Finite Group Databasebeta

← C7×#19024 #13196240

Subgroups — C7×#19024 #13196240

Order 1792.

948 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic
2C7×#3216C211yesmaximal, characteristic, complemented
3C7×#3172C211yesmaximal, characteristic, complemented
4C7×#3140C211yesmaximal, characteristic, complemented
5C7×#1662C212yesmaximal, complemented
6C7×#1661C212yesmaximal, complemented
7C7×#1659C212yesmaximal, complemented
8C7×#1659C212yesmaximal, complemented
9C7×#1529C211yesmaximal, characteristic, complemented
10C7×#1527C211yesmaximal, characteristic, complemented
11C7×#1525C211yesmaximal, characteristic, complemented
12C7×#562C2212yes—
13C7×H(3,𝔽4)C2212yes—
14C7×#559—21no—
15C7×#548—21no—
16C7×#546—22no—
17C7×#546—21no—
18C7×D42—21no—
19C7×#542—21no—
20C7×#538—22no—
21C7×#538—21no—
22C7×#535—21no—
23C7×#534—21no—
24C7×#533C2211yescharacteristic, complemented
25C7×#532C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#501C2211yescharacteristic, complemented
29C7×#497C2211yescharacteristic, complemented
30C7×#497C2212yescomplemented
31C7×#491C2211yescharacteristic, complemented
32C7×#489C2211yescharacteristic, complemented
33C7×#489C2212yescomplemented
34C7×#489C2212yescomplemented
35C7×#489C2212yescomplemented
36C7×#488—22no—
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, complemented
45C7×#419—21no—
46C7×#418—21no—
47C4×C7×M16C2211yescharacteristic, complemented
48C4×C7×M16C2211yescharacteristic, complemented
49C7×#330C2212yescomplemented
50C7×#329C2212yescomplemented
51C7×#329C2212yescomplemented
52C7×#329C2212yescomplemented
53C7×#329C2212yescomplemented
54C7×#329C2212yescomplemented
55#19024C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41D412yes—
58C22×C7×Q8D411yescharacteristic
59C22×C7×D4—22no—
60C22×C7×D4—21no—
61C22×C7×D4D411yescharacteristic
62C2×C7×Q16—21no—
63C2×C7×Q16—22no—
64C2×C7×SD16—21no—
65C2×C7×SD16—22no—
66C2×C7×M16—21no—
67C2×C7×M16—21no—
68C7×C4⋊Q8D411yescharacteristic
69C7×C4⋊Q8C2311yescharacteristic
70C7×C4⋊Q8C2311yescharacteristic
71C7×#127—21no—
72C7×#127D411yescharacteristic
73C7×#126—22no—
74C7×#126—21no—
75C7×#125—21no—
76C7×#125C2311yescharacteristic
77C7×#124—21no—
78C7×#124—22no—
79C7×#124—22no—
80C7×#124—22no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124D412yes—
84C7×#124C2312yes—
85C7×#124C2311yescharacteristic
86C7×#124C2312yes—
87C7×#123—22no—
88C7×#123—22no—
89C7×#123—21no—
90C7×#123—22no—
91C7×#123—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8D411yescharacteristic
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—22no—
101C7×#121—21no—
102C7×#121—22no—
103C7×#121—22no—
104C7×#121—21no—
105C7×#121—21no—
106C7×#121D411yescharacteristic
107C7×#120—22no—
108C7×#120—22no—
109C7×#120—22no—
110C7×#120—22no—
111C7×#120—22no—
112C4×C7×Q8—21no—
113C4×C7×Q8—21no—
114C4×C7×Q8—21no—
115C4×C7×Q8D411yescharacteristic
116C4×C7×D4—22no—
117C4×C7×D4—22no—
118C4×C7×D4—21no—
119C4×C7×D4—21no—
120C4×C7×D4—21no—
121C4×C7×D4—21no—
122C4×C7×D4D411yescharacteristic
123C7×#117—21no—
124C7×#117—21no—
125C2×C7×C22⋊C4—22no—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C42×C7C2311yescharacteristic
129C7×#107—21no—
130C7×#106—21no—
131C7×C4⋊C8—21no—
132C7×#103—21no—
133C7×#103—22no—
134C7×#103—22no—
135C7×#103—22no—
136C7×#103—21no—
137C7×#103—21no—
138C7×#102—21no—
139C7×#102—22no—
