Finite Group Databasebeta

← C7×#8757 #13185973

Subgroups — C7×#8757 #13185973

Order 1792.

438 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2316C211yesmaximal, characteristic
2C7×#1463C212yesmaximal
3C7×#1305C214yesmaximal
4C2×C7×#117C411yescharacteristic
5C2×C7×#117C411yescharacteristic
6C22×C7×C4⋊C4C2211yescharacteristic
7C7×#407C2214yes—
8C7×#389—22no—
9C7×#389—22no—
10C7×#380C412yes—
11C7×#380C2212yes—
12C7×#324—22no—
13C7×#324—22no—
14#8757C711yesmaximal, characteristic, direct product factor, Sylow
15C23×C4×C7C2×C411yescharacteristic
16C23×C4×C7C2×C411yescharacteristic
17C23×C4×C7C2311yescharacteristic
18C2×C7×M16—24no—
19C7×#117—42no—
20C7×#117—42no—
21C2×C7×C4⋊C4—24no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4D412yes—
27C2×C7×C4⋊C4D412yes—
28C2×C7×C22⋊C4C2×C412yes—
29C2×C7×C22⋊C4C2×C412yes—
30C2×C42×C7—21no—
31C2×C42×C7—21no—
32C7×C22⋊C8—24no—
33C7×C22⋊C8—24no—
34C7×#95—22no—
35C7×#95—22no—
36#2316C2×C711yescharacteristic
37#1463C2×C712yes—
38#1305C2×C714yes—
39C24×C7C22×C411yescharacteristic
40C22×C4×C7—22no—
41C22×C4×C7—22no—
42C22×C4×C7—22no—
43C22×C4×C7—22no—
44C22×C4×C7—22no—
45C22×C4×C7—22no—
46C22×C4×C7—81no—
47C22×C4×C7—81no—
48C22×C4×C7C22⋊C411yescharacteristic
49C22×C4×C7C22⋊C411yescharacteristic
50C22×C4×C7C22⋊C412yes—
51C22×C4×C7C2×D412yes—
52C22×C4×C7#4114yes—
53C7×M16—84no—
54C2×C7×C8—44no—
55C7×C4⋊C4—24no—
56C7×C4⋊C4—24no—
57C7×C4⋊C4—42no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4—82no—
60C7×C4⋊C4—82no—
61C7×C22⋊C4—42no—
62C7×C22⋊C4—42no—
63C42×C7—42no—
64C42×C7—42no—
65C2×#117C4×C711yescharacteristic
66C2×#117C4×C711yescharacteristic
67C22×C4⋊C4C22×C711yescharacteristic
68#407C22×C714yes—
69#389—22no—
70#389—22no—
71#380C4×C712yes—
72#380C22×C712yes—
73#324—22no—
74#324—22no—
75C23×C7—21no—
76C23×C7—21no—
77C23×C7—41no—
78C23×C7—41no—
79C23×C7C2×C22⋊C411yescharacteristic
80C23×C7#11712yes—
81C2×C4×C7—28no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—42no—
88C2×C4×C7—44no—
89C2×C4×C7—42no—
90C2×C4×C7—42no—
91C2×C4×C7—42no—
92C2×C4×C7—42no—
93C2×C4×C7—42no—
94C2×C4×C7—42no—
95C2×C4×C7—42no—
96C2×C4×C7—82no—
97C2×C4×C7#10012yes—
98C2×C4×C7#10112yes—
99C2×C4×C7#12314yes—
100C7×C8—84no—
101C23×C4C2×C4×C711yescharacteristic
102C23×C4C2×C4×C711yescharacteristic
103C23×C4C23×C711yescharacteristic
104C2×M16—24no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4C7×D412yes—
113C2×C4⋊C4C7×D412yes—
114C2×C22⋊C4C2×C4×C712yes—
115C2×C22⋊C4C2×C4×C712yes—
116C2×C42—21no—
117C2×C42—21no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120#95—22no—
121#95—22no—
122C22×C7—21no—
123C22×C7—21no—
124C22×C7—22no—
125C22×C7—22no—
126C22×C7—22no—
127C22×C7—42no—
128C22×C7—42no—
129C22×C7#38111yescharacteristic
130C22×C7C2×#10011yescharacteristic
131C22×C7C2×#10111yescharacteristic
132C4×C7—22no—
133C4×C7—22no—
134C4×C7—24no—
135C4×C7—42no—
136C4×C7—82no—
137C4×C7—82no—
138C24C22×C4×C711yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4C7×C22⋊C411yescharacteristic
148C22×C4C7×C22⋊C411yescharacteristic
149C22×C4C7×C22⋊C412yes—
150C22×C4C2×C7×D412yes—
151C22×C4C7×#4114yes—
152M16—84no—
153C2×C8—44no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C42—42no—
163C42—42no—
164C2×C7—21no—
165C2×C7—21no—
166C2×C7—42no—
167C2×C7#182111yescharacteristic
168C2×C7#182811yescharacteristic
169C2×C7#183011yescharacteristic
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23C2×C7×C22⋊C411yescharacteristic
175C23C7×#11712yes—
176C2×C4—28no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4C7×#10012yes—
193C2×C4C7×#10112yes—
194C2×C4C7×#12314yes—
195C8—84no—
196C7#875711yescharacteristic, direct product factor, Sylow
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22C7×#38111yescharacteristic
205C22C2×C7×#10011yescharacteristic
206C22C2×C7×#10111yescharacteristic
207C4—22no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—82no—
212C4—82no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C7×#182111yescharacteristic
217C2C7×#182811yescharacteristic
218C2C7×#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
14. #8757C7direct (N × H)
196. C7#8757direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23162
C7×#14632
C7×#13052
#87577