic
238C3C7×#649511yescharacteristic, direct product factor, Sylow
239C2—21no—
240C2—42no—
241C2—281no—
242C2C2×C3×C7×C22⋊D711yescharacteristic
subgroupG/H
centerC2×C3×C7C2×C22⋊D7
commutator, cosocleC4×C7C23×C3×C7
FrattiniC4C22×C3×C7×D7
FittingC2×C3×C72×D4C2
socleC2×C3×C72C2×D4
hypercenterC2×C3×C7×D4D7
hypocenterC7C3×C7×AGL(1,ℤ/8)
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3×C7 ◁ C2×C3×C4×C7 ◁ C2×C3×C7×D4

Complements

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

NH ≅ G/Ntype
1. C3×C7×#1141C2semidirect
2. C3×C7×#1142C2semidirect
3. C3×C7×#1157C2semidirect
4. C3×C7×#1136C2semidirect
5. C2×C3×C72×D4C2semidirect
6. C7×#6495C3direct (N × H)
12. C3×C7×D28C22semidirect
14. C3×C7×Dic14C22semidirect
15. C3×C7×C7⋊C8C22semidirect
16. C2×C72×D4C2×C3semidirect
17. C7×#1157C2×C3semidirect
18. C7×#1142C2×C3semidirect
19. C7×#1141C2×C3semidirect
20. C7×#1136C2×C3semidirect
22. C3×#6495C7direct (N × H)
35. C7×D28C22×C3semidirect
37. C7×Dic14C22×C3semidirect
38. C7×C7⋊C8C22×C3semidirect
41. C2×C3×C7×D4D7semidirect
42. C2×C3×C7×D4C2×C7semidirect
43. C3×#1157C2×C7semidirect
47. C3×#1142C2×C7semidirect
48. C3×#1141C2×C7semidirect
49. C3×#1136C2×C7semidirect
55. #6495C3×C7direct (N × H)
80. C3×D28C22×C7semidirect
82. C3×Dic14C22×C7semidirect
84. C3×C7⋊C8C22×C7semidirect
85. C3×C72AGL(1,ℤ/8)semidirect
88. C2×C7×D4C3×D7semidirect
89. C2×C7×D4C2×C3×C7semidirect
90. #1157C2×C3×C7semidirect
94. #1142C2×C3×C7semidirect
95. #1141C2×C3×C7semidirect
96. #1136C2×C3×C7semidirect
138. D28C22×C3×C7semidirect
140. Dic14C22×C3×C7semidirect
142. C7⋊C8C22×C3×C7semidirect
143. C72C3×AGL(1,ℤ/8)semidirect
145. C2×C3×D4C7×D7semidirect
190. C3×C7#6495direct (N × H)
191. C3×C7C7×AGL(1,ℤ/8)semidirect
193. C2×D4C3×C7×D7semidirect
225. C7C3×#6495direct (N × H)
226. C7C3×C7×AGL(1,ℤ/8)semidirect
238. C3C7×#6495direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C7×#11412
C3×C7×#11422
C3×C7×#11572
C3×C7×#11362
C2×C3×C72×D42
C7×#64953
C3×C7×AGL(1,ℤ/8)7
C3×#64957