Finite Group Databasebeta

← C7×#24177 #13201393

Subgroups — C7×#24177 #13201393

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3507C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic, complemented
3C7×#3036C211yesmaximal, characteristic
4C4×C7×#124C211yesmaximal, characteristic, complemented
5C7×#2038C212yesmaximal, complemented
6C7×#2003C212yesmaximal, complemented
7C7×#1994C211yesmaximal, characteristic, complemented
8C7×#1962C211yesmaximal, characteristic, complemented
9C7×#1883C212yesmaximal, complemented
10C7×#1860C212yesmaximal, complemented
11C7×#1801C211yesmaximal, characteristic, complemented
12C2×C7×AGL(1,ℤ/8)C2212yes—
13C7×#559—21no—
14C7×#557—21no—
15C7×#556—21no—
16C7×#556—22no—
17C7×#554—21no—
18C7×#552—22no—
19C7×#539—22no—
20C7×#536C2212yes—
21C7×#533C2211yescharacteristic
22C7×#532C2211yescharacteristic
23C7×#529—22no—
24C2×C7×#124C2211yescharacteristic
25C7×#520C2212yes—
26C2×C4×C7×Q8C2211yescharacteristic
27C2×C4×C7×D4C2211yescharacteristic
28C7×#478—22no—
29C7×#476—21no—
30C7×#474—21no—
31C7×#463—22no—
32C7×#461—21no—
33C7×#459—21no—
34C7×#428C2212yes—
35C7×#423C2211yescharacteristic
36C7×#423C2211yescharacteristic
37C2×C7×#104C2212yes—
38C2×C7×#104C2212yes—
39C7×#418C2212yes—
40C7×#417C2212yes—
41C2×C7×#101C2212yes—
42C7×#391C2211yescharacteristic
43C7×#390C2211yescharacteristic
44C7×#388—22no—
45C7×#382C2211yescharacteristic
46C4×C7×C22⋊C4—22no—
47C43×C7C2211yescharacteristic
48C7×#343C2212yes—
49C7×#339C2212yes—
50C7×#339C2212yes—
51#24177C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41C2312yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4C2311yescharacteristic
55C23×C4×C7—21no—
56C7×AGL(1,ℤ/8)—24no—
57C7×AGL(1,ℤ/8)—42no—
58C2×C7×SD16—22no—
59C2×C7×D8—22no—
60C2×C7×M16C2312yes—
61C2×C7×M16C2312yes—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8D411yescharacteristic
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8D411yescharacteristic
66C7×#127D411yescharacteristic
67C7×#126—24no—
68C7×#126—24no—
69C7×#125—22no—
70C7×#125—24no—
71C7×#125D412yes—
72C7×#124—22no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#124—22no—
76C7×#124D412yes—
77C7×#123—24no—
78C7×#123—24no—
79C7×C22⋊Q8—24no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—21no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×#121—21no—
85C7×#121—21no—
86C4×C7×Q8—22no—
87C4×C7×Q8—22no—
88C4×C7×Q8—21no—
89C4×C7×Q8—21no—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8—21no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4—24no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—21no—
103C7×#117—24no—
104C7×#117—22no—
105C7×#117C2312yes—
106C2×C7×C4⋊C4—22no—
107C2×C7×C4⋊C4D412yes—
108C2×C7×C4⋊C4C2311yescharacteristic
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C22⋊C4—22no—
111C2×C7×C22⋊C4—22no—
112C2×C7×C22⋊C4—21no—
113C2×C42×C7—21no—
114C2×C42×C7—21no—
115C2×C42×C7D412yes—
116C2×C42×C7C2311yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C2×C42×C7C2311yescharacteristic
119C7×#107—22no—
120C7×#106—22no—
121C7×C4⋊C8—21no—
122C7×C4⋊C8—21no—
123C7×C4⋊C8—21no—
124C7×C4⋊C8—21no—
125C7×#104—24no—
126C7×#104—42no—
127C7×#103—22no—
128C7×#102—22no—
129C7×#102—22no—
130C7×#102—22no—
131C7×#101—42no—
132C7×#95—22no—
133C7×#95—22no—
134#3507C2×C711yescharacteristic, complemented
135#3077C2×C711yescharacteristic, complemented
136#3036C2×C711yescharacteristic
137C4×#124C2×C711yescharacteristic, complemented
138#2038C2×C712yescomplemented
139#2003C2×C712yescomplemented
140#1994C2×C711yescharacteristic, complemented
141#1962C2×C711yescharacteristic, complemented
142#1883C2×C712yescomplemented
143#1860C2×C712yescomplemented
144#1801C2×C711yescharacteristic, complemented
145C24×C7—21no—
146C7×#41—22no—
147C7×#41—22no—
148C7×#41—22no—
149C7×#41—22no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8C2×D412yes—
