Finite Group Databasebeta

← C7×#5752 #13061468

Subgroups — C7×#5752 #13061468

Order 1344.

536 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 214 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×#1009C212yesmaximal, complemented
2C3×C7×#131C211yesmaximal, characteristic, complemented
3C7×#908C216yesmaximal, complemented
4C7×#904C213yesmaximal, complemented
5C2×C7×D24C213yesmaximal, complemented
6C7×#576—31nomaximal, Hall
7C3×C7×#41C2211yescharacteristic
8C7×#239—22no—
9D3×C7×D4—26no—
10C7×#235C2216yescomplemented
11C2×C7×D12C2216yescomplemented
12C3×C7×M16C2213yescomplemented
13C2×C3×C7×C8C2213yescomplemented
14C7×Dic12C2211yescharacteristic, complemented
15C7×D24C2213yescomplemented
16C7×D24C2216yescomplemented
17C7×#204C2216yescomplemented
18C7×H(5,𝔽2)—32no—
19C7×AGL(1,ℤ/8)—36no—
20C7×#135—33no—
21C2×C7×D8—33no—
22C7×#131D311yescharacteristic, complemented
23#5752C711yesmaximal, characteristic, direct product factor, Hall
24C22×D3×C7—26no—
25C3×C7×Q8D411yescharacteristic
26C3×C7×D4D413yes—
27C2×C3×C4×C7C2313yes—
28C7×#67—26no—
29C7×D12—26no—
30C7×D12C2316yes—
31C4×D3×C7—26no—
32C7×Dic6C2312yes—
33C3×C7×C8C2313yes—
34C3×C7×C8C2311yescharacteristic
35C7×#41—36no—
36C7×#41—62no—
37C7×#41C2×D311yescharacteristic
38C2×C7×D4—36no—
39C2×C7×D4—66no—
40C7×Q16—31no—
41C7×SD16—36no—
42C7×D8—33no—
43C7×D8—36no—
44C7×M16C2×D313yescomplemented
45C2×C7×C8C2×D313yescomplemented
46#1009C2×C712yescomplemented
47C3×#131C2×C711yescharacteristic, complemented
48#908C2×C716yescomplemented
49#904C2×C713yescomplemented
50C2×D24C2×C713yescomplemented
51C22×C3×C7C2×D413yes—
52C2×D3×C7—26no—
53C2×D3×C7—86no—
54C3×C4×C7C2×D413yes—
55C3×C4×C7C2411yescharacteristic
56C7×Dic3—22no—
57#576—31noSylow
58C23×C7—66no—
59C7×Q8—32no—
60C7×Q8D1211yescharacteristic
61C7×D4—36no—
62C7×D4—66no—
63C7×D4—66no—
64C7×D4D1213yes—
65C2×C4×C7—66no—
66C2×C4×C7C22×D313yes—
67C7×C8C22×D311yescharacteristic
68C7×C8C22×D313yes—
69C3×#41C22×C711yescharacteristic
70#239—22no—
71D3×D4—26no—
72#235C22×C716yescomplemented
73C2×D12C22×C716yescomplemented
74C3×M16C22×C713yescomplemented
75C2×C3×C8C22×C713yescomplemented
76Dic12C22×C711yescharacteristic, complemented
77D24C22×C713yescomplemented
78D24C22×C716yescomplemented
79#204C22×C716yescomplemented
80C2×C3×C7—23no—
81C2×C3×C7C22×D411yescharacteristic
82D3×C7—46no—
83H(5,𝔽2)—32no—
84AGL(1,ℤ/8)—36no—
85#135—33no—
86C2×D8—33no—
87#131D3×C711yescharacteristic, complemented
88C22×C7—66no—
89C22×C7—246no—
90C22×C7C2×D1213yes—
91C4×C7—62no—
92C4×C7C2×D1213yes—
93C4×C7C23×D311yescharacteristic
94C22×D3—26no—
95C3×Q8C7×D411yescharacteristic
96C3×D4C7×D413yes—
97C2×C3×C4C23×C713yes—
98#67—26no—
99D12—26no—
100D12C23×C716yes—
101C4×D3—26no—
102Dic6C23×C712yes—
103C3×C8C23×C713yes—
104C3×C8C23×C711yescharacteristic
105C3×C7#57611yescharacteristic, complemented, Hall
106#41—36no—
107#41—62no—
108#41C2×D3×C711yescharacteristic
109C2×D4—36no—
110C2×D4—66no—
111Q16—31no—
112SD16—36no—
113D8—33no—
114D8—36no—
115M16C2×D3×C713yescomplemented
116C2×C8C2×D3×C713yescomplemented
117C2×C7—23no—
118C2×C7—126no—
119C2×C7C22×D1211yescharacteristic
120C22×C3C2×C7×D413yes—
121C2×D3—26no—
122C2×D3—86no—
123C3×C4C2×C7×D413yes—
124C3×C4C24×C711yescharacteristic
125Dic3—22no—
126C23—66no—
127Q8—32no—
128Q8C7×D1211yescharacteristic
129D4—36no—
130D4—66no—
131D4—66no—
132D4C7×D1213yes—
133C2×C4—66no—
134C2×C4C22×D3×C713yes—
135C8C22×D3×C711yescharacteristic
136C8C22×D3×C713yes—
137C7#575211yescharacteristic, direct product factor, Sylow
138C2×C3—23no—
139C2×C3C22×C7×D411yescharacteristic
140D3—46no—
141C22—66no—
142C22—246no—
143C22C2×C7×D1213yes—
144C4—62no—
145C4C2×C7×D1213yes—
146C4C23×D3×C711yescharacteristic
147C3C7×#57611yescharacteristic, complemented, Sylow
148C2—23no—
149C2—126no—
150C2C22×C7×D1211yescharacteristic
subgroupG/H
centerC2×C7C22×D12
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×#131C2
socleC2×C3×C7C22×D4
hypercenterC7×#131D3
hypocenterC3C7×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C7×#41 ◁ C7×#131

Complements

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

NH ≅ G/Ntype
1. C7×#1009C2semidirect
2. C3×C7×#131C2semidirect
3. C7×#908C2semidirect
4. C7×#904C2semidirect
5. C2×C7×D24C2semidirect
10. C7×#235C22semidirect
11. C2×C7×D12C22semidirect
12. C3×C7×M16C22semidirect
13. C2×C3×C7×C8C22semidirect
14. C7×Dic12C22semidirect
15. C7×D24C22semidirect
16. C7×D24C22semidirect
17. C7×#204C22semidirect
22. C7×#131D3semidirect
23. #5752C7direct (N × H)
44. C7×M16C2×D3semidirect
45. C2×C7×C8C2×D3semidirect
46. #1009C2×C7semidirect
47. C3×#131C2×C7semidirect
48. #908C2×C7semidirect
49. #904C2×C7semidirect
50. C2×D24C2×C7semidirect
72. #235C22×C7semidirect
73. C2×D12C22×C7semidirect
74. C3×M16C22×C7semidirect
75. C2×C3×C8C22×C7semidirect
76. Dic12C22×C7semidirect
77. D24C22×C7semidirect
78. D24C22×C7semidirect
79. #204C22×C7semidirect
87. #131D3×C7semidirect
105. C3×C7#576semidirect
115. M16C2×D3×C7semidirect
116. C2×C8C2×D3×C7semidirect
137. C7#5752direct (N × H)
147. C3C7×#576semidirect

Maximal subgroups

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

subgroupindex
C7×#10092
C3×C7×#1312
C7×#9082
C7×#9042
C2×C7×D242
C7×#5763
#57527