Finite Group Databasebeta

← C7×#19016 #13196232

Subgroups — C7×#19016 #13196232

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic, complemented
3C7×#3151C211yesmaximal, characteristic, complemented
4C2×C7×#490C211yesmaximal, characteristic, complemented
5C7×#1657C214yesmaximal, complemented
6C7×#1656C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C2×C7×AGL(1,ℤ/8)—22no—
11C7×#563C2212yescomplemented
12C7×#562C2212yescomplemented
13C7×#554—21no—
14C7×#553—21no—
15C7×#553—21no—
16C7×#547—22no—
17C7×#547—21no—
18C7×#544—21no—
19C7×#540—21no—
20C7×#539—22no—
21C7×#539—21no—
22C7×#537—21no—
23C7×#532C2211yescharacteristic
24C2×C7×#127C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C7×#496C2212yes—
28C7×#496C2214yes—
29C7×#493—21no—
30C7×#492—21no—
31C7×#490—22no—
32C7×#490C2212yes—
33C7×#490C2214yes—
34C7×#460—21no—
35C7×#459—21no—
36C7×#450—21no—
37C7×#447—21no—
38C7×#432C2211yescharacteristic
39C2×C7×#102—21no—
40C2×C7×#102—21no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yescomplemented
44C7×#330C2214yescomplemented
45C7×#329C2212yescomplemented
46C7×#329C2214yescomplemented
47#19016C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41D412yes—
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C22×C7×D4—21no—
53C7×AGL(1,ℤ/8)—48no—
54C2×C7×SD16—24no—
55C2×C7×SD16—22no—
56C2×C7×D8—24no—
57C2×C7×D8—22no—
58C2×C7×M16—21no—
59C22×C7×C8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8D411yescharacteristic
63C7×#127—21no—
64C7×#127—21no—
65C7×#127—21no—
66C7×#127D411yescharacteristic
67C7×#127C2312yes—
68C7×#126—22no—
69C7×#126—22no—
70C7×#126—21no—
71C7×#125—22no—
72C7×#125—21no—
73C7×#125—21no—
74C7×#125—21no—
75C7×#125D412yes—
76C7×#125C2312yes—
77C7×#125C2311yescharacteristic
78C7×#124C2311yescharacteristic
79C7×#124C2312yes—
80C7×#123—22no—
81C7×#123—22no—
82C7×#123—22no—
83C7×#123—22no—
84C7×#123—21no—
85C7×#123—22no—
86C7×#123—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8D411yescharacteristic
94C7×#121—22no—
95C7×#121—22no—
96C7×#121—21no—
97C7×#121—21no—
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—21no—
101C7×#121—21no—
102C7×#121D411yescharacteristic
103C4×C7×Q8D411yescharacteristic
104C4×C7×D4—22no—
105C4×C7×D4—22no—
106C4×C7×D4—21no—
107C4×C7×D4—21no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4—21no—
111C4×C7×D4D411yescharacteristic
112C7×#117—21no—
113C7×#117—21no—
114C2×C7×C4⋊C4—22no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C4⋊C4—21no—
117C2×C7×C4⋊C4—21no—
118C2×C7×C4⋊C4D411yescharacteristic
119C2×C7×C4⋊C4D411yescharacteristic
120C2×C42×C7C2311yescharacteristic
121C7×C4⋊C8—21no—
122C7×#102—24no—
123C7×#102—24no—
124C7×#102—22no—
125C7×#102—22no—
126C7×C22⋊C8—21no—
127C7×#97—21no—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C7×#97C2312yes—
131C4×C7×C8—21no—
132#3559C2×C711yescharacteristic, complemented
133#3217C2×C711yescharacteristic, complemented
134#3151C2×C711yescharacteristic, complemented
135C2×#490C2×C711yescharacteristic, complemented
136#1657C2×C714yescomplemented
137#1656C2×C714yescomplemented
138#1530C2×C711yescharacteristic, complemented
139#1528C2×C711yescharacteristic, complemented
140#1524C2×C711yescharacteristic, complemented
141C24×C7—41no—
142C7×#41—44no—
143C7×#41—84no—
144C2×C7×Q8—22no—
145C2×C7×Q8C2×D411yescharacteristic
146C2×C7×Q8C2×D412yes—
147C2×C7×D4—22no—
148C2×C7×D4—21no—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—44no—
157C2×C7×D4—44no—
158C2×C7×D4—41no—
159C2×C7×D4—41no—
160C2×C7×D4—41no—
161C2×C7×D4—82no—
162C2×C7×D4C2×D411yescharacteristic
163C2×C7×D4C2×D412yes—
