Finite Group Databasebeta

← C7×#9460 #13186676

Subgroups — C7×#9460 #13186676

Order 1792.

512 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2503C211yesmaximal, characteristic
2C7×#2313C211yesmaximal, characteristic
3C7×#1821C211yesmaximal, characteristic
4C7×#1511C212yesmaximal
5C7×#1284C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407C2212yes—
9C7×#400—22no—
10C7×#391C412yescomplemented
11C7×#387—24no—
12C7×#382C412yescomplemented
13C7×#381—21no—
14C7×#381—21no—
15C7×#381C2212yes—
16C7×#380—21no—
17C4×C7×C22⋊C4—21no—
18C4×C7×C22⋊C4C2211yescharacteristic
19C7×#343—22no—
20C7×#328—24no—
21#9460C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C2×C7×M16—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C412yes—
35C2×C7×C4⋊C4D412yes—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C42×C7—22no—
40C2×C42×C7C2×C412yes—
41C7×C22⋊C8—22no—
42C7×C22⋊C8—22no—
43C7×#95—22no—
44C7×#95—22no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—21no—
50C7×#95—22no—
51C7×#95D414yes—
52#2503C2×C711yescharacteristic
53#2313C2×C711yescharacteristic
54#1821C2×C711yescharacteristic
55#1511C2×C712yes—
56#1284C2×C712yes—
57C24×C7C2×D411yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—42no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—24no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D412yes—
76C22×C4×C7#4112yes—
77C7×M16—44no—
78C2×C7×C8—42no—
79C7×C4⋊C4—24no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—82no—
84C7×C22⋊C4—44no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C42×C7—42no—
89C42×C7—82no—
90C2×C22⋊Q8C22×C711yescharacteristic
91C22×C4⋊C4C22×C711yescharacteristic
92#407C22×C712yes—
93#400—22no—
94#391C4×C712yescomplemented
95#387—24no—
96#382C4×C712yescomplemented
97#381—21no—
98#381—21no—
99#381C22×C712yes—
100#380—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4C22×C711yescharacteristic
103#343—22no—
104#328—24no—
105C23×C7—22no—
106C23×C7—22no—
107C23×C7—41no—
108C23×C7C2×C22⋊C411yescharacteristic
109C23×C7#12011yescharacteristic
110C23×C7#12311yescharacteristic
111C7×Q8—82no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—44no—
124C2×C4×C7—42no—
125C2×C4×C7—44no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—44no—
132C2×C4×C7—81no—
133C2×C4×C7—81no—
134C2×C4×C7C4×D412yes—
135C2×C4×C7#12112yes—
136C7×C8—82no—
137C22×Q8C2×C4×C711yescharacteristic
138C23×C4C7×D411yescharacteristic
139C23×C4C7×D411yescharacteristic
140C23×C4C23×C711yescharacteristic
141C2×M16—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C2×C4×C711yescharacteristic
149C2×C4⋊C4C2×C4×C712yes—
150C2×C4⋊C4C7×D412yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C42—22no—
155C2×C42C2×C4×C712yes—
156C22⋊C8—22no—
157C22⋊C8—22no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—21no—
165#95—22no—
166#95C7×D414yes—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—44no—
175C22×C7#38611yescharacteristic
176C22×C7#41012yes—
177C22×C7#45512yes—
178C4×C7—22no—
179C4×C7—22no—
180C4×C7—42no—
181C4×C7—42no—
182C4×C7—84no—
183C4×C7—82no—
184C4×C7—81no—
185C4×C7—81no—
186C24C2×C7×D411yescharacteristic
187C2×Q8—22no—
188C2×Q8—42no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C7×C22⋊C412yes—
201C22×C4C7×C22⋊C412yes—
202C22×C4C22×C4×C711yescharacteristic
203C22×C4C2×C7×D411yescharacteristic
204C22×C4C2×C7×D412yes—
205C22×C4C7×#4112yes—
206M16—44no—
207C2×C8—42no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C22⋊C4—44no—
214C22⋊C4—42no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—42no—
218C42—82no—
219C2×C7—22no—
220C2×C7—22no—
221C2×C7—41no—
222C2×C7#188511yescharacteristic
223C2×C7#189911yescharacteristic
224C2×C7#191211yescharacteristic
225C23—22no—
226C23—22no—
227C23—41no—
228C23C2×C7×C22⋊C411yescharacteristic
229C23C7×#12011yescharacteristic
230C23C7×#12311yescharacteristic
231Q8—82no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4C4×C7×D412yes—
255C2×C4C7×#12112yes—
256C8—82no—
257C7#946011yescharacteristic, direct product factor, Sylow
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22C7×#38611yescharacteristic
267C22C7×#41012yes—
268C22C7×#45512yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—84no—
274C4—82no—
275C4—81no—
276C4—81no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2C7×#188511yescharacteristic
281C2C7×#189911yescharacteristic
282C2C7×#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×#391C4semidirect
12. C7×#382C4semidirect
21. #9460C7direct (N × H)
94. #391C4×C7semidirect
96. #382C4×C7semidirect
257. C7#9460direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25032
C7×#23132
C7×#18212
C7×#15112
C7×#12842
#94607