Finite Group Databasebeta

← C7×#10621415 #72269398

Subgroups — C7×#10621415 #72269398

Order 4704.

434 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C7×#66632C212yesmaximal, complemented
2C7×#66633C212yesmaximal, complemented
3C7×#66728C211yesmaximal, characteristic, complemented
4C7×#66627C211yesmaximal, characteristic, complemented
5C2×C3×C72×D4C211yesmaximal, characteristic, complemented
6C7×#6495—31nomaximal, Hall
7C23×C3×C72—21no—
8C3×C72×D4—21no—
9C3×C72×D4C2212yes—
10C2×C3×C4×C72C2211yescharacteristic
11C7×#4268—21no—
12C7×D84C2211yescharacteristic, complemented
13C4×C7×D21—21no—
14C7×Dic42C2211yescharacteristic, complemented
15C7×C3⋊(C7⋊C8)C2212yescomplemented
16C2×C72×D4D311yescharacteristic, complemented
17C7×#1157—31no—
18C7×#1142—32no—
19C7×#1141—32no—
20C7×#1136—31no—
21#10621415C711yesmaximal, characteristic, direct product factor
22C7×#932—71nomaximal
23C22×C3×C72—22no—
24C22×C3×C72—42no—
25C22×C3×C72D411yescharacteristic
26C2×C7×D21—21no—
27C3×C4×C72D411yescharacteristic
28C3×C4×C72C2311yescharacteristic
29C7×Dic21—21no—
30C23×C72—21no—
31C72×D4—21no—
32C72×D4C2×D312yes—
33C2×C4×C72C2×D311yescharacteristic
34C7×C22⋊D7—61no—
35C7×D28—31no—
36C4×C7×D7—61no—
37C7×Dic14—31no—
38C7×C7⋊C8—32no—
39C2×C3×C7×D4—23no—
40C2×C3×C7×D4D711yescharacteristic, complemented
41C2×C3×C7×D4C2×C711yescharacteristic, complemented
42#66728C2×C711yescharacteristic, complemented
43C7×#235—71no—
44#66633C2×C712yescomplemented
45#66632C2×C712yescomplemented
46#66627C2×C711yescharacteristic, complemented
47C7×#214—72no—
48C7×#213—72no—
49C7×#208—71no—
50C2×C3×C72—21no—
51C2×C3×C72—42no—
52C2×C3×C72C2×D411yescharacteristic
53C7×D21—41no—
54C7×AGL(1,ℤ/8)—211no—
55#6495—31no—
56C22×C72—22no—
57C22×C72—42no—
58C22×C72#6711yescharacteristic
59C2×C7×D7—61no—
60C4×C72#6711yescharacteristic
61C4×C72C22×D311yescharacteristic
62C7×Dic7—61no—
63C23×C3×C7—26no—
64C23×C3×C7—21no—
65C23×C3×C7—21no—
66C3×C7×D4—26no—
67C3×C7×D4—21no—
68C3×C7×D4—21no—
69C3×C7×D4—26no—
70C3×C7×D4C2×D712yes—
71C3×C7×D4C22×C712yes—
72C2×C3×C4×C7—23no—
73C2×C3×C4×C7C2×D711yescharacteristic
74C2×C3×C4×C7C22×C711yescharacteristic
75#4268—21no—
76D84C22×C711yescharacteristic, complemented
77C4×D21—21no—
78Dic42C22×C711yescharacteristic, complemented
79C7×#67—141no—
80C7×D12—71no—
81C4×D3×C7—141no—
82C7×Dic6—71no—
83C3⋊(C7⋊C8)C22×C712yescomplemented
84C7×C3⋊C8—72no—
85C3×C72AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
86C7×#41—211no—
87C2×C7×D4—23no—
88C2×C7×D4D3×C711yescharacteristic, complemented
89C2×C7×D4D2111yescharacteristic, complemented
90#1157—31no—
91C7×SD16—212no—
92C7×D8—212no—
93C7×M16—211no—
94#1142—32no—
95#1141—32no—
96#1136—31no—
97C2×C72—21no—
98C2×C72—42no—
99C2×C72C2×#6711yescharacteristic
100C7×D7—121no—
101#932—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×C7C22⋊D711yescharacteristic
110C22×C3×C7C7×D411yescharacteristic
111C2×D21—21no—
112C2×D3×C7—141no—
113C3×C4×C7—23no—
114C3×C4×C7—23no—
115C3×C4×C7C22⋊D711yescharacteristic
116C3×C4×C7C7×D411yescharacteristic
117C3×C4×C7C22×D711yescharacteristic
118C3×C4×C7C23×C711yescharacteristic
119Dic21—21no—
120C7×Dic3—141no—
121C23×C7—26no—
122C23×C7—21no—
123C23×C7—21no—
124C7×Q8—211no—
125C7×D4—26no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—26no—
129C7×D4—211no—
130C7×D4—421no—
131C7×D4C2×D3×C712yes—
132C7×D4C2×D2112yes—
133C2×C4×C7—23no—
134C2×C4×C7—421no—
135C2×C4×C7C2×D3×C711yescharacteristic
136C2×C4×C7C2×D2111yescharacteristic
137C22⋊D7—61no—
138D28—31no—
139C4×D7—61no—
140Dic14—31no—
141C7×C8—212no—
142C7⋊C8—32no—
143C72#93211yescharacteristic, complemented, Sylow
144C2×C3×D4C7×D711yescharacteristic, complemented
145#235—71no—
146#214—72no—
147#213—72no—
148#208—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×C7C2×C22⋊D711yescharacteristic
157C2×C3×C7C2×C7×D411yescharacteristic
158D21—41no—
159D3×C7—281no—
160AGL(1,ℤ/8)—211noSylow
161C22×C7—212no—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—23no—
165C22×C7—412no—
166C22×C7—42no—
167C22×C7—42no—
168C22×C7—421no—
169C22×C7C7×#6711yescharacteristic
170C22×C7#426811yescharacteristic
171C2×D7—61no—
172C4×C7—23no—
173C4×C7—23no—
174C4×C7—421no—
175C4×C7C7×#6711yescharacteristic
176C4×C7#426811yescharacteristic
177C4×C7C22×D3×C711yescharacteristic
178C4×C7C22×D2111yescharacteristic
179Dic7—61no—
180C23×C3—21no—
181C3×D4—21no—
182C3×D4C2×C7×D712yes—
183C2×C3×C4C2×C7×D711yescharacteristic
184#67—141no—
185D12—71no—
186C4×D3—141no—
187Dic6—71no—
188C3⋊C8—72no—
189C3×C7—23no—
190C3×C7#649511yescharacteristic, complemented
191C3×C7C7×AGL(1,ℤ/8)11yescharacteristic, complemented
192#41—211no—
193C2×D4C7×D2111yescharacteristic, complemented
194SD16—212no—
195D8—212no—
196M16—211no—
197C2×C7—26no—
198C2×C7—21no—
199C2×C7—21no—
200C2×C7—23no—
201C2×C7—412no—
202C2×C7—42no—
203C2×C7—42no—
204C2×C7—841no—
205C2×C7C2×C7×#6711yescharacteristic
206C2×C7C2×#426811yescharacteristic
207D7—121no—
208C22×C3—22no—
209C22×C3—42no—
210C22×C3C7×C22⋊D711yescharacteristic
211C2×D3—141no—
212C3×C4C7×C22⋊D711yescharacteristic
213C3×C4C22×C7×D711yescharacteristic
214Dic3—141no—
215C23—21no—
216Q8—211no—
217D4—21no—
218D4—211no—
219D4—421no—
220D4C2×C7×D2112yes—
221C2×C4—421no—
222C2×C4C2×C7×D2111yescharacteristic
223C8—212no—
224C7—23no—
225C7C7×#93211yescharacteristic, complemented
226C7#1062141511yescharacteristic, direct product factor
227C2×C3—21no—
228C2×C3—42no—
229C2×C3C2×C7×C22⋊D711yescharacteristic
230D3—281no—
231C22—22no—
232C22—42no—
233C22—421no—
234C22C7×#426811yescharacteristic
235C4—421no—
236C4C7×#426811yescharacteristic
237C4C22×C7×D2111yescharacteristic
238C3C7×#649511yescharacteristic, complemented, Sylow
239C2—21no—
240C2—42no—
241C2—841no—
242C2C2×C7×#426811yescharacteristic
subgroupG/H
centerC2×C7C2×#4268
commutator, cosocleC3×C4×C7C23×C7
FrattiniC4C22×C7×D21
FittingC2×C3×C72×D4C2
socleC2×C3×C72C2×D4
hypercenterC2×C7×D4D21
hypocenterC3×C7C7×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×D4

