Finite Group Databasebeta

← C7×#24415 #13201631

Subgroups — C7×#24415 #13201631

Order 1792.

1402 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3485C211yesmaximal, characteristic
2C7×#3056C212yesmaximal, complemented
3C2×C7×#455C211yesmaximal, characteristic, complemented
4C2×C7×#413C211yesmaximal, characteristic, complemented
5C7×#2432C211yesmaximal, characteristic, complemented
6C7×#2038C212yesmaximal, complemented
7C7×#2005C212yesmaximal, complemented
8C7×#1885C211yesmaximal, characteristic, complemented
9C7×#1871C212yesmaximal, complemented
10C7×#1856C211yesmaximal, characteristic, complemented
11C7×#1792C211yesmaximal, characteristic, complemented
12C2×C7×H(5,𝔽2)C2211yescharacteristic
13C22×C7×#41C2211yescharacteristic
14C2×C7×#137C2212yes—
15C22×C7×M16C2212yescomplemented
16C7×#547—22no—
17C7×#547—22no—
18C7×#546—22no—
19C7×#546—22no—
20C7×#536C2212yes—
21C7×#534—22no—
22C2×C7×C4⋊Q8C2211yescharacteristic, complemented
23C2×C7×#124C2211yescharacteristic
24C7×#525C2212yes—
25C2×C7×#123—22no—
26C2×C7×C22⋊Q8C2211yescharacteristic, complemented
27C7×#518C2211yescharacteristic
28C7×#470—22no—
29C7×#468—22no—
30C7×#467—22no—
31C7×#465—22no—
32C7×#455—24no—
33C7×#428C2212yescomplemented
34C2×C7×#104C2212yes—
35C2×C7×#104C2212yes—
36C7×#419C2212yescomplemented
37C7×#417C2212yes—
38C7×#413—24no—
39C2×C7×#101C2212yescomplemented
40C2×C7×#100C2211yescharacteristic
41C2×C7×#100C2211yescharacteristic
42C7×#394—22no—
43C7×#391C2211yescharacteristic, complemented
44C7×#390C2211yescharacteristic
45C7×#385—22no—
46C4×C7×C22⋊C4C2211yescharacteristic, complemented
47C7×#339C2212yescomplemented
48C7×#339C2212yescomplemented
49#24415C711yesmaximal, characteristic, direct product factor, Sylow
50C7×H(5,𝔽2)—22no—
51C7×H(5,𝔽2)—22no—
52C7×H(5,𝔽2)—42no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41—22no—
56C2×C7×#41—22no—
57C2×C7×#41—22no—
58C2×C7×#41—42no—
59C2×C7×#41C2312yes—
60C22×C7×Q8C2311yescharacteristic
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4—22no—
63C22×C7×D4—22no—
64C22×C7×D4D412yes—
65C22×C7×D4C2311yescharacteristic
66C23×C4×C7D412yes—
67C23×C4×C7C2311yescharacteristic
68C7×#137—24no—
69C7×#137—42no—
70C2×C7×Q16—22no—
71C2×C7×SD16—22no—
72C2×C7×M16—22no—
73C2×C7×M16—24no—
74C2×C7×M16—42no—
75C2×C7×M16C2312yes—
76C2×C7×M16C2312yes—
77C22×C7×C8—22no—
78C7×C4⋊Q8—22no—
79C7×C4⋊Q8—42no—
80C7×#124—22no—
81C7×#124—22no—
82C7×#124—22no—
83C7×#123—24no—
84C7×#123—24no—
85C7×#123—24no—
86C7×#123—22no—
87C7×#123—24no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8D412yes—
93C7×C22⋊Q8D412yes—
94C7×#121—24no—
95C7×#121—24no—
96C7×#121D412yes—
97C7×#121D412yes—
98C7×#120—24no—
99C7×#120—22no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C4×C7×D4—22no—
104C7×#117C2312yes—
105C2×C7×C4⋊C4—22no—
106C2×C7×C4⋊C4—22no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4—22no—
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C22⋊C4—22no—
111C2×C7×C22⋊C4—22no—
112C2×C7×C22⋊C4—22no—
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—22no—
115C2×C7×C22⋊C4—22no—
116C2×C7×C22⋊C4—21no—
117C2×C42×C7C2311yescharacteristic
118C2×C42×C7C2311yescharacteristic
119C7×#107—22no—
120C7×#106—22no—
121C7×#104—24no—
122C7×#104—42no—
123C7×#103—22no—
124C7×#103—22no—
125C7×#103—22no—
126C7×#102—22no—
127C7×#101—24no—
