Finite Group Databasebeta

← C7×#9462 #13186678

Subgroups — C7×#9462 #13186678

Order 1792.

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

normal subgroups only

This group has 700 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×#2434C211yesmaximal, characteristic
2C7×#2310C211yesmaximal, characteristic
3C7×#1784C211yesmaximal, characteristic
4C7×#1511C212yesmaximal
5C7×#1284C212yesmaximal
6C23×C7×Q8C2211yescharacteristic
7C2×C7×C4⋊Q8C412yescomplemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#117—21no—
10C2×C7×#101—22no—
11C7×#407C2212yes—
12C7×#395—22no—
13C7×#391—24no—
14C7×#391C412yescomplemented
15C7×#385—21no—
16C7×#385—21no—
17C7×#381—21no—
18C7×#381C2212yes—
19C4×C7×C22⋊C4C2211yescharacteristic
20C7×#328—24no—
21#9462C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C411yescharacteristic
26C22×C7×Q8C2×C412yes—
27C22×C7×Q8D412yes—
28C23×C4×C7D411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C2×C7×M16—22no—
32C7×C4⋊Q8—44no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—22no—
35C7×#117—44no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—21no—
44C2×C42×C7—22no—
45C2×C42×C7C2×C412yes—
46C7×#101—44no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×#95—24no—
50C7×#95—22no—
51C7×#95—22no—
52C7×#95—21no—
53C7×#95—22no—
54C7×#95D414yes—
55#2434C2×C711yescharacteristic
56#2310C2×C711yescharacteristic
57#1784C2×C711yescharacteristic
58#1511C2×C712yes—
59#1284C2×C712yes—
60C24×C7C2×D411yescharacteristic
61C2×C7×Q8—24no—
62C2×C7×Q8—24no—
63C2×C7×Q8—24no—
64C2×C7×Q8—24no—
65C2×C7×Q8—22no—
66C2×C7×Q8—42no—
67C2×C7×Q8—42no—
68C2×C7×Q8—44no—
69C2×C7×Q8—44no—
70C2×C7×Q8—48no—
71C2×C7×Q8—42no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—24no—
75C22×C4×C7—22no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—24no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7C22⋊C412yes—
84C22×C4×C7C22⋊C412yes—
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D412yes—
88C22×C4×C7#4112yes—
89C7×M16—44no—
90C2×C7×C8—42no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—44no—
93C7×C4⋊C4—44no—
94C7×C4⋊C4—44no—
95C7×C4⋊C4—42no—
96C7×C22⋊C4—24no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C42×C7—42no—
101C42×C7—44no—
102C23×Q8C22×C711yescharacteristic
103C2×C4⋊Q8C4×C712yescomplemented
104C2×C22⋊Q8C22×C711yescharacteristic
105C2×#117—21no—
106C2×#101—22no—
107#407C22×C712yes—
108#395—22no—
109#391—24no—
110#391C4×C712yescomplemented
111#385—21no—
112#385—21no—
113#381—21no—
114#381C22×C712yes—
115C4×C22⋊C4C22×C711yescharacteristic
116#328—24no—
117C23×C7—22no—
118C23×C7—22no—
119C23×C7—41no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#12011yescharacteristic
122C23×C7#12311yescharacteristic
123C7×Q8—48no—
124C7×Q8—48no—
125C7×Q8—82no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—24no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—24no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—44no—
144C2×C4×C7—44no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—42no—
149C2×C4×C7—44no—
150C2×C4×C7#9914yes—
151C2×C4×C7C4×D412yes—
152C2×C4×C7#12112yes—
153C7×C8—82no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—44no—
157C22×Q8C2×C4×C711yescharacteristic
158C22×Q8C2×C4×C712yes—
159C22×Q8C7×D412yes—
160C23×C4C7×D411yescharacteristic
161C23×C4C7×D411yescharacteristic
162C23×C4C23×C711yescharacteristic
163C2×M16—22no—
164C4⋊Q8—44no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×C4×C711yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C42—22no—
177C2×C42C2×C4×C712yes—
178#101—44no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#95—24no—
182#95—22no—
183#95—22no—
184#95—21no—
185#95—22no—
186#95C7×D414yes—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—44no—
195C22×C7#38611yescharacteristic
196C22×C7C2×#9912yes—
197C22×C7#45612yes—
198C4×C7—22no—
199C4×C7—24no—
200C4×C7—22no—
201C4×C7—44no—
202C4×C7—42no—
203C4×C7—84no—
204C4×C7—81no—
205C4×C7—81no—
206C24C2×C7×D411yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—48no—
217C2×Q8—42no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C7×C22⋊C412yes—
230C22×C4C7×C22⋊C412yes—
231C22×C4C22×C4×C711yescharacteristic
232C22×C4C2×C7×D411yescharacteristic
233C22×C4C2×C7×D412yes—
234C22×C4C7×#4112yes—
235M16—44no—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C42—42no—
247C42—44no—
248C2×C7—22no—
249C2×C7—22no—
250C2×C7—41no—
251C2×C7#188611yescharacteristic
252C2×C7#189811yescharacteristic
253C2×C7#191211yescharacteristic
254C23—22no—
255C23—22no—
256C23—41no—
257C23C2×C7×C22⋊C411yescharacteristic
258C23C7×#12011yescharacteristic
259C23C7×#12311yescharacteristic
260Q8—48no—
261Q8—48no—
262Q8—82no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4C7×#9914yes—
288C2×C4C4×C7×D412yes—
289C2×C4C7×#12112yes—
290C8—82no—
291C7#946211yescharacteristic, direct product factor, Sylow
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22C7×#38611yescharacteristic
301C22C2×C7×#9912yes—
302C22C7×#45612yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—84no—
309C4—81no—
310C4—81no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2C7×#188611yescharacteristic
315C2C7×#189811yescharacteristic
316C2C7×#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
7. C2×C7×C4⋊Q8C4semidirect
14. C7×#391C4semidirect
21. #9462C7direct (N × H)
103. C2×C4⋊Q8C4×C7semidirect
110. #391C4×C7semidirect
291. C7#9462direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24342
C7×#23102
C7×#17842
C7×#15112
C7×#12842
#94627