Complements

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

NH ≅ G/Ntype
1. C7×#66632C2semidirect
2. C7×#66633C2semidirect
3. C7×#66728C2semidirect
4. C7×#66627C2semidirect
5. C2×C3×C72×D4C2semidirect
12. C7×D84C22semidirect
14. C7×Dic42C22semidirect
15. C7×C3⋊(C7⋊C8)C22semidirect
16. C2×C72×D4D3semidirect
21. #10621415C7direct (N × H)
40. C2×C3×C7×D4D7semidirect
41. C2×C3×C7×D4C2×C7semidirect
42. #66728C2×C7semidirect
44. #66633C2×C7semidirect
45. #66632C2×C7semidirect
46. #66627C2×C7semidirect
76. D84C22×C7semidirect
78. Dic42C22×C7semidirect
83. C3⋊(C7⋊C8)C22×C7semidirect
85. C3×C72AGL(1,ℤ/8)semidirect
88. C2×C7×D4D3×C7semidirect
89. C2×C7×D4D21semidirect
143. C72#932semidirect
144. C2×C3×D4C7×D7semidirect
190. C3×C7#6495semidirect
191. C3×C7C7×AGL(1,ℤ/8)semidirect
193. C2×D4C7×D21semidirect
225. C7C7×#932semidirect
226. C7#10621415direct (N × H)
238. C3C7×#6495semidirect

Maximal subgroups

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

subgroupindex
C7×#666322
C7×#666332
C7×#667282
C7×#666272
C2×C3×C72×D42
C7×#64953
#106214157
C7×#9327