Finite Group Databasebeta

← C7×#24321 #13201537

Subgroups — C7×#24321 #13201537

Order 1792.

1036 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3503C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic, complemented
3C4×C7×AGL(1,ℤ/8)C211yesmaximal, characteristic
4C7×#2563C211yesmaximal, characteristic, complemented
5C7×#2061C212yesmaximal, complemented
6C7×#2027C212yesmaximal, complemented
7C7×#1997C211yesmaximal, characteristic, complemented
8C7×#1986C211yesmaximal, characteristic, complemented
9C7×#1883C212yesmaximal
10C7×#1860C212yesmaximal, complemented
11C7×#1806C211yesmaximal, characteristic, complemented
12C2×C7×AGL(1,ℤ/8)C2211yescharacteristic
13C7×Q82—21no—
14C7×#557—21no—
15C7×#554—21no—
16C7×#554—22no—
17C7×D4×Q8—22no—
18C7×D4×Q8—21no—
19C7×#544—22no—
20C7×#544C2212yes—
21C7×#533—21no—
22C7×#533C2211yescharacteristic
23C2×C7×C4⋊Q8—21no—
24C2×C7×C4⋊Q8C2211yescharacteristic
25C7×#520C2212yes—
26C4×C7×#41C2211yescharacteristic
27C2×C4×C7×D4C2211yescharacteristic
28C7×#478—22no—
29C7×#476—21no—
30C7×#474—21no—
31C7×#442—21no—
32C7×#440—21no—
33C4×C7×SD16—21no—
34C4×C7×D8—21no—
35C7×#428C2211yescharacteristic
36C7×#428C2212yes—
37C7×#423C2211yescharacteristic
38C2×C7×#104C2212yes—
39C7×#418C2211yescharacteristic
40C7×#418C2212yes—
41C7×#417C2211yescharacteristic
42C4×C7×M16C2211yescharacteristic
43C7×#393—22no—
44C7×#390C2212yes—
45C7×#382C2212yes—
46C7×#380—22no—
47C4×C7×C4⋊C4C2211yescharacteristic
48C7×#343C2212yes—
49C7×#343C2212yes—
50C7×#339C2212yes—
51C7×#339C2212yes—
52C7×#339C2212yes—
53#24321C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41C2311yescharacteristic
55C22×C7×Q8—22no—
56C22×C7×Q8D412yes—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C7×AGL(1,ℤ/8)—22no—
60C7×AGL(1,ℤ/8)Q814yescomplemented
61C2×C7×SD16—21no—
62C2×C7×D8—21no—
63C2×C7×M16C2311yescharacteristic
64C2×C7×M16C2312yes—
65C2×C7×M16C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—24no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8—21no—
73C7×C4⋊Q8—21no—
74C7×C4⋊Q8—21no—
75C7×C4⋊Q8D411yescharacteristic
76C7×C4⋊Q8D411yescharacteristic
77C7×#125—24no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125D412yes—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—24no—
84C7×C22⋊Q8—24no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—24no—
88C4×C7×Q8—22no—
89C4×C7×Q8—22no—
90C4×C7×Q8—22no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8D411yescharacteristic
94C4×C7×Q8D411yescharacteristic
95C4×C7×D4—22no—
96C4×C7×D4—21no—
97C4×C7×D4—21no—
98C4×C7×D4—24no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—21no—
103C4×C7×D4D411yescharacteristic
104C4×C7×D4D411yescharacteristic
105C7×#117—22no—
106C7×#117—21no—
107C7×#117C2311yescharacteristic
108C7×#117C2312yes—
109C2×C7×C4⋊C4—22no—
110C2×C7×C4⋊C4—22no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4D412yes—
113C2×C7×C4⋊C4C2311yescharacteristic
114C2×C7×C4⋊C4C2312yes—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—24no—
117C2×C42×C7—21no—
118C2×C42×C7C2312yes—
119C2×C42×C7C2311yescharacteristic
120C7×#107—22no—
121C7×#107—21no—
122C7×#106—22no—
123C7×#106—21no—
124C7×C4⋊C8—21no—
125C7×C4⋊C8—21no—
126C7×#104—24no—
127C7×#103—21no—
128C7×#102—21no—
129C7×#102—22no—
130C7×#102—22no—