159C2×C7×D4—22no—
160C2×C7×D4—21no—
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×D412yes—
168C2×C7×D4#4112yes—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—24no—
172C22×C4×C7—22no—
173C22×C4×C7—21no—
174C22×C4×C7—44no—
175C22×C4×C7—41no—
176C22×C4×C7—41no—
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C7×SD16—44no—
183C7×D8—44no—
184C7×M16—22no—
185C7×M16—22no—
186C7×M16—22no—
187C7×M16—22no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4C2×D412yes—
206C7×C4⋊C4C2×D412yes—
207C7×C22⋊C4—28no—
208C7×C22⋊C4—24no—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—44no—
215C7×C22⋊C4—41no—
216C7×C22⋊C4—41no—
217C42×C7—24no—
218C42×C7—22no—
219C42×C7—22no—
220C42×C7—22no—
221C42×C7—21no—
222C42×C7—21no—
223C42×C7—21no—
224C42×C7—42no—
225C42×C7—42no—
226C42×C7C2×D411yescharacteristic
227C42×C7C2×D411yescharacteristic
228C42×C7C2×D411yescharacteristic
229C42×C7C2×D411yescharacteristic
230C42×C7#4112yes—
231C2×AGL(1,ℤ/8)C22×C712yes—
232#559—21no—
233#557—21no—
234#556—21no—
235#556—22no—
236#554—21no—
237#552—22no—
238#539—22no—
239#536C22×C712yes—
240#533C22×C711yescharacteristic
241#532C22×C711yescharacteristic
242#529—22no—
243C2×#124C22×C711yescharacteristic
244#520C22×C712yes—
245C2×C4×Q8C22×C711yescharacteristic
246C2×C4×D4C22×C711yescharacteristic
247#478—22no—
248#476—21no—
249#474—21no—
250#463—22no—
251#461—21no—
252#459—21no—
253#428C22×C712yes—
254#423C22×C711yescharacteristic
255#423C22×C711yescharacteristic
256C2×#104C22×C712yes—
257C2×#104C22×C712yes—
258#418C22×C712yes—
259#417C22×C712yes—
260C2×#101C22×C712yes—
261#391C22×C711yescharacteristic
262#390C22×C711yescharacteristic
263#388—22no—
264#382C22×C711yescharacteristic
265C4×C22⋊C4—22no—
266C43C22×C711yescharacteristic
267#343C22×C712yes—
268#339C22×C712yes—
269#339C22×C712yes—
270C23×C7—22no—
271C23×C7—22no—
272C23×C7—42no—
273C23×C7—42no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7C22×D411yescharacteristic
277C7×Q8—22no—
278C7×Q8—22no—
279C7×Q8—41no—
280C7×Q8—41no—
281C7×Q8—41no—
282C7×Q8—41no—
283C7×Q8—42no—
284C7×Q8—81no—
285C7×D4—22no—
286C7×D4—22no—
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—42no—
290C7×D4—42no—
291C7×D4—41no—
292C7×D4—41no—
293C7×D4—42no—
294C7×D4—81no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—41no—
315C2×C4×C7—82no—
316C2×C4×C7—84no—
317C2×C4×C7—82no—
318C2×C4×C7#12011yescharacteristic
319C2×C4×C7#12011yescharacteristic
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7H(5,𝔽2)11yescharacteristic
327C2×C4×C72−1+411yescharacteristic
328C7×C8—42no—
329C7×C8—42no—
330C2×#41C23×C712yes—
331C22×Q8C23×C711yescharacteristic
332C22×D4C23×C711yescharacteristic
333C23×C4—21no—
334AGL(1,ℤ/8)—24no—
335AGL(1,ℤ/8)—42no—
336C2×SD16—22no—
337C2×D8—22no—
338C2×M16C23×C712yes—
339C2×M16C23×C712yes—
340C4⋊Q8—22no—
341C4⋊Q8C7×D411yescharacteristic
342C4⋊Q8C7×D411yescharacteristic
343C4⋊Q8C7×D411yescharacteristic
344#127C7×D411yescharacteristic
345#126—24no—
346#126—24no—
347#125—22no—
348#125—24no—
349#125C7×D412yes—
350#124—22no—
351#124—22no—
352#124—22no—
353#124—22no—
354#124C7×D412yes—
355#123—24no—
356#123—24no—
357C22⋊Q8—24no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362#121—21no—
363#121—21no—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—24no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381#117—24no—
382#117—22no—
383#117C23×C712yes—
384C2×C4⋊C4—22no—
385C2×C4⋊C4C7×D412yes—
386C2×C4⋊C4C23×C711yescharacteristic
387C2×C4⋊C4C23×C711yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C7×D412yes—
394C2×C42C23×C711yescharacteristic
395C2×C42C23×C711yescharacteristic
396C2×C42C23×C711yescharacteristic
397#107—22no—
398#106—22no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403#104—24no—
404#104—42no—
405#103—22no—