164C22×C4×C7—21no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C7×SD16—48no—
175C7×SD16—82no—
176C7×D8—48no—
177C7×D8—82no—
178C7×M16—24no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C2×C7×C8—42no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4C2×D412yes—
200C7×C4⋊C4C2×D412yes—
201C7×C4⋊C4C2×D411yescharacteristic
202C7×C4⋊C4C2×D411yescharacteristic
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—21no—
209C7×C22⋊C4—21no—
210C7×C22⋊C4—21no—
211C7×C22⋊C4—21no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—21no—
215C42×C7—22no—
216C42×C7—21no—
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219C42×C7C2×D411yescharacteristic
220C42×C7C2411yescharacteristic
221C2×AGL(1,ℤ/8)—22no—
222#563C22×C712yescomplemented
223#562C22×C712yescomplemented
224#554—21no—
225#553—21no—
226#553—21no—
227#547—22no—
228#547—21no—
229#544—21no—
230#540—21no—
231#539—22no—
232#539—21no—
233#537—21no—
234#532C22×C711yescharacteristic
235C2×#127C22×C711yescharacteristic
236#529C22×C711yescharacteristic
237C2×#125C22×C711yescharacteristic
238#496C22×C714yes—
239#496C22×C712yes—
240#493—21no—
241#492—21no—
242#490—22no—
243#490C22×C714yes—
244#490C22×C712yes—
245#460—21no—
246#459—21no—
247#450—21no—
248#447—21no—
249#432C22×C711yescharacteristic
250C2×#102—21no—
251C2×#102—21no—
252C4×M16C22×C711yescharacteristic
253C2×#97C22×C711yescharacteristic
254#330C22×C714yescomplemented
255#330C22×C712yescomplemented
256#329C22×C712yescomplemented
257#329C22×C714yescomplemented
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—42no—
261C23×C7—82no—
262C23×C7—82no—
263C23×C7—82no—
264C23×C7#12011yescharacteristic
265C7×Q8—42no—
266C7×Q8—41no—
267C7×Q8—82no—
268C7×D4—42no—
269C7×D4—41no—
270C7×D4—44no—
271C7×D4—42no—
272C7×D4—42no—
273C7×D4—44no—
274C7×D4—44no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—81no—
278C7×D4—82no—
279C7×D4—82no—
280C7×D4—82no—
281C7×D4—82no—
282C7×D4—82no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—41no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—82no—
295C2×C4×C7—81no—
296C2×C4×C7—82no—
297C2×C4×C7—82no—
298C2×C4×C7—82no—
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7#12011yescharacteristic
301C2×C4×C7#12011yescharacteristic
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D411yescharacteristic
305C7×C8—24no—
306C7×C8—42no—
307C7×C8—42no—
308C2×#41—22no—
309C2×#41C7×D412yes—
310C22×D4—21no—
311C22×D4—21no—
312C22×D4—21no—
313AGL(1,ℤ/8)—48no—
314C2×SD16—24no—
315C2×SD16—22no—
316C2×D8—24no—
317C2×D8—22no—
318C2×M16—21no—
319C22×C8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—22no—
322C4⋊Q8C7×D411yescharacteristic
323#127—21no—
324#127—21no—
325#127—21no—
326#127C7×D411yescharacteristic
327#127C23×C712yes—
328#126—22no—
329#126—22no—
330#126—21no—
331#125—22no—
332#125—21no—
333#125—21no—
334#125—21no—
335#125C7×D412yes—
336#125C23×C712yes—
337#125C23×C711yescharacteristic
338#124C23×C711yescharacteristic
339#124C23×C712yes—
340#123—22no—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—21no—
345#123—22no—
346#123—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8C7×D411yescharacteristic
354#121—22no—
355#121—22no—
356#121—21no—
357#121—21no—
358#121—22no—
359#121—22no—
360#121—21no—
361#121—21no—
362#121C7×D411yescharacteristic
363C4×Q8C7×D411yescharacteristic
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4C7×D411yescharacteristic
372#117—21no—
373#117—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C7×D411yescharacteristic
379C2×C4⋊C4C7×D411yescharacteristic
380C2×C42C23×C711yescharacteristic
381C4⋊C8—21no—
382#102—24no—
383#102—24no—
