Finite Group Databasebeta

← C7×#9445 #13186661

Subgroups — C7×#9445 #13186661

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2879C211yesmaximal, characteristic, complemented
2C7×#2431C211yesmaximal, characteristic
3C7×#1782C211yesmaximal, characteristic, complemented
4C7×#1510C212yesmaximal, complemented
5C7×#1284C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C22×C7×M16—21no—
8C2×C7×#127C411yescharacteristic, complemented
9C2×C7×#124C411yescharacteristic, complemented
10C2×C7×#121C2211yescharacteristic
11C2×C7×#99—21no—
12C2×C7×#99—21no—
13C7×#408—24no—
14C7×#407—21no—
15C7×#407C2212yes—
16C7×#392—21no—
17C7×#392—21no—
18C7×#391C411yescharacteristic, complemented
19C7×#390C411yescharacteristic, complemented
20C7×#386—21no—
21C7×#386—21no—
22C7×#381C2212yes—
23C4×C7×C22⋊C4C2211yescharacteristic
24C7×#327—24no—
25#9445C711yesmaximal, characteristic, direct product factor, Sylow
26C2×C7×#41—22no—
27C2×C7×#41—22no—
28C2×C7×#41—42no—
29C2×C7×#41—42no—
30C22×C7×Q8C2×C411yescharacteristic
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4C2×C411yescharacteristic
35C22×C7×D4C2×C411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C22×C7×D4D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C23×C4×C7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C2×C7×M16—22no—
42C2×C7×M16—22no—
43C2×C7×M16—42no—
44C2×C7×M16—42no—
45C22×C7×C8—22no—
46C7×#127—42no—
47C7×#124—42no—
48C7×#121—22no—
49C7×#121—22no—
50C2×C7×C4⋊C4C2×C411yescharacteristic
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4—21no—
58C2×C42×C7C2×C411yescharacteristic
59C2×C42×C7C2×C411yescharacteristic
60C7×#99—42no—
61C7×#99—42no—
62C7×C22⋊C8—24no—
63C7×C22⋊C8—22no—
64C7×C22⋊C8D414yescomplemented
65C7×#95—21no—
66C7×#95—21no—
67C7×#95—21no—
68C7×#95—22no—
69C7×#95—22no—
70#2879C2×C711yescharacteristic, complemented
71#2431C2×C711yescharacteristic
72#1782C2×C711yescharacteristic, complemented
73#1510C2×C712yescomplemented
74#1284C2×C712yescomplemented
75C24×C7—21no—
76C24×C7C22⋊C411yescharacteristic
77C24×C7C22⋊C411yescharacteristic
78C24×C7C2×D411yescharacteristic
79C7×#41—44no—
80C7×#41—44no—
81C7×#41—48no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8—41no—
85C2×C7×D4—22no—
86C2×C7×D4—22no—
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—42no—
96C2×C7×D4—42no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—42no—
100C2×C7×D4—42no—
101C2×C7×D4—42no—
102C2×C7×D4—81no—
103C2×C7×D4—81no—
104C2×C7×D4—81no—
105C2×C7×D4—81no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—41no—
116C22×C4×C7—41no—
117C22×C4×C7—41no—
118C22×C4×C7—41no—
119C22×C4×C7—42no—
120C22×C4×C7C22⋊C411yescharacteristic
121C22×C4×C7C22⋊C411yescharacteristic
122C22×C4×C7C22×C411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D412yes—
125C22×C4×C7#4112yes—
126C7×M16—44no—
127C7×M16—42no—
128C7×M16—42no—
129C2×C7×C8—24no—
130C2×C7×C8—44no—
131C7×C4⋊C4—22no—
132C7×C22⋊C4—42no—
133C7×C22⋊C4—42no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C7×C22⋊C4—42no—
138C7×C22⋊C4—42no—
139C7×C22⋊C4—81no—
140C7×C22⋊C4—81no—
141C42×C7—41no—
142C42×C7—41no—
143C22×#41C22×C711yescharacteristic
144C22×M16—21no—
145C2×#127C4×C711yescharacteristic, complemented
146C2×#124C4×C711yescharacteristic, complemented
147C2×#121C22×C711yescharacteristic
148C2×#99—21no—
149C2×#99—21no—
150#408—24no—
151#407—21no—
152#407C22×C712yes—
153#392—21no—
154#392—21no—
155#391C4×C711yescharacteristic, complemented
156#390C4×C711yescharacteristic, complemented
157#386—21no—
158#386—21no—
159#381C22×C712yes—
160C4×C22⋊C4C22×C711yescharacteristic
161#327—24no—
162C23×C7—21no—
163C23×C7—21no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7—21no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7—41no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7#9912yes—
183C23×C7C2×C22⋊C411yescharacteristic
184C23×C7#12112yes—
185C7×Q8—44no—
186C7×D4—44no—
187C7×D4—44no—
188C7×D4—44no—
189C7×D4—82no—
190C7×D4—84no—
191C7×D4—84no—
192C7×D4—84no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—22no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—82no—