140C7×#102—22no—
141C7×#102—22no—
142C7×C22⋊C8—21no—
143C7×#97C2312yes—
144C7×#97C2311yescharacteristic
145C7×#97C2312yes—
146C7×#97C2311yescharacteristic
147C4×C7×C8—21no—
148C4×C7×C8—21no—
149#3558C2×C711yescharacteristic
150#3216C2×C711yescharacteristic, complemented
151#3172C2×C711yescharacteristic, complemented
152#3140C2×C711yescharacteristic, complemented
153#1662C2×C712yescomplemented
154#1661C2×C712yescomplemented
155#1659C2×C712yescomplemented
156#1659C2×C712yescomplemented
157#1529C2×C711yescharacteristic, complemented
158#1527C2×C711yescharacteristic, complemented
159#1525C2×C711yescharacteristic, complemented
160C24×C7—22no—
161C24×C7—21no—
162C7×#41—82no—
163C7×#41—82no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—42no—
168C2×C7×Q8C2×D412yes—
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×Q8C2×D412yes—
171C2×C7×D4—22no—
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—22no—
175C2×C7×D4—22no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—21no—
179C2×C7×D4—21no—
180C2×C7×D4—21no—
181C2×C7×D4—42no—
182C2×C7×D4—82no—
183C2×C7×D4—84no—
184C2×C7×D4C2×D412yes—
185C2×C7×D4C2×D411yescharacteristic
186C2×C7×D4C2×D412yes—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—21no—
190C22×C4×C7—21no—
191C22×C4×C7—22no—
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C7×Q16—42no—
196C7×Q16—82no—
197C7×SD16—42no—
198C7×SD16—82no—
199C7×M16—42no—
200C7×M16—42no—
201C2×C7×C8—21no—
202C2×C7×C8—22no—
203C2×C7×C8—21no—
204C2×C7×C8—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
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—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4—41no—
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D411yescharacteristic
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—22no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—22no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—41no—
238C42×C7—22no—
239C42×C7—21no—
240C42×C7—41no—
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2411yescharacteristic
245#562C22×C712yes—
246H(3,𝔽4)C22×C712yes—
247#559—21no—
248#548—21no—
249#546—22no—
250#546—21no—
251D42—21no—
252#542—21no—
253#538—22no—
254#538—21no—
255#535—21no—
256#534—21no—
257#533C22×C711yescharacteristic, complemented
258#532C22×C711yescharacteristic
259#529C22×C711yescharacteristic
260C2×#124C22×C711yescharacteristic
261#501C22×C711yescharacteristic, complemented
262#497C22×C711yescharacteristic, complemented
263#497C22×C712yescomplemented
264#491C22×C711yescharacteristic, complemented
265#489C22×C712yescomplemented
266#489C22×C712yescomplemented
267#489C22×C711yescharacteristic, complemented
268#489C22×C712yescomplemented
269#488—22no—
270#470—21no—
271#468—21no—
272#462—21no—
273#460—21no—
274#452—22no—
275C4×Q16—21no—
276C4×SD16—21no—
277#432C22×C711yescharacteristic, complemented
278#419—21no—
279#418—21no—
280C4×M16C22×C711yescharacteristic, complemented
281C4×M16C22×C711yescharacteristic, complemented
282#330C22×C712yescomplemented
283#329C22×C712yescomplemented
284#329C22×C712yescomplemented
285#329C22×C712yescomplemented
286#329C22×C712yescomplemented
287#329C22×C712yescomplemented
288C23×C7—21no—
289C23×C7—22no—
290C23×C7—22no—
291C23×C7—42no—
292C23×C7—81no—
293C23×C7—81no—
294C23×C7—82no—
295C23×C7—82no—
296C23×C7—82no—
297C23×C7#12011yescharacteristic
