Finite Group Databasebeta

← C7×#19036 #13196252

Subgroups — C7×#19036 #13196252

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic
2C7×#3216C211yesmaximal, characteristic, complemented
3C7×#3172C211yesmaximal, characteristic, complemented
4C7×#3166C211yesmaximal, characteristic, complemented
5C7×#1664C212yesmaximal, complemented
6C7×#1661C212yesmaximal, complemented
7C7×#1661C212yesmaximal, complemented
8C7×#1660C212yesmaximal, complemented
9C7×#1529C211yesmaximal, characteristic, complemented
10C7×#1528C211yesmaximal, characteristic, complemented
11C7×#1525C211yesmaximal, characteristic, complemented
12C7×#563C2212yes—
13C7×#562C2212yes—
14C7×#554—21no—
15C7×#553—21no—
16C7×#553—21no—
17C7×#547—22no—
18C7×#547—21no—
19C7×#544—21no—
20C7×#540—21no—
21C7×#539—22no—
22C7×#539—21no—
23C7×#537—21no—
24C7×#533C2211yescharacteristic
25C7×#532C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#125C2211yescharacteristic
28C7×#501C2212yes—
29C7×#501C2212yes—
30C7×#501C2211yescharacteristic
31C7×#499—22no—
32C7×#497C2211yescharacteristic
33C7×#497C2212yes—
34C7×#491C2211yescharacteristic
35C7×#489C2211yescharacteristic
36C7×#489C2212yes—
37C7×#470—21no—
38C7×#468—21no—
39C7×#462—21no—
40C7×#460—21no—
41C7×#452—22no—
42C4×C7×Q16—21no—
43C4×C7×SD16—21no—
44C7×#432C2211yescharacteristic
45C7×#428—21no—
46C7×#428—21no—
47C4×C7×M16C2211yescharacteristic
48C4×C7×M16C2211yescharacteristic
49C7×#330C2212yes—
50C7×#330C2212yes—
51C7×#330C2212yes—
52C7×#329C2212yes—
53C7×#329C2212yes—
54C7×#329C2212yes—
55#19036C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—22no—
57C2×C7×#41D412yes—
58C2×C7×Q16—21no—
59C2×C7×Q16—22no—
60C2×C7×SD16—21no—
61C2×C7×SD16—22no—
62C2×C7×M16—21no—
63C2×C7×M16—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8D411yescharacteristic
67C7×C4⋊Q8C2311yescharacteristic
68C7×C4⋊Q8C2311yescharacteristic
69C7×#127D411yescharacteristic
70C7×#126—22no—
71C7×#126—22no—
72C7×#126—21no—
73C7×#125—22no—
74C7×#125—21no—
75C7×#125—21no—
76C7×#125—21no—
77C7×#125—21no—
78C7×#125D412yes—
79C7×#125C2311yescharacteristic
80C7×#125C2312yes—
81C7×#124C2311yescharacteristic
82C7×#124C2312yes—
83C7×#123—22no—
84C7×#123—22no—
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—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8D411yescharacteristic
99C7×#121—22no—
100C7×#121—22no—
101C7×#121—21no—
102C7×#121—21no—
103C7×#121—22no—
104C7×#121—22no—
105C7×#121—21no—
106C7×#121—21no—
107C7×#121D411yescharacteristic
108C4×C7×Q8—21no—
109C4×C7×Q8—21no—
110C4×C7×Q8D411yescharacteristic
111C4×C7×D4—22no—
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—21no—
116C4×C7×D4—21no—
117C4×C7×D4—21no—
118C4×C7×D4D411yescharacteristic
119C7×#117—21no—
120C7×#117—21no—
121C7×#117—21no—
122C2×C7×C4⋊C4—22no—
123C2×C7×C4⋊C4—21no—
124C2×C7×C4⋊C4—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4D411yescharacteristic
127C2×C7×C4⋊C4D411yescharacteristic
128C2×C42×C7C2311yescharacteristic
129C7×#107—21no—
130C7×#107—22no—
131C7×#107—22no—
132C7×#106—21no—
133C7×#106—22no—
134C7×#106—22no—
135C7×C4⋊C8—21no—
136C7×#103—21no—
137C7×#103—22no—
138C7×#103—21no—
139C7×#103—21no—
140C7×#102—21no—
141C7×#102—22no—
142C7×C22⋊C8—21no—
143C7×#97C2311yescharacteristic
144C7×#97C2312yes—
145C7×#97C2312yes—
146C7×#97C2311yescharacteristic
147C4×C7×C8—21no—
148C4×C7×C8—21no—
149#3559C2×C711yescharacteristic
150#3216C2×C711yescharacteristic, complemented
151#3172C2×C711yescharacteristic, complemented
152#3166C2×C711yescharacteristic, complemented
153#1664C2×C712yescomplemented
154#1661C2×C712yescomplemented
155#1661C2×C712yescomplemented
156#1660C2×C712yescomplemented