128C7×#100—22no—
129C7×#100D414yescomplemented
130C7×#95—21no—
131C7×#95—22no—
132#3485C2×C711yescharacteristic
133#3056C2×C712yescomplemented
134C2×#455C2×C711yescharacteristic, complemented
135C2×#413C2×C711yescharacteristic, complemented
136#2432C2×C711yescharacteristic, complemented
137#2038C2×C712yescomplemented
138#2005C2×C712yescomplemented
139#1885C2×C711yescharacteristic, complemented
140#1871C2×C712yescomplemented
141#1856C2×C711yescharacteristic, complemented
142#1792C2×C711yescharacteristic, complemented
143C24×C7—22no—
144C24×C7C2×D411yescharacteristic
145C24×C7C2×D412yes—
146C7×#41—22no—
147C7×#41—22no—
148C7×#41—22no—
149C7×#41—22no—
150C7×#41—44no—
151C7×#41—44no—
152C7×#41—44no—
153C7×#41—44no—
154C7×#41—44no—
155C7×#41—44no—
156C2×C7×Q8—21no—
157C2×C7×Q8—22no—
158C2×C7×Q8—21no—
159C2×C7×Q8—22no—
160C2×C7×Q8—22no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—22no—
164C2×C7×Q8C2×D412yes—
165C2×C7×D4—24no—
166C2×C7×D4—24no—
167C2×C7×D4—24no—
168C2×C7×D4—22no—
169C2×C7×D4—22no—
170C2×C7×D4—22no—
171C2×C7×D4—22no—
172C2×C7×D4—22no—
173C2×C7×D4—42no—
174C2×C7×D4—44no—
175C2×C7×D4—44no—
176C2×C7×D4—44no—
177C2×C7×D4—44no—
178C2×C7×D4C2×D412yes—
179C2×C7×D4C2×D412yes—
180C2×C7×D4#4112yes—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—22no—
184C22×C4×C7—22no—
185C22×C4×C7—24no—
186C22×C4×C7—21no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—22no—
190C22×C4×C7—22no—
191C22×C4×C7—22no—
192C22×C4×C7—21no—
193C22×C4×C7—21no—
194C22×C4×C7—21no—
195C22×C4×C7—42no—
196C22×C4×C7C2×D412yes—
197C22×C4×C7C2×D411yescharacteristic
198C22×C4×C7C2×D412yes—
199C22×C4×C7C2×D412yes—
200C22×C4×C7C2411yescharacteristic
201C7×Q16—44no—
202C7×SD16—44no—
203C7×M16—22no—
204C7×M16—22no—
205C7×M16—22no—
206C7×M16—44no—
207C7×M16C2×D414yes—
208C2×C7×C8—22no—
209C2×C7×C8—22no—
210C2×C7×C8—44no—
211C7×C4⋊C4—24no—
212C7×C4⋊C4—24no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—22no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—44no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4—41no—
219C7×C4⋊C4C2×D412yes—
220C7×C4⋊C4C2×D412yes—
221C7×C22⋊C4—24no—
222C7×C22⋊C4—24no—
223C7×C22⋊C4—24no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—42no—
230C7×C22⋊C4—44no—
231C7×C22⋊C4—42no—
232C7×C22⋊C4—44no—
233C42×C7—22no—
234C42×C7—21no—
235C42×C7—21no—
236C42×C7—22no—
237C2×H(5,𝔽2)C22×C711yescharacteristic
238C22×#41C22×C711yescharacteristic
239C2×#137C22×C712yes—
240C22×M16C22×C712yescomplemented
241#547—22no—
242#547—22no—
243#546—22no—
244#546—22no—
245#536C22×C712yes—
246#534—22no—
247C2×C4⋊Q8C22×C711yescharacteristic, complemented
248C2×#124C22×C711yescharacteristic
249#525C22×C712yes—
250C2×#123—22no—
251C2×C22⋊Q8C22×C711yescharacteristic, complemented
252#518C22×C711yescharacteristic
253#470—22no—
254#468—22no—
255#467—22no—
256#465—22no—
257#455—24no—
258#428C22×C712yescomplemented
259C2×#104C22×C712yes—
260C2×#104C22×C712yes—
261#419C22×C712yescomplemented
262#417C22×C712yes—
263#413—24no—
264C2×#101C22×C712yescomplemented
265C2×#100C22×C711yescharacteristic
266C2×#100C22×C711yescharacteristic
267#394—22no—
268#391C22×C711yescharacteristic, complemented
269#390C22×C711yescharacteristic
270#385—22no—
271C4×C22⋊C4C22×C711yescharacteristic, complemented
272#339C22×C712yescomplemented
273#339C22×C712yescomplemented
