Finite Group Databasebeta

← C7×#9478 #13186694

Subgroups — C7×#9478 #13186694

Order 1792.

610 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2484C211yesmaximal, characteristic
2C7×#2274C211yesmaximal, characteristic
3C7×#1826C212yesmaximal
4C7×#1463C211yesmaximal, characteristic
5C7×#1284C212yesmaximal
6C2×C7×#117—21no—
7C2×C7×#117C2211yescharacteristic
8C22×C7×C4⋊C4C2211yescharacteristic
9C7×#407C2212yes—
10C7×#400C412yescomplemented
11C7×#398C412yescomplemented
12C7×#387—22no—
13C7×#385—21no—
14C7×#381—21no—
15C7×#381C2211yescharacteristic
16C4×C7×C4⋊C4—24no—
17C4×C7×C22⋊C4—21no—
18C4×C7×C22⋊C4C2212yes—
19C7×#339—24no—
20C7×#324—22no—
21#9478C711yesmaximal, characteristic, direct product factor, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C2×C7×M16—22no—
26C7×#117—28no—
27C7×#117—22no—
28C7×#117—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4D412yes—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C42×C7—22no—
41C2×C42×C7—21no—
42C2×C42×C7—22no—
43C2×C42×C7D414yes—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—22no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—22no—
51C7×#95—21no—
52C7×#95C2×C412yes—
53#2484C2×C711yescharacteristic
54#2274C2×C711yescharacteristic
55#1826C2×C712yes—
56#1463C2×C711yescharacteristic
57#1284C2×C712yes—
58C24×C7C2×D411yescharacteristic
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—24no—
62C22×C4×C7—24no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7C22⋊C412yes—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D412yes—
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7#4112yes—
76C7×M16—44no—
77C2×C7×C8—42no—
78C7×C4⋊C4—28no—
79C7×C4⋊C4—24no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—44no—
84C7×C22⋊C4—28no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C42×C7—28no—
89C42×C7—44no—
90C42×C7—42no—
91C42×C7—82no—
92C2×#117—21no—
93C2×#117C22×C711yescharacteristic
94C22×C4⋊C4C22×C711yescharacteristic
95#407C22×C712yes—
96#400C4×C712yescomplemented
97#398C4×C712yescomplemented
98#387—22no—
99#385—21no—
100#381—21no—
101#381C22×C711yescharacteristic
102C4×C4⋊C4—24no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4C22×C712yes—
105#339—24no—
106#324—22no—
107C23×C7—22no—
108C23×C7—22no—
109C23×C7—41no—
110C23×C7C2×C22⋊C411yescharacteristic
111C23×C7#12011yescharacteristic
112C23×C7#12311yescharacteristic
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—24no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—24no—
119C2×C4×C7—28no—
120C2×C4×C7—22no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—44no—
125C2×C4×C7—44no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—81no—
136C2×C4×C7#10414yes—
137C2×C4×C7C4×D412yes—
138C2×C4×C7#12112yes—
139C7×C8—82no—
140C23×C4C7×D411yescharacteristic
141C23×C4C7×D411yescharacteristic
142C23×C4C23×C711yescharacteristic
143C2×M16—22no—
144#117—28no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C2×C4×C711yescharacteristic
152C2×C4⋊C4C2×C4×C711yescharacteristic
153C2×C4⋊C4C2×C4×C712yes—
154C2×C4⋊C4C7×D412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42C7×D414yes—
162C22⋊C8—22no—
163C22⋊C8—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95C2×C4×C712yes—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—44no—
179C22×C7#38611yescharacteristic
180C22×C7C2×#10412yes—
181C22×C7#45812yes—
182C4×C7—22no—
183C4×C7—22no—
184C4×C7—24no—
185C4×C7—48no—
186C4×C7—42no—
187C4×C7—82no—
188C4×C7—81no—
189C4×C7—81no—
190C24C2×C7×D411yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4C7×C22⋊C412yes—
203C22×C4C7×C22⋊C412yes—
204C22×C4C22×C4×C711yescharacteristic
205C22×C4C2×C7×D412yes—
206C22×C4C2×C7×D411yescharacteristic
207C22×C4C7×#4112yes—
208M16—44no—
209C2×C8—42no—
210C4⋊C4—28no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—44no—
216C22⋊C4—28no—
217C22⋊C4—42no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C42—28no—
221C42—44no—
222C42—42no—
223C42—82no—
224C2×C7—22no—
225C2×C7—22no—
226C2×C7—41no—
227C2×C7#189011yescharacteristic
228C2×C7#189611yescharacteristic
229C2×C7#191211yescharacteristic
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×C7×C22⋊C411yescharacteristic
234C23C7×#12011yescharacteristic
235C23C7×#12311yescharacteristic
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—28no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4C7×#10414yes—
260C2×C4C4×C7×D412yes—
261C2×C4C7×#12112yes—
262C8—82no—
263C7#947811yescharacteristic, direct product factor, Sylow
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22C7×#38611yescharacteristic
273C22C2×C7×#10412yes—
274C22C7×#45812yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4—81no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2C7×#189011yescharacteristic
287C2C7×#189611yescharacteristic
288C2C7×#191211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
10. C7×#400C4semidirect
11. C7×#398C4semidirect
21. #9478C7direct (N × H)
96. #400C4×C7semidirect
97. #398C4×C7semidirect
263. C7#9478direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24842
C7×#22742
C7×#18262
C7×#14632
C7×#12842
#94787