157#1529C2×C711yescharacteristic, complemented
158#1528C2×C711yescharacteristic, complemented
159#1525C2×C711yescharacteristic, complemented
160C7×#41—84no—
161C7×#41—82no—
162C2×C7×Q8—22no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8C2×D411yescharacteristic
166C2×C7×Q8C2×D412yes—
167C2×C7×D4—22no—
168C2×C7×D4—21no—
169C2×C7×D4—22no—
170C2×C7×D4—22no—
171C2×C7×D4—21no—
172C2×C7×D4—21no—
173C2×C7×D4C2×D411yescharacteristic
174C2×C7×D4C2×D412yes—
175C22×C4×C7—21no—
176C22×C4×C7—22no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—21no—
180C22×C4×C7—21no—
181C22×C4×C7—22no—
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C7×Q16—42no—
186C7×Q16—82no—
187C7×SD16—42no—
188C7×SD16—82no—
189C7×M16—24no—
190C7×M16—42no—
191C2×C7×C8—22no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—41no—
216C7×C4⋊C4—41no—
217C7×C4⋊C4C2×D411yescharacteristic
218C7×C4⋊C4C2×D411yescharacteristic
219C7×C4⋊C4C2×D412yes—
220C7×C4⋊C4C2×D412yes—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—41no—
234C42×C7—22no—
235C42×C7—21no—
236C42×C7—41no—
237C42×C7C2×D411yescharacteristic
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2411yescharacteristic
241#563C22×C712yes—
242#562C22×C712yes—
243#554—21no—
244#553—21no—
245#553—21no—
246#547—22no—
247#547—21no—
248#544—21no—
249#540—21no—
250#539—22no—
251#539—21no—
252#537—21no—
253#533C22×C711yescharacteristic
254#532C22×C711yescharacteristic
255#529C22×C711yescharacteristic
256C2×#125C22×C711yescharacteristic
257#501C22×C712yes—
258#501C22×C712yes—
259#501C22×C711yescharacteristic
260#499—22no—
261#497C22×C711yescharacteristic
262#497C22×C712yes—
263#491C22×C711yescharacteristic
264#489C22×C711yescharacteristic
265#489C22×C712yes—
266#470—21no—
267#468—21no—
268#462—21no—
269#460—21no—
270#452—22no—
271C4×Q16—21no—
272C4×SD16—21no—
273#432C22×C711yescharacteristic
274#428—21no—
275#428—21no—
276C4×M16C22×C711yescharacteristic
277C4×M16C22×C711yescharacteristic
278#330C22×C712yes—
279#330C22×C712yes—
280#330C22×C712yes—
281#329C22×C712yes—
282#329C22×C712yes—
283#329C22×C712yes—
284C23×C7—21no—
285C23×C7—22no—
286C23×C7#12011yescharacteristic
287C7×Q8—41no—
288C7×Q8—42no—
289C7×Q8—41no—
290C7×Q8—41no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×Q8—82no—
294C7×D4—41no—
295C7×D4—42no—
296C7×D4—44no—
297C7×D4—42no—
298C7×D4—42no—
299C7×D4—81no—
300C7×D4—82no—
301C7×D4—82no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—21no—
308C2×C4×C7—22no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—82no—
316C2×C4×C7—81no—
317C2×C4×C7—82no—
318C2×C4×C7—82no—
319C2×C4×C7—82no—
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C7×C8—24no—
327C7×C8—22no—
328C7×C8—41no—
329C7×C8—42no—
330C2×#41—22no—
331C2×#41C7×D412yes—
332C2×Q16—21no—
333C2×Q16—22no—
334C2×SD16—21no—
335C2×SD16—22no—
336C2×M16—21no—
337C2×M16—21no—
338C4⋊Q8—21no—
339C4⋊Q8—22no—
340C4⋊Q8C7×D411yescharacteristic
341C4⋊Q8C23×C711yescharacteristic
342C4⋊Q8C23×C711yescharacteristic
343#127C7×D411yescharacteristic
344#126—22no—
345#126—22no—
346#126—21no—
347#125—22no—
348#125—21no—
349#125—21no—
350#125—21no—
351#125—21no—
352#125C7×D412yes—
353#125C23×C712yes—
354#125C23×C711yescharacteristic
355#124C23×C711yescharacteristic
356#124C23×C712yes—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361#123—21no—
362#123—22no—
363#123—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8C7×D411yescharacteristic
373#121—22no—
374#121—22no—
375#121—21no—
376#121—21no—
377#121—22no—
378#121—22no—
379#121—21no—
380#121—21no—
381#121C7×D411yescharacteristic
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8C7×D411yescharacteristic