298C7×Q8—41no—
299C7×Q8—42no—
300C7×Q8—42no—
301C7×Q8—41no—
302C7×Q8—41no—
303C7×Q8—41no—
304C7×Q8—41no—
305C7×Q8—42no—
306C7×D4—41no—
307C7×D4—42no—
308C7×D4—42no—
309C7×D4—42no—
310C7×D4—44no—
311C7×D4—42no—
312C7×D4—42no—
313C7×D4—81no—
314C7×D4—82no—
315C7×D4—82no—
316C7×D4—82no—
317C7×D4—82no—
318C7×D4—82no—
319C2×C4×C7—22no—
320C2×C4×C7—21no—
321C2×C4×C7—22no—
322C2×C4×C7—22no—
323C2×C4×C7—21no—
324C2×C4×C7—21no—
325C2×C4×C7—41no—
326C2×C4×C7—41no—
327C2×C4×C7—42no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—82no—
331C2×C4×C7—81no—
332C2×C4×C7—82no—
333C2×C4×C7—82no—
334C2×C4×C7#12011yescharacteristic
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C2×C4×C7C22×D411yescharacteristic
340C7×C8—22no—
341C7×C8—42no—
342C7×C8—41no—
343C7×C8—42no—
344C2×#41—21no—
345C2×#41C7×D412yes—
346C22×Q8C7×D411yescharacteristic
347C22×D4—22no—
348C22×D4—21no—
349C22×D4C7×D411yescharacteristic
350C2×Q16—21no—
351C2×Q16—22no—
352C2×SD16—21no—
353C2×SD16—22no—
354C2×M16—21no—
355C2×M16—21no—
356C4⋊Q8C7×D411yescharacteristic
357C4⋊Q8C23×C711yescharacteristic
358C4⋊Q8C23×C711yescharacteristic
359#127—21no—
360#127C7×D411yescharacteristic
361#126—22no—
362#126—21no—
363#125—21no—
364#125C23×C711yescharacteristic
365#124—21no—
366#124—22no—
367#124—22no—
368#124—22no—
369#124—21no—
370#124—21no—
371#124C7×D412yes—
372#124C23×C712yes—
373#124C23×C711yescharacteristic
374#124C23×C712yes—
375#123—22no—
376#123—22no—
377#123—21no—
378#123—22no—
379#123—22no—
380C22⋊Q8—21no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8C7×D411yescharacteristic
386#121—22no—
387#121—22no—
388#121—22no—
389#121—21no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—21no—
394#121C7×D411yescharacteristic
395#120—22no—
396#120—22no—
397#120—22no—
398#120—22no—
399#120—22no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8C7×D411yescharacteristic
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4C7×D411yescharacteristic
411#117—21no—
412#117—21no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42C23×C711yescharacteristic
417#107—21no—
418#106—21no—
419C4⋊C8—21no—
420#103—22no—
421#103—22no—
422#103—21no—
423#103—22no—
424#103—21no—
425#103—21no—
426#102—22no—
427#102—22no—
428#102—21no—
429#102—22no—
430C22⋊C8—21no—
431#97C23×C711yescharacteristic
432#97C23×C711yescharacteristic
433#97C23×C712yes—
434#97C23×C712yes—
435C4×C8—21no—
436C4×C8—21no—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—21no—
440C22×C7—41no—
441C22×C7—42no—
442C22×C7—42no—
443C22×C7—41no—
444C22×C7—42no—
445C22×C7—42no—
446C22×C7—42no—
447C22×C7—42no—
448C22×C7—41no—
449C22×C7—162no—
450C22×C7—164no—
451C22×C7C2×#12011yescharacteristic
452C4×C7—21no—
453C4×C7—21no—
454C4×C7—21no—
455C4×C7—21no—
456C4×C7—21no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—41no—
460C4×C7—41no—
461C4×C7—42no—
462C4×C7—81no—
463C4×C7—81no—
464C4×C7#45512yes—
465C24—22no—
466C24—21no—
467#41—82no—
468#41—82no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—42no—
473C2×Q8C2×C7×D412yes—
474C2×Q8C2×C7×D411yescharacteristic
475C2×Q8C2×C7×D412yes—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—42no—
487C2×D4—82no—
488C2×D4—84no—
489C2×D4C2×C7×D412yes—
490C2×D4C2×C7×D411yescharacteristic
491C2×D4C2×C7×D412yes—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D411yescharacteristic