384#102—22no—
385#102—22no—
386C22⋊C8—21no—
387#97—21no—
388#97C23×C712yes—
389#97C23×C712yes—
390#97C23×C712yes—
391C4×C8—21no—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—42no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—41no—
400C22×C7—82no—
401C22×C7—82no—
402C22×C7—82no—
403C22×C7—164no—
404C22×C7#45312yes—
405C22×C7C2×#12011yescharacteristic
406C4×C7—21no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—21no—
410C4×C7—21no—
411C4×C7—42no—
412C4×C7—42no—
413C4×C7—41no—
414C4×C7—41no—
415C4×C7—42no—
416C4×C7—41no—
417C4×C7#45412yes—
418C24—41no—
419#41—44no—
420#41—84no—
421C2×Q8—22no—
422C2×Q8C2×C7×D411yescharacteristic
423C2×Q8C2×C7×D412yes—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—82no—
439C2×D4C2×C7×D411yescharacteristic
440C2×D4C2×C7×D412yes—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C2×C7×D411yescharacteristic
450C22×C4C2×C7×D411yescharacteristic
451SD16—48no—
452SD16—82no—
453D8—48no—
454D8—82no—
455M16—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—42no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4C2×C7×D412yes—
477C4⋊C4C2×C7×D412yes—
478C4⋊C4C2×C7×D411yescharacteristic
479C4⋊C4C2×C7×D411yescharacteristic
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C42—22no—
493C42—21no—
494C42C2×C7×D411yescharacteristic
495C42C2×C7×D411yescharacteristic
496C42C2×C7×D411yescharacteristic
497C42C24×C711yescharacteristic
498C2×C7—22no—
499C2×C7—82no—
500C2×C7—81no—
501C2×C7—162no—
502C2×C7C2×#45311yescharacteristic
503C2×C7C2×#45411yescharacteristic
504C2×C7#302111yescharacteristic
505C23—22no—
506C23—21no—
507C23—42no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23C7×#12011yescharacteristic
512Q8—42no—
513Q8—41no—
514Q8—82no—
515D4—42no—
516D4—41no—
517D4—44no—
518D4—42no—
519D4—42no—
520D4—44no—
521D4—44no—
522D4—42no—
523D4—42no—
524D4—81no—
525D4—82no—
526D4—82no—
527D4—82no—
528D4—82no—
529D4—82no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C7×#12011yescharacteristic
547C2×C4C7×#12011yescharacteristic
548C2×C4C7×#12011yescharacteristic
549C2×C4C22×C7×D411yescharacteristic
550C2×C4C22×C7×D411yescharacteristic
551C2×C4C22×C7×D411yescharacteristic
552C8—24no—
553C8—42no—
554C8—42no—
555C7#1901611yescharacteristic, direct product factor, Sylow
556C22—21no—
557C22—21no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—164no—
568C22C7×#45312yes—
569C22C2×C7×#12011yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4C7×#45412yes—
582C2—22no—
583C2—82no—
584C2—81no—
585C2—162no—
586C2C2×C7×#45311yescharacteristic
587C2C2×C7×#45411yescharacteristic
588C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3559C2semidirect
2. C7×#3217C2semidirect
3. C7×#3151C2semidirect
4. C2×C7×#490C2semidirect
5. C7×#1657C2semidirect
6. C7×#1656C2semidirect
7. C7×#1530C2semidirect
8. C7×#1528C2semidirect
9. C7×#1524C2semidirect
11. C7×#563C22semidirect
12. C7×#562C22semidirect
43. C7×#330C22semidirect
44. C7×#330C22semidirect
45. C7×#329C22semidirect
46. C7×#329C22semidirect
47. #19016C7direct (N × H)
132. #3559C2×C7semidirect
133. #3217C2×C7semidirect
134. #3151C2×C7semidirect
135. C2×#490C2×C7semidirect
136. #1657C2×C7semidirect
137. #1656C2×C7semidirect
138. #1530C2×C7semidirect
139. #1528C2×C7semidirect
140. #1524C2×C7semidirect
222. #563C22×C7semidirect
223. #562C22×C7semidirect
254. #330C22×C7semidirect
255. #330C22×C7semidirect
256. #329C22×C7semidirect
257. #329C22×C7semidirect
555. C7#19016direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35592
C7×#32172
C7×#31512
C2×C7×#4902
C7×#16572
C7×#16562
C7×#15302
C7×#15282
C7×#15242
#190167