Finite Group Databasebeta

← C7×#9481 #13186697

Subgroups — C7×#9481 #13186697

Order 1792.

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

normal subgroups only

This group has 614 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×#2482C211yesmaximal, characteristic
2C7×#2312C211yesmaximal, characteristic
3C7×#1826C212yesmaximal, complemented
4C7×#1462C211yesmaximal, characteristic
5C7×#1284C212yesmaximal, complemented
6C2×C7×#117C2211yescharacteristic
7C22×C7×C22⋊C4C2211yescharacteristic
8C7×#407C2212yes—
9C7×#399C412yescomplemented
10C7×#397C412yescomplemented
11C7×#388—22no—
12C7×#388—24no—
13C7×#385—21no—
14C7×#381C2211yescharacteristic
15C7×#380—21no—
16C7×#379—21no—
17C7×#379—21no—
18C4×C7×C22⋊C4C2212yes—
19C7×#339—24no—
20C7×#323—22no—
21#9481C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C2×C7×M16—22no—
26C7×#117—22no—
27C7×#117—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4C2×C412yes—
42C2×C7×C22⋊C4D412yes—
43C2×C42×C7—21no—
44C2×C42×C7D414yes—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—21no—
51C7×#95C2×C412yes—
52#2482C2×C711yescharacteristic
53#2312C2×C711yescharacteristic
54#1826C2×C712yescomplemented
55#1462C2×C711yescharacteristic
56#1284C2×C712yescomplemented
57C24×C7—21no—
58C24×C7—21no—
59C24×C7—42no—
60C24×C7—42no—
61C24×C7—81no—
62C24×C7C22⋊C412yes—
63C24×C7C2×D411yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—24no—
70C22×C4×C7—42no—
71C22×C4×C7—81no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7#4112yes—
77C7×M16—44no—
78C2×C7×C8—42no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—82no—
82C7×C4⋊C4—82no—
83C7×C22⋊C4—24no—
84C7×C22⋊C4—24no—
85C7×C22⋊C4—44no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—44no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—82no—
91C7×C22⋊C4—82no—
92C7×C22⋊C4—82no—
93C42×C7—44no—
94C42×C7—42no—
95C2×#117C22×C711yescharacteristic
96C22×C22⋊C4C22×C711yescharacteristic
97#407C22×C712yes—
98#399C4×C712yescomplemented
99#397C4×C712yescomplemented
100#388—22no—
101#388—24no—
102#385—21no—
103#381C22×C711yescharacteristic
104#380—21no—
105#379—21no—
106#379—21no—
107C4×C22⋊C4C22×C712yes—
108#339—24no—
109#323—22no—
110C23×C7—22no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—41no—
116C23×C7—42no—
117C23×C7—42no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—81no—
123C23×C7—81no—
124C23×C7—88no—
125C23×C7C2×C22⋊C411yescharacteristic
126C23×C7#12011yescharacteristic
127C23×C7#12311yescharacteristic
128C2×C4×C7—24no—
129C2×C4×C7—22no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—44no—
137C2×C4×C7—42no—
138C2×C4×C7—44no—
139C2×C4×C7—42no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—44no—
145C2×C4×C7—81no—
146C2×C4×C7C4×D412yes—
147C2×C4×C7#12112yes—
148C7×C8—82no—
149C25C7×D411yescharacteristic
150C23×C4C7×D411yescharacteristic
151C23×C4C23×C711yescharacteristic
152C2×M16—22no—
153#117—22no—
154#117—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C2×C4×C711yescharacteristic
158C2×C4⋊C4C2×C4×C711yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C2×C4×C712yes—
169C2×C22⋊C4C7×D412yes—
170C2×C42—21no—
171C2×C42C7×D414yes—
172C22⋊C8—22no—
173C22⋊C8—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—21no—
178#95C2×C4×C712yes—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—44no—
187C22×C7—44no—
188C22×C7—42no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7—82no—
192C22×C7—84no—
193C22×C7—84no—
194C22×C7#38611yescharacteristic
195C22×C7#42112yes—
196C22×C7#45812yes—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—42no—
200C4×C7—84no—
201C4×C7—82no—
202C4×C7—81no—
203C4×C7—81no—
204C24—21no—
205C24—21no—
206C24—42no—
207C24—42no—
208C24—81no—
209C24C7×C22⋊C412yes—
210C24C2×C7×D411yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4C7×C22⋊C412yes—
220C22×C4C22×C4×C711yescharacteristic
221C22×C4C2×C7×D412yes—
222C22×C4C2×C7×D411yescharacteristic
223C22×C4C7×#4112yes—
224M16—44no—
225C2×C8—42no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C42—44no—
241C42—42no—
242C2×C7—22no—
243C2×C7—22no—
244C2×C7—41no—
245C2×C7—82no—
246C2×C7#189011yescharacteristic
247C2×C7#189711yescharacteristic
248C2×C7#191111yescharacteristic
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—81no—
262C23—81no—
263C23—88no—
264C23C2×C7×C22⋊C411yescharacteristic
265C23C7×#12011yescharacteristic
266C23C7×#12311yescharacteristic
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—81no—
285C2×C4C4×C7×D412yes—
286C2×C4C7×#12112yes—
287C8—82no—
288C7#948111yescharacteristic, direct product factor, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22C7×#38611yescharacteristic
305C22C7×#42112yes—
306C22C7×#45812yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—84no—
311C4—82no—
312C4—81no—
313C4—81no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2C7×#189011yescharacteristic
319C2C7×#189711yescharacteristic
320C2C7×#191111yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7D4
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
3. C7×#1826C2semidirect
5. C7×#1284C2semidirect
9. C7×#399C4semidirect
10. C7×#397C4semidirect
21. #9481C7direct (N × H)
54. #1826C2×C7semidirect
56. #1284C2×C7semidirect
98. #399C4×C7semidirect
99. #397C4×C7semidirect
288. C7#9481direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24822
C7×#23122
C7×#18262
C7×#14622
C7×#12842
#94817