Finite Group Databasebeta

← C71×#2901 #22585463

Subgroups — C71×#2901 #22585463

Order 9088.

738 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 340 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
1#585×C71C211yesmaximal, characteristic
2#568×C71C211yesmaximal, characteristic, complemented
3C4×#41×C71C211yesmaximal, characteristic, complemented
4#421×C71C216yesmaximal, complemented
5C2×#104×C71C216yesmaximal, complemented
62−1+4×C71—21no—
72−1+4×C71C414yescomplemented
8H(5,𝔽2)×C71—23no—
9H(5,𝔽2)×C71C414yescomplemented
10C2×#41×C71—24no—
11C2×#41×C71C2216yes—
12C2×#41×C71C2211yescharacteristic
13#131×C71—24no—
14C2×M16×C71C2216yescomplemented
15C4×Q8×C71—21no—
16C4×D4×C71—23no—
17#117×C71C2213yescomplemented
18C2×C42×C71C2213yescomplemented
19#104×C71C2214yescomplemented
20#104×C71C22112yescomplemented
21#41×C71—24no—
22#41×C71—212no—
23#41×C71—26no—
24#41×C71—26no—
25#41×C71C2×C414yes—
26#41×C71C2×C4112yescomplemented
27#41×C71D414yes—
28#41×C71C2314yes—
29C2×Q8×C71—24no—
30C2×Q8×C71C2×C416yescomplemented
31C2×Q8×C71D411yescharacteristic
32C2×D4×C71—26no—
33C2×D4×C71—212no—
34C2×D4×C71C2×C416yescomplemented
35C2×D4×C71D413yes—
36C22×C4×C71—26no—
37C22×C4×C71—23no—
38C22×C4×C71C2313yes—
39M16×C71—26no—
40M16×C71C2314yes—
41C2×C8×C71—26no—
42C4⋊C4×C71—23no—
43C22⋊C4×C71—23no—
44C42×C71—23no—
45C42×C71C2311yescharacteristic
46C42×C71C2313yes—
47C23×C71—26no—
48C23×C71C2×D413yes—
49Q8×C71—26no—
50Q8×C71C22⋊C414yes—
51Q8×C71C22×C414yes—
52D4×C71—212no—
53D4×C71—24no—
54D4×C71—26no—
55D4×C71C22⋊C4112yes—
56D4×C71C22×C414yes—
57C2×C4×C71—26no—
58C2×C4×C71—21no—
59C2×C4×C71—212no—
60C2×C4×C71—24no—
61C2×C4×C71—23no—
62C2×C4×C71—46no—
63C2×C4×C71C22×C416yes—
64C2×C4×C71C2×D413yes—
65C2×C4×C71C2×D416yes—
66C2×C4×C71C2411yescharacteristic
67C8×C71—24no—
68C22×C71—24no—
69C22×C71—43no—
70C22×C71—86no—
71C22×C71C2×C22⋊C416yes—
72C22×C71C22×D411yescharacteristic
73C4×C71—22no—
74C4×C71—24no—
75C4×C71—43no—
76C4×C71C2×C22⋊C416yes—
77C4×C71C23×C411yescharacteristic
78C4×C71C22×D411yescharacteristic
79C2×C71—21no—
80C2×C71—26no—
81C2×C71—44no—
82C2×C71C22×C22⋊C411yescharacteristic
83#2901C7111yesmaximal, characteristic, direct product factor, Sylow
84C71#290111yescharacteristic, direct product factor, Sylow
85#585C2×C7111yescharacteristic
86#568C2×C7111yescharacteristic, complemented
87C4×#41C2×C7111yescharacteristic, complemented
88#421C2×C7116yescomplemented
89C2×#104C2×C7116yescomplemented
902−1+4—21no—
912−1+4C4×C7114yescomplemented
92H(5,𝔽2)—23no—
93H(5,𝔽2)C4×C7114yescomplemented
94C2×#41—24no—
95C2×#41C22×C7116yes—
96C2×#41C22×C7111yescharacteristic
97#131—24no—
