Finite Group Databasebeta

← C7×#19026 #13196242

Subgroups — C7×#19026 #13196242

Order 1792.

1012 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3561C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic, complemented
3C2×C7×#496C211yesmaximal, characteristic, complemented
4C7×#3138C211yesmaximal, characteristic, complemented
5C7×#1657C214yesmaximal, complemented
6C7×#1656C214yesmaximal, complemented
7C7×#1531C211yesmaximal, characteristic, complemented
8C7×#1527C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C22×C7×SD16—21no—
11C22×C7×D8—21no—
12C7×#563C2212yescomplemented
13C7×#562C2212yescomplemented
14C7×#556—21no—
15C7×#551—22no—
16C7×D4×Q8—21no—
17C7×#548—21no—
18C7×#546—21no—
19C7×#543—21no—
20C7×#542—21no—
21C7×#542—22no—
22C7×#538—21no—
23C7×#536—21no—
24C7×#533C2211yescharacteristic
25C2×C7×#127C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#496—22no—
29C7×#496C2214yes—
30C7×#496C2212yescomplemented
31C7×#490C2212yescomplemented
32C7×#490C2214yescomplemented
33C7×#486—21no—
34C7×#486—21no—
35C7×#460—21no—
36C7×#459—21no—
37C7×#450—21no—
38C7×#447—21no—
39C7×#432C2211yescharacteristic
40C7×#418—22no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yescomplemented
44C7×#330C2214yescomplemented
45C7×#329C2212yescomplemented
46C7×#329C2214yescomplemented
47#19026C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4—21no—
52C22×C7×D4—21no—
53C22×C7×D4—21no—
54C22×C7×D4D411yescharacteristic
55C2×C7×SD16—24no—
56C2×C7×SD16—22no—
57C2×C7×SD16—44no—
58C2×C7×D8—24no—
59C2×C7×D8—22no—
60C2×C7×D8—44no—
61C2×C7×M16—21no—
62C22×C7×C8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8D411yescharacteristic
66C7×C4⋊Q8D411yescharacteristic
67C7×#127—21no—
68C7×#127—21no—
69C7×#127—21no—
70C7×#127C2312yes—
71C7×#126—22no—
72C7×#126—22no—
73C7×#126—22no—
74C7×#126—21no—
75C7×#125C2311yescharacteristic
76C7×#125C2312yes—
77C7×#124—21no—
78C7×#124—22no—
79C7×#124—22no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#124D412yes—
83C7×#124C2312yes—
84C7×#124C2311yescharacteristic
85C7×#123—22no—
86C7×#123—22no—
87C7×#123—21no—
88C7×#123—22no—
89C7×#123—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8D411yescharacteristic
101C7×#121—21no—
102C7×#121—22no—
103C7×#121—22no—
104C7×#121D411yescharacteristic
105C7×#120—22no—
106C4×C7×Q8—21no—
107C4×C7×Q8—21no—
108C4×C7×Q8—21no—
109C4×C7×Q8D411yescharacteristic
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4D411yescharacteristic
115C7×#117—22no—
116C7×#117—21no—
117C7×#117D412yes—
118C2×C7×C22⋊C4—22no—
119C2×C7×C22⋊C4—21no—
120C2×C7×C22⋊C4—21no—
121C2×C42×C7C2311yescharacteristic
122C7×C4⋊C8—21no—
123C7×#102—24no—
124C7×#102—24no—
125C7×#102—22no—
126C7×#102—22no—
127C7×C22⋊C8—21no—
128C7×#97—21no—
129C7×#97C2312yes—
130C7×#97C2312yes—
131C7×#97C2312yes—
132C4×C7×C8—21no—
133#3561C2×C711yescharacteristic, complemented
134#3217C2×C711yescharacteristic, complemented
135C2×#496C2×C711yescharacteristic, complemented
136#3138C2×C711yescharacteristic, complemented
137#1657C2×C714yescomplemented
138#1656C2×C714yescomplemented
139#1531C2×C711yescharacteristic, complemented
140#1527C2×C711yescharacteristic, complemented
141#1524C2×C711yescharacteristic, complemented
142C24×C7—21no—
143C24×C7—41no—
144C7×#41—82no—
145C2×C7×Q8—22no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—42no—
150C2×C7×Q8—82no—
151C2×C7×Q8C2×D411yescharacteristic
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—42no—
161C2×C7×D4—42no—
162C2×C7×D4—42no—
163C2×C7×D4—44no—
164C2×C7×D4—41no—
165C2×C7×D4—41no—
166C2×C7×D4—41no—
167C2×C7×D4—82no—
168C2×C7×D4C2×D411yescharacteristic
169C2×C7×D4C2×D412yes—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C7×SD16—48no—