406#102—22no—
407#102—22no—
408#102—22no—
409#101—42no—
410#95—22no—
411#95—22no—
412C22×C7—21no—
413C22×C7—21no—
414C22×C7—44no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—82no—
420C22×C7—81no—
421C22×C7—81no—
422C22×C7C2×#12011yescharacteristic
423C22×C7#52511yescharacteristic
424C22×C7#53711yescharacteristic
425C4×C7—21no—
426C4×C7—21no—
427C4×C7—21no—
428C4×C7—42no—
429C4×C7—42no—
430C4×C7—42no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7—81no—
436C4×C7#45412yes—
437C4×C7#54611yescharacteristic
438C4×C7#54711yescharacteristic
439C4×C7#54811yescharacteristic
440C4×C7#55011yescharacteristic
441C24—21no—
442#41—22no—
443#41—22no—
444#41—22no—
445#41—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C2×C7×D412yes—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C2×C7×D412yes—
464C2×D4C7×#4112yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4C2×C7×D412yes—
474C22×C4C2×C7×D411yescharacteristic
475C22×C4C2×C7×D412yes—
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C24×C711yescharacteristic
478SD16—44no—
479D8—44no—
480M16—22no—
481M16—22no—
482M16—22no—
483M16—22no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4C2×C7×D412yes—
502C4⋊C4C2×C7×D412yes—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—44no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—42no—
522C42C2×C7×D411yescharacteristic
523C42C2×C7×D411yescharacteristic
524C42C2×C7×D411yescharacteristic
525C42C2×C7×D411yescharacteristic
526C42C7×#4112yes—
527C2×C7—21no—
528C2×C7—21no—
529C2×C7—82no—
530C2×C7—82no—
531C2×C7#240711yescharacteristic
532C2×C7C2×#45411yescharacteristic
533C2×C7#301811yescharacteristic
534C23—22no—
535C23—22no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23C22×C7×D411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—42no—
548Q8—81no—
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—42no—
554D4—42no—
555D4—41no—
556D4—41no—
557D4—42no—
558D4—81no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—82no—
582C2×C4C7×#12011yescharacteristic
583C2×C4C7×#12011yescharacteristic
584C2×C4C7×#12011yescharacteristic
585C2×C4C7×#12011yescharacteristic
586C2×C4C22×C7×D411yescharacteristic
587C2×C4C22×C7×D411yescharacteristic
588C2×C4C2×C7×#4111yescharacteristic
589C2×C4C2×C7×#4111yescharacteristic
590C2×C4C7×H(5,𝔽2)11yescharacteristic
591C2×C4C7×2−1+411yescharacteristic
592C8—42no—
593C8—42no—
594C7#2417711yescharacteristic, direct product factor, Sylow
595C22—21no—
596C22—21no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×C7×#12011yescharacteristic
606C22C7×#52511yescharacteristic
607C22C7×#53711yescharacteristic
608C4—21no—
609C4—21no—
610C4—21no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4C7×#45412yes—
620C4C7×#54611yescharacteristic
621C4C7×#54711yescharacteristic
622C4C7×#54811yescharacteristic
623C4C7×#55011yescharacteristic
624C2—21no—
625C2—21no—
626C2—82no—
627C2—82no—
628C2C7×#240711yescharacteristic
629C2C2×C7×#45411yescharacteristic
630C2C7×#301811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3036C2
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×#3507C2semidirect
2. C7×#3077C2semidirect
4. C4×C7×#124C2semidirect
5. C7×#2038C2semidirect
6. C7×#2003C2semidirect
7. C7×#1994C2semidirect
8. C7×#1962C2semidirect
9. C7×#1883C2semidirect
10. C7×#1860C2semidirect
11. C7×#1801C2semidirect
51. #24177C7direct (N × H)
134. #3507C2×C7semidirect
135. #3077C2×C7semidirect
137. C4×#124C2×C7semidirect
138. #2038C2×C7semidirect
139. #2003C2×C7semidirect
140. #1994C2×C7semidirect
141. #1962C2×C7semidirect
142. #1883C2×C7semidirect
143. #1860C2×C7semidirect
144. #1801C2×C7semidirect
594. C7#24177direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35072
C7×#30772
C7×#30362
C4×C7×#1242
C7×#20382
C7×#20032
C7×#19942
C7×#19622
C7×#18832
C7×#18602
C7×#18012
#241777