Finite Group Databasebeta

← C7×#11566 #13188782

Subgroups — C7×#11566 #13188782

Order 1792.

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

normal subgroups only

This group has 494 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×#2544C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic, complemented
3C7×#1459C211yesmaximal, characteristic, complemented
4C7×#1319C212yesmaximal, complemented
5C7×#1318C212yesmaximal, complemented
6C2×C7×#127C411yescharacteristic
7C2×C7×#124C411yescharacteristic
8C2×C7×#124C411yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#407—21no—
11C2×C7×C22⋊C8—21no—
12C7×#394—21no—
13C7×#394—21no—
14C7×#391C2211yescharacteristic
15C7×#390C2211yescharacteristic
16C7×#386—21no—
17C7×#386—21no—
18C4×C7×C4⋊C4C411yescharacteristic
19C7×#324C2212yes—
20C7×#324C2212yes—
21C7×#323—21no—
22C7×#323—21no—
23#11566C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—41no—
32C22×C7×C8—41no—
33C7×#127—42no—
34C7×#124—41no—
35C7×#124—41no—
36C7×#124—42no—
37C4×C7×D4—41no—
38C4×C7×D4—41no—
39C2×C7×C4⋊C4—21no—
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⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C42×C7C2×C411yescharacteristic
50C2×C42×C7C2×C411yescharacteristic
51C2×C42×C7C2×C411yescharacteristic
52C7×C22⋊C8—22no—
53C7×C22⋊C8—22no—
54C7×C22⋊C8—41no—
55C7×C22⋊C8—41no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59#2544C2×C711yescharacteristic
60#1465C2×C711yescharacteristic, complemented
61#1459C2×C711yescharacteristic, complemented
62#1319C2×C712yescomplemented
63#1318C2×C712yescomplemented
64C24×C7—21no—
65C24×C7C22⋊C411yescharacteristic
66C24×C7C22⋊C411yescharacteristic
67C2×C7×Q8—22no—
68C2×C7×Q8—41no—
69C2×C7×Q8—41no—
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—22no—
73C2×C7×D4—41no—
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—81no—
81C2×C7×D4—81no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—81no—
90C22×C4×C7—81no—
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22×C411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C7×M16—82no—
97C2×C7×C8—42no—
98C2×C7×C8—42no—
99C2×C7×C8—82no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—42no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—81no—
110C7×C22⋊C4—81no—
111C42×C7—21no—
112C42×C7—21no—
113C42×C7—41no—
114C42×C7—41no—
115C2×#127C4×C711yescharacteristic
116C2×#124C4×C711yescharacteristic
117C2×#124C4×C711yescharacteristic
118C2×C4×D4C22×C711yescharacteristic
119#407—21no—
120C2×C22⋊C8—21no—
121#394—21no—
122#394—21no—
123#391C22×C711yescharacteristic
124#390C22×C711yescharacteristic
125#386—21no—
126#386—21no—
127C4×C4⋊C4C4×C711yescharacteristic
128#324C22×C712yes—
129#324C22×C712yes—
130#323—21no—
131#323—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7#10012yes—
149C23×C7C2×C22⋊C411yescharacteristic
150C7×Q8—82no—
151C7×D4—44no—
152C7×D4—82no—
153C7×D4—84no—
154C7×D4—84no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—22no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7#9911yescharacteristic
178C2×C4×C7#9911yescharacteristic
179C2×C4×C7#13112yes—
180C7×C8—82no—
181C7×C8—82no—
182C22×Q8C2×C4×C711yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C2×C4×C711yescharacteristic
186C22×D4C2×C4×C711yescharacteristic
187C23×C4C7×D411yescharacteristic
188C23×C4C7×D411yescharacteristic
189C2×M16—41no—
190C22×C8—41no—
191#127—42no—
192#124—41no—
193#124—41no—
194#124—42no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C23×C711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4C7×D411yescharacteristic
206C2×C22⋊C4C7×D411yescharacteristic
207C2×C42C2×C4×C711yescharacteristic
208C2×C42C2×C4×C711yescharacteristic
209C2×C42C2×C4×C711yescharacteristic
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214#95—21no—
215#95—21no—
216#95—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—22no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7—81no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7#40811yescharacteristic
237C22×C7C2×#9911yescharacteristic
238C22×C7C2×#10011yescharacteristic
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—22no—
242C4×C7—42no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—81no—
247C4×C7—81no—
248C24—21no—
249C24C7×C22⋊C411yescharacteristic
250C24C7×C22⋊C411yescharacteristic
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—81no—
265C2×D4—81no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4C7×C22⋊C411yescharacteristic
276C22×C4C7×C22⋊C411yescharacteristic
277C22×C4C22×C4×C711yescharacteristic
278C22×C4C2×C7×D411yescharacteristic
279C22×C4C2×C7×D411yescharacteristic
280M16—82no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C2×C7—21no—
300C2×C7—21no—
301C2×C7—42no—
302C2×C7—81no—
303C2×C7—161no—
304C2×C7#147111yescharacteristic
305C2×C7#212711yescharacteristic
306C2×C7#213211yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C7×#10012yes—
324C23C2×C7×C22⋊C411yescharacteristic
325Q8—82no—
326D4—44no—
327D4—82no—
328D4—84no—
329D4—84no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C7×#9911yescharacteristic
353C2×C4C7×#9911yescharacteristic
354C2×C4C7×#13112yes—
355C8—82no—
356C8—82no—
357C7#1156611yescharacteristic, direct product factor, Sylow
358C22—21no—
359C22—21no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22C7×#40811yescharacteristic
378C22C2×C7×#9911yescharacteristic
379C22C2×C7×#10011yescharacteristic
380C4—22no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—81no—
393C2—161no—
394C2C7×#147111yescharacteristic
395C2C7×#212711yescharacteristic
396C2C7×#213211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC2×C7×#127C4
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
2. C7×#1465C2semidirect
3. C7×#1459C2semidirect
4. C7×#1319C2semidirect
5. C7×#1318C2semidirect
23. #11566C7direct (N × H)
60. #1465C2×C7semidirect
61. #1459C2×C7semidirect
62. #1319C2×C7semidirect
63. #1318C2×C7semidirect
357. C7#11566direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25442
C7×#14652
C7×#14592
C7×#13192
C7×#13182
#115667