179C7×SD16—82no—
180C7×D8—48no—
181C7×D8—82no—
182C7×M16—42no—
183C2×C7×C8—24no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
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—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—21no—
213C7×C22⋊C4—21no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—21no—
217C7×C22⋊C4—21no—
218C42×C7—22no—
219C42×C7—22no—
220C42×C7—21no—
221C42×C7C2×D411yescharacteristic
222C42×C7C2×D411yescharacteristic
223C42×C7C2×D411yescharacteristic
224C42×C7C2411yescharacteristic
225C22×SD16—21no—
226C22×D8—21no—
227#563C22×C712yescomplemented
228#562C22×C712yescomplemented
229#556—21no—
230#551—22no—
231D4×Q8—21no—
232#548—21no—
233#546—21no—
234#543—21no—
235#542—21no—
236#542—22no—
237#538—21no—
238#536—21no—
239#533C22×C711yescharacteristic
240C2×#127C22×C711yescharacteristic
241#529C22×C711yescharacteristic
242C2×#124C22×C711yescharacteristic
243#496—22no—
244#496C22×C714yes—
245#496C22×C712yescomplemented
246#490C22×C714yescomplemented
247#490C22×C712yescomplemented
248#486—21no—
249#486—21no—
250#460—21no—
251#459—21no—
252#450—21no—
253#447—21no—
254#432C22×C711yescharacteristic
255#418—22no—
256C4×M16C22×C711yescharacteristic
257C2×#97C22×C711yescharacteristic
258#330C22×C714yescomplemented
259#330C22×C712yescomplemented
260#329C22×C712yescomplemented
261#329C22×C714yescomplemented
262C23×C7—22no—
263C23×C7—21no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7—81no—
267C23×C7—81no—
268C23×C7—82no—
269C23×C7—82no—
270C23×C7—82no—
271C23×C7#12011yescharacteristic
272C7×Q8—42no—
273C7×Q8—41no—
274C7×Q8—42no—
275C7×Q8—42no—
276C7×Q8—81no—
277C7×Q8—82no—
278C7×D4—42no—
279C7×D4—41no—
280C7×D4—42no—
281C7×D4—42no—
282C7×D4—42no—
283C7×D4—42no—
284C7×D4—44no—
285C7×D4—42no—
286C7×D4—42no—
287C7×D4—82no—
288C7×D4—82no—
289C7×D4—82no—
290C7×D4—82no—
291C7×D4—82no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—41no—
301C2×C4×C7—42no—
302C2×C4×C7—81no—
303C2×C4×C7—82no—
304C2×C4×C7—82no—
305C2×C4×C7—82no—
306C2×C4×C7#12011yescharacteristic
307C2×C4×C7#12011yescharacteristic
308C2×C4×C7#12011yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C7×C8—24no—
313C7×C8—42no—
314C7×C8—42no—
315C2×#41—21no—
316C22×Q8—21no—
317C22×Q8C7×D411yescharacteristic
318C22×D4—21no—
319C22×D4—21no—
320C22×D4—21no—
321C22×D4C7×D411yescharacteristic
322C2×SD16—24no—
323C2×SD16—22no—
324C2×SD16—44no—
325C2×D8—24no—
326C2×D8—22no—
327C2×D8—44no—
328C2×M16—21no—
329C22×C8—21no—
330C4⋊Q8—21no—
331C4⋊Q8—22no—
332C4⋊Q8C7×D411yescharacteristic
333C4⋊Q8C7×D411yescharacteristic
334#127—21no—
335#127—21no—
336#127—21no—
337#127C23×C712yes—
338#126—22no—
339#126—22no—
340#126—22no—
341#126—21no—
342#125C23×C711yescharacteristic
343#125C23×C712yes—
344#124—21no—
345#124—22no—
346#124—22no—
347#124—21no—
348#124—21no—
349#124C7×D412yes—
350#124C23×C712yes—
351#124C23×C711yescharacteristic
352#123—22no—
353#123—22no—
354#123—21no—
355#123—22no—
356#123—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—21no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8C7×D411yescharacteristic
368#121—21no—
369#121—22no—
370#121—22no—
371#121C7×D411yescharacteristic
372#120—22no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8C7×D411yescharacteristic
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4C7×D411yescharacteristic
382#117—22no—
383#117—21no—
384#117C7×D412yes—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C42C23×C711yescharacteristic
389C4⋊C8—21no—
390#102—24no—
391#102—24no—
392#102—22no—
393#102—22no—
394C22⋊C8—21no—
395#97—21no—
396#97C23×C712yes—
397#97C23×C712yes—
398#97C23×C712yes—
399C4×C8—21no—