131C7×#102—21no—
132C7×#102—21no—
133C7×#97—21no—
134C4×C7×C8—21no—
135C7×#95—21no—
136C7×#95—24no—
137#3503C2×C711yescharacteristic, complemented
138#3077C2×C711yescharacteristic, complemented
139C4×AGL(1,ℤ/8)C2×C711yescharacteristic
140#2563C2×C711yescharacteristic, complemented
141#2061C2×C712yescomplemented
142#2027C2×C712yescomplemented
143#1997C2×C711yescharacteristic, complemented
144#1986C2×C711yescharacteristic, complemented
145#1883C2×C712yes—
146#1860C2×C712yescomplemented
147#1806C2×C711yescharacteristic, complemented
148C24×C7—21no—
149C7×#41—21no—
150C7×#41—21no—
151C7×#41C2×Q812yes—
152C7×#41#4112yes—
153C2×C7×Q8—24no—
154C2×C7×Q8—21no—
155C2×C7×Q8—22no—
156C2×C7×Q8—21no—
157C2×C7×Q8—22no—
158C2×C7×Q8—82no—
159C2×C7×Q8—84no—
160C2×C7×Q8C2×D411yescharacteristic
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4C2×D411yescharacteristic
168C2×C7×D4C2×Q812yescomplemented
169C22×C4×C7—21no—
170C22×C4×C7—24no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—22no—
175C22×C4×C7—44no—
176C22×C4×C7—41no—
177C22×C4×C7—41no—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D414yes—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C7×SD16—24no—
183C7×D8—24no—
184C7×M16—21no—
185C7×M16—22no—
186C7×M16—21no—
187C7×M16—21no—
188C7×M16—22no—
189C7×M16C2×Q812yescomplemented
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8—22no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—24no—
196C7×C4⋊C4—24no—
197C7×C4⋊C4—21no—
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—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—42no—
210C7×C4⋊C4C2×D412yes—
211C7×C4⋊C4C2×D411yescharacteristic
212C7×C4⋊C4C2×D412yes—
213C7×C4⋊C4C2×D411yescharacteristic
214C7×C22⋊C4—24no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—21no—
217C7×C22⋊C4—21no—
218C7×C22⋊C4—21no—
219C7×C22⋊C4—41no—
220C7×C22⋊C4—41no—
221C7×C22⋊C4—84no—
222C42×C7—24no—
223C42×C7—21no—
224C42×C7—21no—
225C42×C7—21no—
226C42×C7—22no—
227C42×C7—22no—
228C42×C7—22no—
229C42×C7C2×D411yescharacteristic
230C42×C7C2×D411yescharacteristic
231C42×C7C2×D411yescharacteristic
232C42×C7C2×D411yescharacteristic
233C2×AGL(1,ℤ/8)C22×C711yescharacteristic
234Q82—21no—
235#557—21no—
236#554—21no—
237#554—22no—
238D4×Q8—22no—
239D4×Q8—21no—
240#544—22no—
241#544C22×C712yes—
242#533—21no—
243#533C22×C711yescharacteristic
244C2×C4⋊Q8—21no—
245C2×C4⋊Q8C22×C711yescharacteristic
246#520C22×C712yes—
247C4×#41C22×C711yescharacteristic
248C2×C4×D4C22×C711yescharacteristic
249#478—22no—
250#476—21no—
251#474—21no—
252#442—21no—
253#440—21no—
254C4×SD16—21no—
255C4×D8—21no—
256#428C22×C712yes—
257#428C22×C711yescharacteristic
258#423C22×C711yescharacteristic
259C2×#104C22×C712yes—
260#418C22×C712yes—
261#418C22×C711yescharacteristic
262#417C22×C711yescharacteristic
263C4×M16C22×C711yescharacteristic
264#393—22no—
265#390C22×C712yes—
266#382C22×C712yes—
267#380—22no—
268C4×C4⋊C4C22×C711yescharacteristic
269#343C22×C712yes—
270#343C22×C712yes—
271#339C22×C712yes—
272#339C22×C712yes—
273#339C22×C712yes—
274C23×C7—21no—
275C23×C7—22no—
276C23×C7—42no—
277C23×C7—42no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C22×D411yescharacteristic
281C7×Q8—21no—
282C7×Q8—44no—
283C7×Q8—42no—
284C7×Q8—42no—
285C7×Q8—82no—
286C7×Q8—84no—
