Finite Group Databasebeta

← C7×#11572 #13188788

Subgroups — C7×#11572 #13188788

Order 1792.

416 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2566C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1325C212yesmaximal, complemented
5C7×#1324C212yesmaximal, complemented
6C2×C7×#125C411yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#396—21no—
11C7×#396—21no—
12C7×#390C411yescharacteristic
13C7×#390C411yescharacteristic
14C7×#385—21no—
15C7×#385—21no—
16C7×#384C2211yescharacteristic
17C7×#382C2211yescharacteristic
18C4×C7×C4⋊C4C411yescharacteristic
19C7×#324C2212yes—
20C7×#324C2212yes—
21C7×#323—21no—
22C7×#323—21no—
23#11572C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C2×C7×M16—41no—
28C22×C7×C8—41no—
29C7×#125—41no—
30C7×#125—41no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4C2311yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C42×C7C2×C411yescharacteristic
48C2×C42×C7C2×C411yescharacteristic
49C2×C42×C7C2×C411yescharacteristic
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×C22⋊C8—41no—
53C7×C22⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57#2566C2×C711yescharacteristic
58#1465C2×C711yescharacteristic
59#1459C2×C711yescharacteristic
60#1325C2×C712yescomplemented
61#1324C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C24×C7C22⋊C411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—81no—
76C22×C4×C7—81no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×M16—82no—
83C2×C7×C8—42no—
84C2×C7×C8—42no—
85C2×C7×C8—82no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—81no—
94C7×C4⋊C4—81no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—81no—
100C7×C22⋊C4—81no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—22no—
104C42×C7—22no—
105C2×#125C4×C711yescharacteristic
106C2×C4×D4C22×C711yescharacteristic
107#407—21no—
108C2×C22⋊C8—21no—
109#396—21no—
110#396—21no—
111#390C4×C711yescharacteristic
112#390C4×C711yescharacteristic
113#385—21no—
114#385—21no—
115#384C22×C711yescharacteristic
116#382C22×C711yescharacteristic
117C4×C4⋊C4C4×C711yescharacteristic
118#324C22×C712yes—
119#324C22×C712yes—
120#323—21no—
121#323—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7#10112yes—
132C23×C7C2×C22⋊C411yescharacteristic
133C7×D4—44no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—41no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—81no—
158C2×C4×C7—81no—
159C2×C4×C7#9911yescharacteristic
160C2×C4×C7#9911yescharacteristic
161C2×C4×C7#13112yes—
162C7×C8—82no—
163C7×C8—82no—
164C22×D4C2×C4×C711yescharacteristic
165C23×C4C7×D411yescharacteristic
166C23×C4C7×D411yescharacteristic
167C2×M16—41no—
168C22×C8—41no—
169#125—41no—
170#125—41no—
171C4×D4—41no—
172C4×D4—41no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C2×C4×C711yescharacteristic
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C4⋊C4C23×C711yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4C7×D411yescharacteristic
186C2×C22⋊C4C7×D411yescharacteristic
187C2×C42C2×C4×C711yescharacteristic
188C2×C42C2×C4×C711yescharacteristic
189C2×C42C2×C4×C711yescharacteristic
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#95—21no—
195#95—21no—
196#95—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—22no—
200C22×C7—42no—
201C22×C7—42no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7#40811yescharacteristic
210C22×C7C2×#9911yescharacteristic
211C22×C7C2×#10111yescharacteristic
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—22no—
215C4×C7—42no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—81no—
220C4×C7—81no—
221C4×C7—81no—
222C24C7×C22⋊C411yescharacteristic
223C24C7×C22⋊C411yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—41no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4C7×C22⋊C411yescharacteristic
238C22×C4C7×C22⋊C411yescharacteristic
239C22×C4C22×C4×C711yescharacteristic
240C22×C4C2×C7×D411yescharacteristic
241C22×C4C2×C7×D411yescharacteristic
242M16—82no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—82no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42—22no—
265C2×C7—21no—
266C2×C7—21no—
267C2×C7—42no—
268C2×C7—81no—
269C2×C7#147211yescharacteristic
270C2×C7#212711yescharacteristic
271C2×C7#213211yescharacteristic
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C7×#10112yes—
282C23C2×C7×C22⋊C411yescharacteristic
283D4—44no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C7×#9911yescharacteristic
310C2×C4C7×#9911yescharacteristic
311C2×C4C7×#13112yes—
312C8—82no—
313C8—82no—
314C7#1157211yescharacteristic, direct product factor, Sylow
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22C7×#40811yescharacteristic
328C22C2×C7×#9911yescharacteristic
329C22C2×C7×#10111yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2C7×#147211yescharacteristic
345C2C7×#212711yescharacteristic
346C2C7×#213211yescharacteristic
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×#1325C2semidirect
5. C7×#1324C2semidirect
23. #11572C7direct (N × H)
60. #1325C2×C7semidirect
61. #1324C2×C7semidirect
314. C7#11572direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25662
C7×#14652
C7×#14592
C7×#13252
C7×#13242
#115727