274C23×C7—22no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—22no—
278C23×C7—22no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—42no—
282C23×C7—42no—
283C23×C7—44no—
284C23×C7—82no—
285C23×C7—82no—
286C23×C7#12012yes—
287C23×C7#12114yes—
288C23×C7C22×D411yescharacteristic
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—42no—
292C7×Q8—42no—
293C7×Q8—42no—
294C7×Q8—42no—
295C7×D4—24no—
296C7×D4—22no—
297C7×D4—24no—
298C7×D4—22no—
299C7×D4—48no—
300C7×D4—48no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—82no—
304C7×D4—82no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—44no—
319C2×C4×C7—44no—
320C2×C4×C7—41no—
321C2×C4×C7—44no—
322C2×C4×C7—41no—
323C2×C4×C7—42no—
324C2×C4×C7—42no—
325C2×C4×C7—41no—
326C2×C4×C7—42no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—41no—
333C2×C4×C7—82no—
334C2×C4×C7—82no—
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D412yes—
339C2×C4×C7C2×#4111yescharacteristic
340C2×C4×C7H(5,𝔽2)12yes—
341C7×C8—24no—
342C7×C8—42no—
343H(5,𝔽2)—22no—
344H(5,𝔽2)—22no—
345H(5,𝔽2)—42no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—42no—
352C2×#41C23×C712yes—
353C22×Q8C23×C711yescharacteristic
354C22×Q8C23×C711yescharacteristic
355C22×D4—22no—
356C22×D4—22no—
357C22×D4C7×D412yes—
358C22×D4C23×C711yescharacteristic
359C23×C4C7×D412yes—
360C23×C4C23×C711yescharacteristic
361#137—24no—
362#137—42no—
363C2×Q16—22no—
364C2×SD16—22no—
365C2×M16—22no—
366C2×M16—24no—
367C2×M16—42no—
368C2×M16C23×C712yes—
369C2×M16C23×C712yes—
370C22×C8—22no—
371C4⋊Q8—22no—
372C4⋊Q8—42no—
373#124—22no—
374#124—22no—
375#124—22no—
376#123—24no—
377#123—24no—
378#123—24no—
379#123—22no—
380#123—24no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8C7×D412yes—
386C22⋊Q8C7×D412yes—
387#121—24no—
388#121—24no—
389#121C7×D412yes—
390#121C7×D412yes—
391#120—24no—
392#120—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397#117C23×C712yes—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4C23×C711yescharacteristic
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—22no—
410C2×C42C23×C711yescharacteristic
411C2×C42C23×C711yescharacteristic
412#107—22no—
413#106—22no—
414#104—24no—
415#104—42no—
416#103—22no—
417#103—22no—
418#103—22no—
419#102—22no—
420#101—24no—
421#100—22no—
422#100C7×D414yescomplemented
423#95—21no—
424#95—22no—
425C22×C7—21no—
426C22×C7—21no—
427C22×C7—22no—
428C22×C7—24no—
429C22×C7—22no—
430C22×C7—22no—
431C22×C7—44no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—42no—
435C22×C7—42no—
436C22×C7—42no—
437C22×C7—81no—
438C22×C7—82no—
439C22×C7—81no—
440C22×C7—164no—
441C22×C7C2×#12011yescharacteristic
442C22×C7C2×#12111yescharacteristic
443C22×C7#53411yescharacteristic
444C4×C7—21no—
445C4×C7—21no—
446C4×C7—42no—
447C4×C7—42no—
448C4×C7—42no—
449C4×C7—42no—
450C4×C7—42no—
451C4×C7—81no—
452C4×C7—81no—
453C4×C7—81no—
454C4×C7D4212yes—
455C4×C7#54612yes—
456C24—22no—
457C24C2×C7×D411yescharacteristic
458C24C2×C7×D412yes—
459#41—22no—
460#41—22no—
461#41—22no—
462#41—22no—
463#41—44no—
464#41—44no—
465#41—44no—
466#41—44no—
467#41—44no—
468#41—44no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8C2×C7×D412yes—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—42no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4C2×C7×D412yes—
492C2×D4C2×C7×D412yes—