287C7×Q8C22⋊Q812yes—
288C7×D4—22no—
289C7×D4—22no—
290C7×D4—22no—
291C7×D4—21no—
292C7×D4—22no—
293C7×D4—41no—
294C7×D4—41no—
295C7×D4C22⋊Q812yes—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—24no—
304C2×C4×C7—22no—
305C2×C4×C7—41no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—44no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—41no—
313C2×C4×C7—42no—
314C2×C4×C7—82no—
315C2×C4×C7—84no—
316C2×C4×C7—82no—
317C2×C4×C7#12011yescharacteristic
318C2×C4×C7#12011yescharacteristic
319C2×C4×C7#12011yescharacteristic
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7C22×D411yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C2×C4×C7C22×Q811yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C72−1+411yescharacteristic
326C2×C4×C72−1+411yescharacteristic
327C7×C8—22no—
328C7×C8—41no—
329C7×C8—42no—
330C2×#41C23×C711yescharacteristic
331C22×Q8—22no—
332C22×Q8C7×D412yes—
333C22×D4C23×C711yescharacteristic
334C23×C4—21no—
335AGL(1,ℤ/8)—22no—
336AGL(1,ℤ/8)C7×Q814yescomplemented
337C2×SD16—21no—
338C2×D8—21no—
339C2×M16C23×C711yescharacteristic
340C2×M16C23×C712yes—
341C2×M16C23×C711yescharacteristic
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C4⋊Q8—24no—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8C7×D411yescharacteristic
352C4⋊Q8C7×D411yescharacteristic
353#125—24no—
354#125—22no—
355#125—22no—
356#125C7×D412yes—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—24no—
360C22⋊Q8—24no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—24no—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8C7×D411yescharacteristic
370C4×Q8C7×D411yescharacteristic
371C4×D4—22no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4—24no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4C7×D411yescharacteristic
380C4×D4C7×D411yescharacteristic
381#117—22no—
382#117—21no—
383#117C23×C711yescharacteristic
384#117C23×C712yes—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4C7×D412yes—
389C2×C4⋊C4C23×C711yescharacteristic
390C2×C4⋊C4C23×C712yes—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—24no—
393C2×C42—21no—
394C2×C42C23×C712yes—
395C2×C42C23×C711yescharacteristic
396#107—21no—
397#107—22no—
398#106—21no—
399#106—22no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402#104—24no—
403#103—21no—
404#102—21no—
405#102—21no—
406#102—21no—
407#102—22no—
408#102—22no—
409#97—21no—
410C4×C8—21no—
411#95—21no—
412#95—24no—
413C22×C7—21no—
414C22×C7—21no—
415C22×C7—22no—
416C22×C7—44no—
417C22×C7—41no—
418C22×C7—41no—
419C22×C7—42no—
420C22×C7—82no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7C2×#12011yescharacteristic
424C22×C7#54811yescharacteristic
425C22×C7D4×Q811yescharacteristic
426C4×C7—22no—
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—41no—
431C4×C7—42no—
432C4×C7—41no—
433C4×C7—44no—
434C4×C7—42no—
435C4×C7—82no—
436C4×C7—81no—
437C4×C7#45412yes—
438C4×C7C2×C22⋊Q811yescharacteristic
439C4×C7#53611yescharacteristic
440C4×C7#54811yescharacteristic
441C4×C7D4×Q811yescharacteristic
442C24—21no—
443#41—21no—
444#41—21no—
445#41C2×C7×Q812yes—
446#41C7×#4112yes—
447C2×Q8—24no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—82no—
453C2×Q8—84no—
454C2×Q8C2×C7×D411yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C7×D411yescharacteristic