500Q16—42no—
501Q16—82no—
502SD16—42no—
503SD16—82no—
504M16—42no—
505M16—42no—
506C2×C8—21no—
507C2×C8—22no—
508C2×C8—21no—
509C2×C8—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4C2×C7×D411yescharacteristic
525C4⋊C4C2×C7×D411yescharacteristic
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—41no—
543C42—22no—
544C42—21no—
545C42—41no—
546C42C2×C7×D411yescharacteristic
547C42C2×C7×D411yescharacteristic
548C42C2×C7×D411yescharacteristic
549C42C24×C711yescharacteristic
550C2×C7—41no—
551C2×C7—82no—
552C2×C7—81no—
553C2×C7—82no—
554C2×C7C2×#45511yescharacteristic
555C2×C7#301811yescharacteristic
556C2×C7#301911yescharacteristic
557C23—21no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23C7×#12011yescharacteristic
567Q8—41no—
568Q8—42no—
569Q8—42no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—42no—
575D4—41no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—81no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—82no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4C7×#12011yescharacteristic
604C2×C4C7×#12011yescharacteristic
605C2×C4C7×#12011yescharacteristic
606C2×C4C22×C7×D411yescharacteristic
607C2×C4C22×C7×D411yescharacteristic
608C2×C4C22×C7×D411yescharacteristic
609C8—22no—
610C8—42no—
611C8—41no—
612C8—42no—
613C7#1902411yescharacteristic, direct product factor, Sylow
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—162no—
627C22—164no—
628C22C2×C7×#12011yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4C7×#45512yes—
642C2—41no—
643C2—81no—
644C2—82no—
645C2—82no—
646C2C2×C7×#45511yescharacteristic
647C2C7×#301811yescharacteristic
648C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3558C2
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×#3140C2semidirect
5. C7×#1662C2semidirect
6. C7×#1661C2semidirect
7. C7×#1659C2semidirect
8. C7×#1659C2semidirect
9. C7×#1529C2semidirect
10. C7×#1527C2semidirect
11. C7×#1525C2semidirect
24. C7×#533C22semidirect
28. C7×#501C22semidirect
29. C7×#497C22semidirect
30. C7×#497C22semidirect
31. C7×#491C22semidirect
32. C7×#489C22semidirect
33. C7×#489C22semidirect
34. C7×#489C22semidirect
35. C7×#489C22semidirect
44. C7×#432C22semidirect
47. C4×C7×M16C22semidirect
48. C4×C7×M16C22semidirect
49. C7×#330C22semidirect
50. C7×#329C22semidirect
51. C7×#329C22semidirect
52. C7×#329C22semidirect
53. C7×#329C22semidirect
54. C7×#329C22semidirect
55. #19024C7direct (N × H)
150. #3216C2×C7semidirect
151. #3172C2×C7semidirect
152. #3140C2×C7semidirect
153. #1662C2×C7semidirect
154. #1661C2×C7semidirect
155. #1659C2×C7semidirect
156. #1659C2×C7semidirect
157. #1529C2×C7semidirect
158. #1527C2×C7semidirect
159. #1525C2×C7semidirect
257. #533C22×C7semidirect
261. #501C22×C7semidirect
262. #497C22×C7semidirect
263. #497C22×C7semidirect
264. #491C22×C7semidirect
265. #489C22×C7semidirect
266. #489C22×C7semidirect
267. #489C22×C7semidirect
268. #489C22×C7semidirect
277. #432C22×C7semidirect
280. C4×M16C22×C7semidirect
281. C4×M16C22×C7semidirect
282. #330C22×C7semidirect
283. #329C22×C7semidirect
284. #329C22×C7semidirect
285. #329C22×C7semidirect
286. #329C22×C7semidirect
287. #329C22×C7semidirect
613. C7#19024direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35582
C7×#32162
C7×#31722
C7×#31402
C7×#16622
C7×#16612
C7×#16592
C7×#15292
C7×#15272
C7×#15252
#190247