Finite Group Databasebeta

← C7×#5968 #13062365

Subgroups — C7×#5968 #13062365

Order 1344.

1620 conjugacy classes of subgroups; only the 886 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 886 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×C7×2−1+4C211yesmaximal, characteristic, complemented
2C7×#1009C2110yesmaximal, complemented
3D3×C7×#41C2110yesmaximal, complemented
4C7×#1007C215yesmaximal, complemented
5C2×C7×#239C215yesmaximal, complemented
6C7×#585—31nomaximal, Hall
7C3×C7×#41C22110yescomplemented
8C2×C3×C7×Q8C2215yescomplemented
9C7×#239C22110yescomplemented
10C7×#239C22120yescomplemented
11D3×C7×Q8C22110yescomplemented
12C7×#237C22110yescomplemented
13D3×C7×D4C22130yescomplemented
14C7×#235C22130yescomplemented
15C2×C7×D12C22115yescomplemented
16C2×C4×D3×C7C22115yescomplemented
17C7×2−1+4—35no—
18C7×2−1+4D311yescharacteristic, complemented
19C7×H(5,𝔽2)—310no—
20C2×C7×#41—310no—
21C2×C7×#41—35no—
22#5968C711yesmaximal, characteristic, direct product factor, Hall
23C22×D3×C7C23115yes—
24C3×C7×Q8C23110yes—
25C3×C7×D4C23110yes—
26C2×C3×C4×C7C23115yes—
27C7×#67C23120yes—
28C2×C7×Dic3C2315yes—
29C7×D12C23130yes—
30C4×D3×C7C23110yes—
31C4×D3×C7C23130yes—
32C7×Dic6C23110yes—
33C7×#41—310no—
34C7×#41—320no—
35C7×#41—310no—
36C7×#41—330no—
37C7×#41C2×D3110yescomplemented
38C2×C7×Q8—310no—
39C2×C7×Q8C2×D315yescomplemented
40C2×C7×D4—330no—
41C2×C7×D4—315no—
42C22×C4×C7—315no—
43C3×2−1+4C2×C711yescharacteristic, complemented
44#1009C2×C7110yescomplemented
45D3×#41C2×C7110yescomplemented
46#1007C2×C715yescomplemented
47C2×#239C2×C715yescomplemented
48C22×C3×C7C2415yes—
49C2×D3×C7—415no—
50C2×D3×C7C24110yes—
51C3×C4×C7C24110yes—
52C7×Dic3C2415yes—
53C7×Dic3C2411yescharacteristic
54#585—31noSylow
55C23×C7—315no—
56C7×Q8—310no—
57C7×Q8C22×D3110yes—
58C7×D4—320no—
59C7×D4—330no—
60C7×D4C22×D3110yes—
61C2×C4×C7—35no—
62C2×C4×C7—310no—
63C2×C4×C7—330no—
64C2×C4×C7C22×D3115yes—
65C3×#41C22×C7110yescomplemented
66C2×C3×Q8C22×C715yescomplemented
67#239C22×C7110yescomplemented
68#239C22×C7120yescomplemented
69D3×Q8C22×C7110yescomplemented
70#237C22×C7110yescomplemented
71D3×D4C22×C7130yescomplemented
72#235C22×C7130yescomplemented
73C2×D12C22×C7115yescomplemented
74C2×C4×D3C22×C7115yescomplemented
75C2×C3×C7—25no—
76C2×C3×C7C2511yescharacteristic
77D3×C7—210no—
782−1+4—35no—
792−1+4D3×C711yescharacteristic, complemented
80H(5,𝔽2)—310no—
81C2×#41—310no—
82C2×#41—35no—
83C22×C7—310no—
84C22×C7—1215no—
85C22×C7C23×D315yes—
86C4×C7—31no—
87C4×C7—35no—
88C4×C7C23×D3110yes—
89C22×D3C23×C7115yes—
90C3×Q8C23×C7110yes—