462C2×D4C2×C7×Q812yescomplemented
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4C2×C7×D411yescharacteristic
473C22×C4C2×C7×D414yes—
474C22×C4C2×C7×D411yescharacteristic
475C22×C4C24×C711yescharacteristic
476SD16—24no—
477D8—24no—
478M16—21no—
479M16—22no—
480M16—21no—
481M16—21no—
482M16—22no—
483M16C2×C7×Q812yescomplemented
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4C2×C7×D412yes—
505C4⋊C4C2×C7×D411yescharacteristic
506C4⋊C4C2×C7×D412yes—
507C4⋊C4C2×C7×D411yescharacteristic
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—84no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42C2×C7×D411yescharacteristic
524C42C2×C7×D411yescharacteristic
525C42C2×C7×D411yescharacteristic
526C42C2×C7×D411yescharacteristic
527C2×C7—21no—
528C2×C7—21no—
529C2×C7—42no—
530C2×C7—82no—
531C2×C7#240911yescharacteristic
532C2×C7C2×#45411yescharacteristic
533C2×C7#301911yescharacteristic
534C23—21no—
535C23—22no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23C22×C7×D411yescharacteristic
541Q8—21no—
542Q8—44no—
543Q8—42no—
544Q8—42no—
545Q8—82no—
546Q8—84no—
547Q8C7×C22⋊Q812yes—
548D4—22no—
549D4—22no—
550D4—22no—
551D4—21no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4C7×C22⋊Q812yes—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—82no—
577C2×C4C7×#12011yescharacteristic
578C2×C4C7×#12011yescharacteristic
579C2×C4C7×#12011yescharacteristic
580C2×C4C7×#12011yescharacteristic
581C2×C4C22×C7×D411yescharacteristic
582C2×C4C22×C7×D411yescharacteristic
583C2×C4C22×C7×Q811yescharacteristic
584C2×C4C2×C7×#4111yescharacteristic
585C2×C4C7×2−1+411yescharacteristic
586C2×C4C7×2−1+411yescharacteristic
587C8—22no—
588C8—41no—
589C8—42no—
590C7#2432111yescharacteristic, direct product factor, Sylow
591C22—21no—
592C22—21no—
593C22—22no—
594C22—44no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—82no—
599C22—81no—
600C22—81no—
601C22C2×C7×#12011yescharacteristic
602C22C7×#54811yescharacteristic
603C22C7×D4×Q811yescharacteristic
604C4—22no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—81no—
615C4C7×#45412yes—
616C4C2×C7×C22⋊Q811yescharacteristic
617C4C7×#53611yescharacteristic
618C4C7×#54811yescharacteristic
619C4C7×D4×Q811yescharacteristic
620C2—21no—
621C2—21no—
622C2—42no—
623C2—82no—
624C2C7×#240911yescharacteristic
625C2C2×C7×#45411yescharacteristic
626C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3503C2semidirect
2. C7×#3077C2semidirect
4. C7×#2563C2semidirect
5. C7×#2061C2semidirect
6. C7×#2027C2semidirect
7. C7×#1997C2semidirect
8. C7×#1986C2semidirect
10. C7×#1860C2semidirect
11. C7×#1806C2semidirect
53. #24321C7direct (N × H)
60. C7×AGL(1,ℤ/8)Q8semidirect
137. #3503C2×C7semidirect
138. #3077C2×C7semidirect
140. #2563C2×C7semidirect
141. #2061C2×C7semidirect
142. #2027C2×C7semidirect
143. #1997C2×C7semidirect
144. #1986C2×C7semidirect
146. #1860C2×C7semidirect
147. #1806C2×C7semidirect
168. C2×C7×D4C2×Q8semidirect
189. C7×M16C2×Q8semidirect
336. AGL(1,ℤ/8)C7×Q8semidirect
462. C2×D4C2×C7×Q8semidirect
483. M16C2×C7×Q8semidirect
590. C7#24321direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35032
C7×#30772
C4×C7×AGL(1,ℤ/8)2
C7×#25632
C7×#20612
C7×#20272
C7×#19972
C7×#19862
C7×#18832
C7×#18602
C7×#18062
#243217