98C2×M16C22×C7116yescomplemented
99C4×Q8—21no—
100C4×D4—23no—
101#117C22×C7113yescomplemented
102C2×C42C22×C7113yescomplemented
103#104C22×C7114yescomplemented
104#104C22×C71112yescomplemented
105#41—24no—
106#41—212no—
107#41—26no—
108#41—26no—
109#41C2×C4×C7114yes—
110#41C2×C4×C71112yescomplemented
111#41D4×C7114yes—
112#41C23×C7114yes—
113C2×Q8—24no—
114C2×Q8C2×C4×C7116yescomplemented
115C2×Q8D4×C7111yescharacteristic
116C2×D4—26no—
117C2×D4—212no—
118C2×D4C2×C4×C7116yescomplemented
119C2×D4D4×C7113yes—
120C22×C4—26no—
121C22×C4—23no—
122C22×C4C23×C7113yes—
123M16—26no—
124M16C23×C7114yes—
125C2×C8—26no—
126C4⋊C4—23no—
127C22⋊C4—23no—
128C42—23no—
129C42C23×C7111yescharacteristic
130C42C23×C7113yes—
131C23—26no—
132C23C2×D4×C7113yes—
133Q8—26no—
134Q8C22⋊C4×C7114yes—
135Q8C22×C4×C7114yes—
136D4—212no—
137D4—24no—
138D4—26no—
139D4C22⋊C4×C71112yes—
140D4C22×C4×C7114yes—
141C2×C4—26no—
142C2×C4—21no—
143C2×C4—212no—
144C2×C4—24no—
145C2×C4—23no—
146C2×C4—46no—
147C2×C4C22×C4×C7116yes—
148C2×C4C2×D4×C7113yes—
149C2×C4C2×D4×C7116yes—
150C2×C4C24×C7111yescharacteristic
151C8—24no—
152C22—24no—
153C22—43no—
154C22—86no—
155C22C2×C22⋊C4×C7116yes—
156C22C22×D4×C7111yescharacteristic
157C4—22no—
158C4—24no—
159C4—43no—
160C4C2×C22⋊C4×C7116yes—
161C4C23×C4×C7111yescharacteristic
162C4C22×D4×C7111yescharacteristic
163C2—21no—
164C2—26no—
165C2—44no—
166C2C22×C22⋊C4×C7111yescharacteristic
subgroupG/H
centerC4×C71C22×D4
commutatorC4C23×C4×C71
Frattini, cosocleC2×C4C24×C71
socleC2×C71C22×C22⋊C4
prime radical#585×C71C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C71 ◁ C2×#41×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. #568×C71C2semidirect
3. C4×#41×C71C2semidirect
4. #421×C71C2semidirect
5. C2×#104×C71C2semidirect
7. 2−1+4×C71C4semidirect
9. H(5,𝔽2)×C71C4semidirect
14. C2×M16×C71C22semidirect
17. #117×C71C22semidirect
18. C2×C42×C71C22semidirect
19. #104×C71C22semidirect
20. #104×C71C22semidirect
26. #41×C71C2×C4semidirect
30. C2×Q8×C71C2×C4semidirect
34. C2×D4×C71C2×C4semidirect
83. #2901C71direct (N × H)
84. C71#2901direct (N × H)
86. #568C2×C71semidirect
87. C4×#41C2×C71semidirect
88. #421C2×C71semidirect
89. C2×#104C2×C71semidirect
91. 2−1+4C4×C71semidirect
93. H(5,𝔽2)C4×C71semidirect
98. C2×M16C22×C71semidirect
101. #117C22×C71semidirect
102. C2×C42C22×C71semidirect
103. #104C22×C71semidirect
104. #104C22×C71semidirect
110. #41C2×C4×C71semidirect
114. C2×Q8C2×C4×C71semidirect
118. C2×D4C2×C4×C71semidirect

Maximal subgroups

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

subgroupindex
#585×C712
#568×C712
C4×#41×C712
#421×C712
C2×#104×C712
#290171