Finite Group Databasebeta

← C7×#11569 #13188785

Subgroups — C7×#11569 #13188785

Order 1792.

424 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2563C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1324C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C4×C7×D4C2211yescharacteristic
7C7×#407—21no—
8C2×C7×C22⋊C8—21no—
9C7×#393—21no—
10C7×#393—21no—
11C7×#390C412yes—
12C7×#382C2212yes—
13C7×#380—21no—
14C7×#380—21no—
15C4×C7×C4⋊C4C411yescharacteristic
16C7×#324C2212yes—
17C7×#324C2212yes—
18C7×#323—21no—
19C7×#323—21no—
20#11569C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C2×C7×M16—41no—
26C22×C7×C8—41no—
27C7×C4⋊Q8—41no—
28C7×C4⋊Q8—41no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C42×C7C2×C412yes—
40C2×C42×C7C2×C411yescharacteristic
41C7×C22⋊C8—22no—
42C7×C22⋊C8—22no—
43C7×C22⋊C8—41no—
44C7×C22⋊C8—41no—
45C7×#95—21no—
46C7×#95—22no—
47C7×#95—22no—
48#2563C2×C711yescharacteristic
49#1465C2×C711yescharacteristic
50#1459C2×C711yescharacteristic
51#1324C2×C714yescomplemented
52C24×C7C22⋊C411yescharacteristic
53C24×C7C22⋊C411yescharacteristic
54C2×C7×Q8—42no—
55C2×C7×Q8—82no—
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4—41no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C7×M16—82no—
73C2×C7×C8—42no—
74C2×C7×C8—42no—
75C2×C7×C8—82no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—82no—
84C7×C22⋊C4—82no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—24no—
88C2×C4⋊Q8C4×C711yescharacteristic
89C2×C4×D4C22×C711yescharacteristic
90#407—21no—
91C2×C22⋊C8—21no—
92#393—21no—
93#393—21no—
94#390C4×C712yes—
95#382C22×C712yes—
96#380—21no—
97#380—21no—
98C4×C4⋊C4C4×C711yescharacteristic
99#324C22×C712yes—
100#324C22×C712yes—
101#323—21no—
102#323—21no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—41no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7#10112yes—
113C23×C7C2×C22⋊C411yescharacteristic
114C7×Q8—84no—
115C7×D4—44no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—81no—
134C2×C4×C7—81no—
135C2×C4×C7#9911yescharacteristic
136C2×C4×C7#9911yescharacteristic
137C2×C4×C7#13112yes—
138C7×C8—82no—
139C7×C8—82no—
140C22×Q8—21no—
141C22×D4C2×C4×C711yescharacteristic
142C23×C4C7×D411yescharacteristic
143C23×C4C7×D411yescharacteristic
144C2×M16—41no—
145C22×C8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4×D4—41no—
149C4×D4—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4C2×C4×C712yes—
153C2×C4⋊C4C23×C711yescharacteristic
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4C7×D411yescharacteristic
157C2×C22⋊C4C7×D411yescharacteristic
158C2×C42C2×C4×C712yes—
159C2×C42C2×C4×C711yescharacteristic
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#95—21no—
165#95—22no—
166#95—22no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7—22no—
170C22×C7—42no—
171C22×C7—42no—
172C22×C7—41no—
173C22×C7—41no—
174C22×C7—41no—
175C22×C7—41no—
176C22×C7—41no—
177C22×C7—81no—
178C22×C7—81no—
179C22×C7#40811yescharacteristic
180C22×C7C2×#9911yescharacteristic
181C22×C7C2×#10111yescharacteristic
182C4×C7—21no—
183C4×C7—22no—
184C4×C7—41no—
185C4×C7—41no—
186C4×C7—41no—
187C4×C7—82no—
188C4×C7—82no—
189C4×C7#35412yes—
190C24C7×C22⋊C411yescharacteristic
191C24C7×C22⋊C411yescharacteristic
192C2×Q8—42no—
193C2×Q8—82no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—41no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4C7×C22⋊C411yescharacteristic
206C22×C4C7×C22⋊C411yescharacteristic
207C22×C4C22×C4×C711yescharacteristic
208C22×C4C2×C7×D411yescharacteristic
209C22×C4C2×C7×D411yescharacteristic
210M16—82no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—82no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C42—21no—
224C42—21no—
225C42—24no—
226C2×C7—21no—
227C2×C7—21no—
228C2×C7—42no—
229C2×C7—81no—
230C2×C7#147211yescharacteristic
231C2×C7C2×#35411yescharacteristic
232C2×C7#213411yescharacteristic
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23C7×#10112yes—
243C23C2×C7×C22⋊C411yescharacteristic
244Q8—84no—
245D4—44no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4C7×#9911yescharacteristic
266C2×C4C7×#9911yescharacteristic
267C2×C4C7×#13112yes—
268C8—82no—
269C8—82no—
270C7#1156911yescharacteristic, direct product factor, Sylow
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22C7×#40811yescharacteristic
284C22C2×C7×#9911yescharacteristic
285C22C2×C7×#10111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4C7×#35412yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2C7×#147211yescharacteristic
299C2C2×C7×#35411yescharacteristic
300C2C7×#213411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1324C2semidirect
20. #11569C7direct (N × H)
51. #1324C2×C7semidirect
270. C7#11569direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25632
C7×#14652
C7×#14592
C7×#13242
#115697