219C2×C4×C7—82no—
220C2×C4×C7—82no—
221C2×C4×C7#9912yes—
222C2×C4×C7C4×D412yes—
223C2×C4×C7#12011yescharacteristic
224C2×C4×C7#12311yescharacteristic
225C7×C8—44no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—42no—
229C2×#41—42no—
230C22×Q8C2×C4×C711yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2×C4×C711yescharacteristic
235C22×D4C2×C4×C711yescharacteristic
236C22×D4C7×D411yescharacteristic
237C22×D4C7×D411yescharacteristic
238C23×C4C7×D411yescharacteristic
239C23×C4C7×D411yescharacteristic
240C23×C4C23×C711yescharacteristic
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—42no—
244C2×M16—42no—
245C22×C8—22no—
246#127—42no—
247#124—42no—
248#121—22no—
249#121—22no—
250C2×C4⋊C4C2×C4×C711yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C42C2×C4×C711yescharacteristic
259C2×C42C2×C4×C711yescharacteristic
260#99—42no—
261#99—42no—
262C22⋊C8—24no—
263C22⋊C8—22no—
264C22⋊C8C7×D414yescomplemented
265#95—21no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—22no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—22no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7—22no—
276C22×C7—42no—
277C22×C7—42no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7—41no—
281C22×C7—41no—
282C22×C7—41no—
283C22×C7—81no—
284C22×C7—81no—
285C22×C7—81no—
286C22×C7—81no—
287C22×C7—81no—
288C22×C7—81no—
289C22×C7—81no—
290C22×C7—81no—
291C22×C7—81no—
292C22×C7—81no—
293C22×C7—81no—
294C22×C7#38611yescharacteristic
295C22×C7C2×#9911yescharacteristic
296C22×C7C2×#9911yescharacteristic
297C4×C7—21no—
298C4×C7—21no—
299C4×C7—22no—
300C4×C7—22no—
301C4×C7—42no—
302C4×C7—41no—
303C4×C7—81no—
304C4×C7—81no—
305C4×C7—162no—
306C24—21no—
307C24C7×C22⋊C411yescharacteristic
308C24C7×C22⋊C411yescharacteristic
309C24C2×C7×D411yescharacteristic
310#41—44no—
311#41—44no—
312#41—48no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—81no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4—81no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4C7×C22⋊C411yescharacteristic
352C22×C4C7×C22⋊C411yescharacteristic
353C22×C4C22×C4×C711yescharacteristic
354C22×C4C2×C7×D411yescharacteristic
355C22×C4C2×C7×D412yes—
356C22×C4C7×#4112yes—
357M16—44no—
358M16—42no—
359M16—42no—
360C2×C8—24no—
361C2×C8—44no—
362C4⋊C4—22no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—81no—
371C22⋊C4—81no—
372C42—41no—
373C42—41no—
374C2×C7—21no—
375C2×C7—21no—
376C2×C7—42no—
377C2×C7—42no—
378C2×C7—81no—
379C2×C7—161no—
380C2×C7#188411yescharacteristic
381C2×C7#189811yescharacteristic
382C2×C7#191011yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C7×#9912yes—
404C23C2×C7×C22⋊C411yescharacteristic
405C23C7×#12112yes—
406Q8—44no—
407D4—44no—
408D4—44no—
409D4—44no—
410D4—82no—
411D4—84no—
412D4—84no—
413D4—84no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4C7×#9912yes—
443C2×C4C4×C7×D412yes—
444C2×C4C7×#12011yescharacteristic
445C2×C4C7×#12311yescharacteristic
446C8—44no—
447C7#944511yescharacteristic, direct product factor, Sylow
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22C7×#38611yescharacteristic
473C22C2×C7×#9911yescharacteristic
474C22C2×C7×#9911yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—162no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—81no—
489C2—161no—
490C2C7×#188411yescharacteristic
491C2C7×#189811yescharacteristic
492C2C7×#191011yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#2431C2
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
1. C7×#2879C2semidirect
3. C7×#1782C2semidirect
4. C7×#1510C2semidirect
5. C7×#1284C2semidirect
8. C2×C7×#127C4semidirect
9. C2×C7×#124C4semidirect
18. C7×#391C4semidirect
19. C7×#390C4semidirect
25. #9445C7direct (N × H)
64. C7×C22⋊C8D4semidirect
70. #2879C2×C7semidirect
72. #1782C2×C7semidirect
73. #1510C2×C7semidirect
74. #1284C2×C7semidirect
145. C2×#127C4×C7semidirect
146. C2×#124C4×C7semidirect
155. #391C4×C7semidirect
156. #390C4×C7semidirect
264. C22⋊C8C7×D4semidirect
447. C7#9445direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28792
C7×#24312
C7×#17822
C7×#15102
C7×#12842
#94457