91C3×D4C23×C7110yes—
92C2×C3×C4C23×C7115yes—
93#67C23×C7120yes—
94C2×Dic3C23×C715yes—
95D12C23×C7130yes—
96C4×D3C23×C7110yes—
97C4×D3C23×C7130yes—
98Dic6C23×C7110yes—
99C3×C7#58511yescharacteristic, complemented, Hall
100#41—310no—
101#41—320no—
102#41—310no—
103#41—330no—
104#41C2×D3×C7110yescomplemented
105C2×Q8—310no—
106C2×Q8C2×D3×C715yescomplemented
107C2×D4—330no—
108C2×D4—315no—
109C22×C4—315no—
110C2×C7—25no—
111C2×C7—610no—
112C2×C7C24×D311yescharacteristic
113C22×C3C24×C715yes—
114C2×D3—415no—
115C2×D3C24×C7110yes—
116C3×C4C24×C7110yes—
117Dic3C24×C715yes—
118Dic3C24×C711yescharacteristic
119C23—315no—
120Q8—310no—
121Q8C22×D3×C7110yes—
122D4—320no—
123D4—330no—
124D4C22×D3×C7110yes—
125C2×C4—35no—
126C2×C4—310no—
127C2×C4—330no—
128C2×C4C22×D3×C7115yes—
129C7#596811yescharacteristic, direct product factor, Sylow
130C2×C3—25no—
131C2×C3C25×C711yescharacteristic
132D3—210no—
133C22—310no—
134C22—1215no—
135C22C23×D3×C715yes—
136C4—31no—
137C4—35no—
138C4C23×D3×C7110yes—
139C3C7×#58511yescharacteristic, complemented, Sylow
140C2—25no—
141C2—610no—
142C2C24×D3×C711yescharacteristic
subgroupG/H
centerC2×C7C24×D3
commutator, cosocleC2×C3C25×C7
FrattiniC2C24×D3×C7
FittingC3×C7×2−1+4C2
socleC2×C3×C7C25
hypercenterC7×2−1+4D3
hypocenterC3C7×#585
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C7×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C7×2−1+4C2semidirect
2. C7×#1009C2semidirect
3. D3×C7×#41C2semidirect
4. C7×#1007C2semidirect
5. C2×C7×#239C2semidirect
7. C3×C7×#41C22semidirect
8. C2×C3×C7×Q8C22semidirect
9. C7×#239C22semidirect
10. C7×#239C22semidirect
11. D3×C7×Q8C22semidirect
12. C7×#237C22semidirect
13. D3×C7×D4C22semidirect
14. C7×#235C22semidirect
15. C2×C7×D12C22semidirect
16. C2×C4×D3×C7C22semidirect
18. C7×2−1+4D3semidirect
22. #5968C7direct (N × H)
37. C7×#41C2×D3semidirect
39. C2×C7×Q8C2×D3semidirect
43. C3×2−1+4C2×C7semidirect
44. #1009C2×C7semidirect
45. D3×#41C2×C7semidirect
46. #1007C2×C7semidirect
47. C2×#239C2×C7semidirect
65. C3×#41C22×C7semidirect
66. C2×C3×Q8C22×C7semidirect
67. #239C22×C7semidirect
68. #239C22×C7semidirect
69. D3×Q8C22×C7semidirect
70. #237C22×C7semidirect
71. D3×D4C22×C7semidirect
72. #235C22×C7semidirect
73. C2×D12C22×C7semidirect
74. C2×C4×D3C22×C7semidirect
79. 2−1+4D3×C7semidirect
99. C3×C7#585semidirect
104. #41C2×D3×C7semidirect
106. C2×Q8C2×D3×C7semidirect
129. C7#5968direct (N × H)
139. C3C7×#585semidirect

Maximal subgroups

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

subgroupindex
C3×C7×2−1+42
C7×#10092
D3×C7×#412
C7×#10072
C2×C7×#2392
C7×#5853
#59687