400C22×C7—21no—
401C22×C7—21no—
402C22×C7—41no—
403C22×C7—42no—
404C22×C7—41no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—41no—
408C22×C7—82no—
409C22×C7—82no—
410C22×C7—82no—
411C22×C7—162no—
412C22×C7—164no—
413C22×C7#45412yes—
414C22×C7C2×#12011yescharacteristic
415C4×C7—21no—
416C4×C7—21no—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—21no—
420C4×C7—41no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7#45312yes—
427C24—21no—
428C24—41no—
429#41—82no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8C2×C7×D411yescharacteristic
437C2×Q8C2×C7×D412yes—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—82no—
453C2×D4C2×C7×D411yescharacteristic
454C2×D4C2×C7×D412yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C2×C7×D411yescharacteristic
462C22×C4C2×C7×D411yescharacteristic
463SD16—48no—
464SD16—82no—
465D8—48no—
466D8—82no—
467M16—42no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4C2×C7×D412yes—
485C4⋊C4C2×C7×D412yes—
486C4⋊C4C2×C7×D411yescharacteristic
487C4⋊C4C2×C7×D411yescharacteristic
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42C2×C7×D411yescharacteristic
507C42C2×C7×D411yescharacteristic
508C42C2×C7×D411yescharacteristic
509C42C24×C711yescharacteristic
510C2×C7—22no—
511C2×C7—81no—
512C2×C7—82no—
513C2×C7—162no—
514C2×C7C2×#45311yescharacteristic
515C2×C7C2×#45411yescharacteristic
516C2×C7#302111yescharacteristic
517C23—22no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—81no—
522C23—81no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23C7×#12011yescharacteristic
527Q8—42no—
528Q8—41no—
529Q8—42no—
530Q8—42no—
531Q8—81no—
532Q8—82no—
533D4—42no—
534D4—41no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—44no—
540D4—42no—
541D4—42no—
542D4—82no—
543D4—82no—
544D4—82no—
545D4—82no—
546D4—82no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4C7×#12011yescharacteristic
562C2×C4C7×#12011yescharacteristic
563C2×C4C7×#12011yescharacteristic
564C2×C4C22×C7×D411yescharacteristic
565C2×C4C22×C7×D411yescharacteristic
566C2×C4C22×C7×D411yescharacteristic
567C8—24no—
568C8—42no—
569C8—42no—
570C7#1902611yescharacteristic, direct product factor, Sylow
571C22—21no—
572C22—21no—
573C22—41no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—162no—
583C22—164no—
584C22C7×#45412yes—
585C22C2×C7×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4C7×#45312yes—
598C2—22no—
599C2—82no—
600C2—81no—
601C2—162no—
602C2C2×C7×#45311yescharacteristic
603C2C2×C7×#45411yescharacteristic
604C2C7×#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×#3561C2semidirect
2. C7×#3217C2semidirect
3. C2×C7×#496C2semidirect
4. C7×#3138C2semidirect
5. C7×#1657C2semidirect
6. C7×#1656C2semidirect
7. C7×#1531C2semidirect
8. C7×#1527C2semidirect
9. C7×#1524C2semidirect
12. C7×#563C22semidirect
13. C7×#562C22semidirect
30. C7×#496C22semidirect
31. C7×#490C22semidirect
32. C7×#490C22semidirect
43. C7×#330C22semidirect
44. C7×#330C22semidirect
45. C7×#329C22semidirect
46. C7×#329C22semidirect
47. #19026C7direct (N × H)
133. #3561C2×C7semidirect
134. #3217C2×C7semidirect
135. C2×#496C2×C7semidirect
136. #3138C2×C7semidirect
137. #1657C2×C7semidirect
138. #1656C2×C7semidirect
139. #1531C2×C7semidirect
140. #1527C2×C7semidirect
141. #1524C2×C7semidirect
227. #563C22×C7semidirect
228. #562C22×C7semidirect
245. #496C22×C7semidirect
246. #490C22×C7semidirect
247. #490C22×C7semidirect
258. #330C22×C7semidirect
259. #330C22×C7semidirect
260. #329C22×C7semidirect
261. #329C22×C7semidirect
570. C7#19026direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35612
C7×#32172
C2×C7×#4962
C7×#31382
C7×#16572
C7×#16562
C7×#15312
C7×#15272
C7×#15242
#190267