Finite Group Databasebeta

← C7×#10339 #13187555

Subgroups — C7×#10339 #13187555

Order 1792.

738 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2483C213yesmaximal
2C7×#1819C213yesmaximal
3C7×#1304C211yesmaximal, characteristic
4C22×C7×C22⋊C4C2213yes—
5C7×#397—26no—
6C7×#385—26no—
7C7×#381—26no—
8C7×#379C2211yescharacteristic
9C7×#379C2213yes—
10C7×#342—26no—
11#10339C711yesmaximal, characteristic, direct product factor, Sylow
12C25×C7C2311yescharacteristic
13C23×C4×C7D416yes—
14C2×C7×C4⋊C4—26no—
15C2×C7×C22⋊C4—23no—
16C2×C7×C22⋊C4—26no—
17C2×C7×C22⋊C4—26no—
18C2×C7×C22⋊C4—26no—
19C2×C7×C22⋊C4—23no—
20C2×C7×C22⋊C4—23no—
21C2×C7×C22⋊C4D416yes—
22C7×#95—26no—
23C7×#95—212no—
24#2483C2×C713yes—
25#1819C2×C713yes—
26#1304C2×C711yescharacteristic
27C24×C7—46no—
28C24×C7C2×D413yes—
29C24×C7C2×D413yes—
30C24×C7#4111yescharacteristic
31C22×C4×C7—26no—
32C22×C4×C7—26no—
33C22×C4×C7—26no—
34C22×C4×C7—26no—
35C22×C4×C7—412no—
36C22×C4×C7—41no—
37C7×C4⋊C4—86no—
38C7×C22⋊C4—412no—
39C7×C22⋊C4—43no—
40C7×C22⋊C4—46no—
41C7×C22⋊C4—43no—
42C7×C22⋊C4—46no—
43C7×C22⋊C4—46no—
44C22×C22⋊C4C22×C713yes—
45#397—26no—
46#385—26no—
47#381—26no—
48#379C22×C711yescharacteristic
49#379C22×C713yes—
50#342—26no—
51C23×C7—212no—
52C23×C7—26no—
53C23×C7—43no—
54C23×C7—46no—
55C23×C7—43no—
56C23×C7—88no—
57C23×C7#12013yes—
58C23×C7#12113yes—
59C23×C7#12711yescharacteristic
60C2×C4×C7—412no—
61C2×C4×C7—412no—
62C2×C4×C7—46no—
63C2×C4×C7—46no—
64C2×C4×C7—412no—
65C2×C4×C7—86no—
66C2×C4×C7—83no—
67C25C23×C711yescharacteristic
68C23×C4C7×D416yes—
69C2×C4⋊C4—26no—
70C2×C22⋊C4—23no—
71C2×C22⋊C4—26no—
72C2×C22⋊C4—26no—
73C2×C22⋊C4—26no—
74C2×C22⋊C4—23no—
75C2×C22⋊C4—23no—
76C2×C22⋊C4C7×D416yes—
77#95—26no—
78#95—212no—
79C22×C7—26no—
80C22×C7—26no—
81C22×C7—23no—
82C22×C7—23no—
83C22×C7—412no—
84C22×C7—86no—
85C22×C7—82no—
86C22×C7#39211yescharacteristic
87C22×C7#45816yes—
88C4×C7—86no—
89C4×C7—86no—
90C4×C7—161no—
91C24—46no—
92C24C2×C7×D413yes—
93C24C2×C7×D413yes—
94C24C7×#4111yescharacteristic
95C22×C4—26no—
96C22×C4—26no—
97C22×C4—26no—
98C22×C4—26no—
99C22×C4—412no—
100C22×C4—41no—
101C4⋊C4—86no—
102C22⋊C4—412no—
103C22⋊C4—43no—
104C22⋊C4—46no—
105C22⋊C4—43no—
106C22⋊C4—46no—
107C22⋊C4—46no—
108C2×C7—26no—
109C2×C7—41no—
110C2×C7—43no—
111C2×C7#202213yes—
112C23—212no—
113C23—26no—
114C23—43no—
115C23—46no—
116C23—43no—
117C23—88no—
118C23C7×#12013yes—
119C23C7×#12113yes—
120C23C7×#12711yescharacteristic
121C2×C4—412no—
122C2×C4—412no—
123C2×C4—46no—
124C2×C4—46no—
125C2×C4—412no—
126C2×C4—86no—
127C2×C4—83no—
128C7#1033911yescharacteristic, direct product factor, Sylow
129C22—26no—
130C22—26no—
131C22—23no—
132C22—23no—
133C22—412no—
134C22—86no—
135C22—82no—
136C22C7×#39211yescharacteristic
137C22C7×#45816yes—
138C4—86no—
139C4—86no—
140C4—161no—
141C2—26no—
142C2—41no—
143C2—43no—
144C2C7×#202213yes—
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC25C23×C7
prime radicalC25×C7C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
11. #10339C7direct (N × H)
128. C7#10339direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24832
C7×#18192
C7×#13042
#103397