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4C7×D411yescharacteristic
393#117—21no—
394#117—21no—
395#117—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4C7×D411yescharacteristic
401C2×C4⋊C4C7×D411yescharacteristic
402C2×C42C23×C711yescharacteristic
403#107—22no—
404#107—22no—
405#107—21no—
406#106—22no—
407#106—22no—
408#106—21no—
409C4⋊C8—21no—
410#103—21no—
411#103—22no—
412#103—21no—
413#103—21no—
414#102—21no—
415#102—22no—
416C22⋊C8—21no—
417#97C23×C712yes—
418#97C23×C711yescharacteristic
419#97C23×C711yescharacteristic
420#97C23×C712yes—
421C4×C8—21no—
422C4×C8—21no—
423C22×C7—21no—
424C22×C7—21no—
425C22×C7—21no—
426C22×C7—41no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—41no—
430C22×C7—42no—
431C22×C7—41no—
432C22×C7C2×#12011yescharacteristic
433C4×C7—21no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—21no—
438C4×C7—41no—
439C4×C7—42no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—42no—
443C4×C7—41no—
444C4×C7—81no—
445C4×C7—81no—
446C4×C7#45612yes—
447#41—84no—
448#41—82no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×C7×D411yescharacteristic
453C2×Q8C2×C7×D412yes—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4C2×C7×D411yescharacteristic
461C2×D4C2×C7×D412yes—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472Q16—42no—
473Q16—82no—
474SD16—42no—
475SD16—82no—
476M16—24no—
477M16—42no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4C2×C7×D411yescharacteristic
505C4⋊C4C2×C7×D411yescharacteristic
506C4⋊C4C2×C7×D412yes—
507C4⋊C4C2×C7×D412yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—41no—
521C42—22no—
522C42—21no—
523C42—41no—
524C42C2×C7×D411yescharacteristic
525C42C2×C7×D411yescharacteristic
526C42C2×C7×D411yescharacteristic
527C42C24×C711yescharacteristic
528C2×C7—41no—
529C2×C7—81no—
530C2×C7—82no—
531C2×C7C2×#45611yescharacteristic
532C2×C7#301811yescharacteristic
533C2×C7#301911yescharacteristic
534C23—21no—
535C23—22no—
536C23C7×#12011yescharacteristic
537Q8—41no—
538Q8—42no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—82no—
544D4—41no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—42no—
549D4—81no—
550D4—82no—
551D4—82no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4C7×#12011yescharacteristic
571C2×C4C7×#12011yescharacteristic
572C2×C4C7×#12011yescharacteristic
573C2×C4C22×C7×D411yescharacteristic
574C2×C4C22×C7×D411yescharacteristic
575C2×C4C22×C7×D411yescharacteristic
576C8—24no—
577C8—22no—
578C8—41no—
579C8—42no—
580C7#1903611yescharacteristic, direct product factor, Sylow
581C22—21no—
582C22—21no—
583C22—21no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22C2×C7×#12011yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4C7×#45612yes—
605C2—41no—
606C2—81no—
607C2—82no—
608C2C2×C7×#45611yescharacteristic
609C2C7×#301811yescharacteristic
610C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3559C2
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
2. C7×#3216C2semidirect
3. C7×#3172C2semidirect
4. C7×#3166C2semidirect
5. C7×#1664C2semidirect
6. C7×#1661C2semidirect
7. C7×#1661C2semidirect
8. C7×#1660C2semidirect
9. C7×#1529C2semidirect
10. C7×#1528C2semidirect
11. C7×#1525C2semidirect
55. #19036C7direct (N × H)
150. #3216C2×C7semidirect
151. #3172C2×C7semidirect
152. #3166C2×C7semidirect
153. #1664C2×C7semidirect
154. #1661C2×C7semidirect
155. #1661C2×C7semidirect
156. #1660C2×C7semidirect
157. #1529C2×C7semidirect
158. #1528C2×C7semidirect
159. #1525C2×C7semidirect
580. C7#19036direct (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×#32162
C7×#31722
C7×#31662
C7×#16642
C7×#16612
C7×#16602
C7×#15292
C7×#15282
C7×#15252
#190367