Finite Group Databasebeta

← C7×#8756 #13185972

Subgroups — C7×#8756 #13185972

Order 1792.

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

normal subgroups only

This group has 546 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×#2315C211yesmaximal, characteristic
2C7×#1462C212yesmaximal
3C7×#1305C214yesmaximal, complemented
4C22×C7×C22⋊C4C411yescharacteristic
5C22×C7×C22⋊C4C411yescharacteristic
6C22×C7×C22⋊C4C2211yescharacteristic
7C7×#407C2214yes—
8C7×#385—22no—
9C7×#385—22no—
10C7×#380C412yes—
11C7×#380C2212yes—
12C7×#323—22no—
13C7×#323—22no—
14#8756C711yesmaximal, characteristic, direct product factor, Sylow
15C25×C7C2×C411yescharacteristic
16C23×C4×C7—21no—
17C23×C4×C7—21no—
18C23×C4×C7C2×C411yescharacteristic
19C23×C4×C7C2311yescharacteristic
20C2×C7×M16—24no—
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4—22no—
23C2×C7×C22⋊C4—24no—
24C2×C7×C22⋊C4—21no—
25C2×C7×C22⋊C4—21no—
26C2×C7×C22⋊C4—42no—
27C2×C7×C22⋊C4—42no—
28C2×C7×C22⋊C4C2×C412yes—
29C2×C7×C22⋊C4C2×C412yes—
30C2×C7×C22⋊C4D412yes—
31C2×C7×C22⋊C4D412yes—
32C7×C22⋊C8—24no—
33C7×C22⋊C8—24no—
34C7×#95—22no—
35C7×#95—22no—
36#2315C2×C711yescharacteristic
37#1462C2×C712yes—
38#1305C2×C714yescomplemented
39C24×C7—22no—
40C24×C7—22no—
41C24×C7—22no—
42C24×C7—81no—
43C24×C7—81no—
44C24×C7C22⋊C411yescharacteristic
45C24×C7C22⋊C411yescharacteristic
46C24×C7C22×C411yescharacteristic
47C22×C4×C7—22no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7—42no—
51C22×C4×C7—42no—
52C22×C4×C7—81no—
53C22×C4×C7—81no—
54C22×C4×C7—81no—
55C22×C4×C7—81no—
56C22×C4×C7—81no—
57C22×C4×C7—81no—
58C22×C4×C7C22⋊C412yes—
59C22×C4×C7C2×D412yes—
60C22×C4×C7#4114yes—
61C7×M16—84no—
62C2×C7×C8—44no—
63C7×C4⋊C4—82no—
64C7×C4⋊C4—82no—
65C7×C22⋊C4—24no—
66C7×C22⋊C4—24no—
67C7×C22⋊C4—42no—
68C7×C22⋊C4—42no—
69C7×C22⋊C4—42no—
70C7×C22⋊C4—42no—
71C22×C22⋊C4C4×C711yescharacteristic
72C22×C22⋊C4C4×C711yescharacteristic
73C22×C22⋊C4C22×C711yescharacteristic
74#407C22×C714yes—
75#385—22no—
76#385—22no—
77#380C4×C712yes—
78#380C22×C712yes—
79#323—22no—
80#323—22no—
81C23×C7—28no—
82C23×C7—21no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7—41no—
89C23×C7—41no—
90C23×C7—41no—
91C23×C7—41no—
92C23×C7—41no—
93C23×C7—41no—
94C23×C7—82no—
95C23×C7—82no—
96C23×C7—84no—
97C23×C7—84no—
98C23×C7#10012yes—
99C23×C7#10112yes—
100C23×C7C2×C22⋊C411yescharacteristic
101C23×C7#11712yes—
102C2×C4×C7—22no—
103C2×C4×C7—42no—
104C2×C4×C7—44no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—82no—
113C2×C4×C7—84no—
114C2×C4×C7—84no—
115C2×C4×C7#12314yes—
116C7×C8—84no—
117C25C2×C4×C711yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×C4×C711yescharacteristic
121C23×C4C23×C711yescharacteristic
122C2×M16—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4C2×C4×C712yes—
131C2×C22⋊C4C2×C4×C712yes—
132C2×C22⋊C4C7×D412yes—
133C2×C22⋊C4C7×D412yes—
134C22⋊C8—24no—
135C22⋊C8—24no—
136#95—22no—
137#95—22no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7—22no—
141C22×C7—22no—
142C22×C7—22no—
143C22×C7—22no—
144C22×C7—22no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7—42no—
149C22×C7—42no—
150C22×C7—42no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—41no—
155C22×C7—88no—
156C22×C7#38111yescharacteristic
157C22×C7C2×#10011yescharacteristic
158C22×C7C2×#10111yescharacteristic
159C4×C7—24no—
160C4×C7—42no—
161C4×C7—82no—
162C4×C7—82no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—81no—
167C24—81no—
168C24C7×C22⋊C411yescharacteristic
169C24C7×C22⋊C411yescharacteristic
170C24C22×C4×C711yescharacteristic
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4C7×C22⋊C412yes—
183C22×C4C2×C7×D412yes—
184C22×C4C7×#4114yes—
185M16—84no—
186C2×C8—44no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7—42no—
198C2×C7—42no—
199C2×C7—42no—
200C2×C7#182111yescharacteristic
201C2×C7#182811yescharacteristic
202C2×C7#182911yescharacteristic
203C23—28no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—82no—
217C23—82no—
218C23—84no—
219C23—84no—
220C23C7×#10012yes—
221C23C7×#10112yes—
222C23C2×C7×C22⋊C411yescharacteristic
223C23C7×#11712yes—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4C7×#12314yes—
238C8—84no—
239C7#875611yescharacteristic, direct product factor, Sylow
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—88no—
258C22C7×#38111yescharacteristic
259C22C2×C7×#10011yescharacteristic
260C22C2×C7×#10111yescharacteristic
261C4—24no—
262C4—42no—
263C4—82no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—42no—
270C2C7×#182111yescharacteristic
271C2C7×#182811yescharacteristic
272C2C7×#182911yescharacteristic
subgroupG/H
center, socleC22×C7#381
commutatorC24C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7C2×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
3. C7×#1305C2semidirect
14. #8756C7direct (N × H)
38. #1305C2×C7semidirect
239. C7#8756direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23152
C7×#14622
C7×#13052
#87567