Finite Group Databasebeta

← C7×#8758 #13185974

Subgroups — C7×#8758 #13185974

Order 1792.

394 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2313C211yesmaximal, characteristic
2C7×#1463C212yesmaximal
3C7×#1305C214yesmaximal
4C22×C7×C4⋊C4C2211yescharacteristic
5C7×#407C2214yes—
6C7×#387—24no—
7C7×#381C412yes—
8C7×#380C2212yes—
9C4×C7×C22⋊C4C412yes—
10C7×#324—24no—
11#8758C711yesmaximal, characteristic, direct product factor, Sylow
12C23×C4×C7C2×C411yescharacteristic
13C23×C4×C7C2×C411yescharacteristic
14C23×C4×C7C2311yescharacteristic
15C2×C7×M16—24no—
16C2×C7×C4⋊C4—24no—
17C2×C7×C4⋊C4—22no—
18C2×C7×C4⋊C4D414yes—
19C2×C7×C22⋊C4C2×C414yes—
20C2×C42×C7—22no—
21C7×C22⋊C8—24no—
22C7×C22⋊C8—24no—
23C7×#95—22no—
24C7×#95—24no—
25C7×#95—22no—
26#2313C2×C711yescharacteristic
27#1463C2×C712yes—
28#1305C2×C714yes—
29C24×C7C22×C411yescharacteristic
30C22×C4×C7—22no—
31C22×C4×C7—22no—
32C22×C4×C7—24no—
33C22×C4×C7—42no—
34C22×C4×C7—81no—
35C22×C4×C7—81no—
36C22×C4×C7C22⋊C412yes—
37C22×C4×C7C22⋊C412yes—
38C22×C4×C7C2×D412yes—
39C22×C4×C7#4114yes—
40C7×M16—84no—
41C2×C7×C8—44no—
42C7×C4⋊C4—28no—
43C7×C4⋊C4—82no—
44C7×C22⋊C4—44no—
45C42×C7—82no—
46C22×C4⋊C4C22×C711yescharacteristic
47#407C22×C714yes—
48#387—24no—
49#381C4×C712yes—
50#380C22×C712yes—
51C4×C22⋊C4C4×C712yes—
52#324—24no—
53C23×C7—22no—
54C23×C7—41no—
55C23×C7—41no—
56C23×C7C2×C22⋊C411yescharacteristic
57C23×C7#11712yes—
58C2×C4×C7—22no—
59C2×C4×C7—22no—
60C2×C4×C7—24no—
61C2×C4×C7—42no—
62C2×C4×C7—44no—
63C2×C4×C7—44no—
64C2×C4×C7—42no—
65C2×C4×C7—44no—
66C2×C4×C7—44no—
67C2×C4×C7—44no—
68C2×C4×C7—82no—
69C2×C4×C7#12314yes—
70C7×C8—84no—
71C23×C4C2×C4×C711yescharacteristic
72C23×C4C2×C4×C711yescharacteristic
73C23×C4C23×C711yescharacteristic
74C2×M16—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C7×D414yes—
78C2×C22⋊C4C2×C4×C714yes—
79C2×C42—22no—
80C22⋊C8—24no—
81C22⋊C8—24no—
82#95—22no—
83#95—24no—
84#95—22no—
85C22×C7—21no—
86C22×C7—21no—
87C22×C7—22no—
88C22×C7—22no—
89C22×C7—22no—
90C22×C7—44no—
91C22×C7#38111yescharacteristic
92C22×C7#41312yes—
93C4×C7—24no—
94C4×C7—42no—
95C4×C7—42no—
96C4×C7—84no—
97C24C22×C4×C711yescharacteristic
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—24no—
101C22×C4—42no—
102C22×C4—81no—
103C22×C4—81no—
104C22×C4C7×C22⋊C412yes—
105C22×C4C7×C22⋊C412yes—
106C22×C4C2×C7×D412yes—
107C22×C4C7×#4114yes—
108M16—84no—
109C2×C8—44no—
110C4⋊C4—28no—
111C4⋊C4—82no—
112C22⋊C4—44no—
113C42—82no—
114C2×C7—22no—
115C2×C7—42no—
116C2×C7#182211yescharacteristic
117C2×C7#182811yescharacteristic
118C2×C7#183011yescharacteristic
119C23—22no—
120C23—41no—
121C23—41no—
122C23C2×C7×C22⋊C411yescharacteristic
123C23C7×#11712yes—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—24no—
127C2×C4—42no—
128C2×C4—44no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—44no—
133C2×C4—44no—
134C2×C4—82no—
135C2×C4C7×#12314yes—
136C8—84no—
137C7#875811yescharacteristic, direct product factor, Sylow
138C22—21no—
139C22—21no—
140C22—22no—
141C22—22no—
142C22—22no—
143C22—44no—
144C22C7×#38111yescharacteristic
145C22C7×#41312yes—
146C4—24no—
147C4—42no—
148C4—42no—
149C4—84no—
150C2—22no—
151C2—42no—
152C2C7×#182211yescharacteristic
153C2C7×#182811yescharacteristic
154C2C7×#183011yescharacteristic
subgroupG/H
center, socleC22×C7#381
commutatorC24C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
11. #8758C7direct (N × H)
137. C7#8758direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23132
C7×#14632
C7×#13052
#87587