493C2×D4C7×#4112yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—42no—
509C22×C4C2×C7×D412yes—
510C22×C4C2×C7×D411yescharacteristic
511C22×C4C2×C7×D412yes—
512C22×C4C2×C7×D412yes—
513C22×C4C24×C711yescharacteristic
514Q16—44no—
515SD16—44no—
516M16—22no—
517M16—22no—
518M16—22no—
519M16—44no—
520M16C2×C7×D414yes—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—44no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4C2×C7×D412yes—
533C4⋊C4C2×C7×D412yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—42no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—22no—
550C2×C7—21no—
551C2×C7—21no—
552C2×C7—42no—
553C2×C7—44no—
554C2×C7—82no—
555C2×C7#241011yescharacteristic
556C2×C7#301811yescharacteristic
557C2×C7#302211yescharacteristic
558C23—22no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—82no—
569C23—82no—
570C23C7×#12012yes—
571C23C7×#12114yes—
572C23C22×C7×D411yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579D4—24no—
580D4—22no—
581D4—24no—
582D4—22no—
583D4—48no—
584D4—48no—
585D4—42no—
586D4—42no—
587D4—82no—
588D4—82no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4C7×#12011yescharacteristic
620C2×C4C7×#12011yescharacteristic
621C2×C4C22×C7×D411yescharacteristic
622C2×C4C22×C7×D412yes—
623C2×C4C2×C7×#4111yescharacteristic
624C2×C4C7×H(5,𝔽2)12yes—
625C8—24no—
626C8—42no—
627C7#2441511yescharacteristic, direct product factor, Sylow
628C22—21no—
629C22—21no—
630C22—22no—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—82no—
642C22—81no—
643C22—164no—
644C22C2×C7×#12011yescharacteristic
645C22C2×C7×#12111yescharacteristic
646C22C7×#53411yescharacteristic
647C4—21no—
648C4—21no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4C7×D4212yes—
658C4C7×#54612yes—
659C2—21no—
660C2—21no—
661C2—42no—
662C2—44no—
663C2—82no—
664C2C7×#241011yescharacteristic
665C2C7×#301811yescharacteristic
666C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3485C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3056C2semidirect
3. C2×C7×#455C2semidirect
4. C2×C7×#413C2semidirect
5. C7×#2432C2semidirect
6. C7×#2038C2semidirect
7. C7×#2005C2semidirect
8. C7×#1885C2semidirect
9. C7×#1871C2semidirect
10. C7×#1856C2semidirect
11. C7×#1792C2semidirect
15. C22×C7×M16C22semidirect
22. C2×C7×C4⋊Q8C22semidirect
26. C2×C7×C22⋊Q8C22semidirect
33. C7×#428C22semidirect
36. C7×#419C22semidirect
39. C2×C7×#101C22semidirect
43. C7×#391C22semidirect
46. C4×C7×C22⋊C4C22semidirect
47. C7×#339C22semidirect
48. C7×#339C22semidirect
49. #24415C7direct (N × H)
129. C7×#100D4semidirect
133. #3056C2×C7semidirect
134. C2×#455C2×C7semidirect
135. C2×#413C2×C7semidirect
136. #2432C2×C7semidirect
137. #2038C2×C7semidirect
138. #2005C2×C7semidirect
139. #1885C2×C7semidirect
140. #1871C2×C7semidirect
141. #1856C2×C7semidirect
142. #1792C2×C7semidirect
240. C22×M16C22×C7semidirect
247. C2×C4⋊Q8C22×C7semidirect
251. C2×C22⋊Q8C22×C7semidirect
258. #428C22×C7semidirect
261. #419C22×C7semidirect
264. C2×#101C22×C7semidirect
268. #391C22×C7semidirect
271. C4×C22⋊C4C22×C7semidirect
272. #339C22×C7semidirect
273. #339C22×C7semidirect
422. #100C7×D4semidirect
627. C7#24415direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34852
C7×#30562
C2×C7×#4552
C2×C7×#4132
C7×#24322
C7×#20382
C7×#20052
C7×#18852
C7×#18712
C7×#18562
C7×#17922
#244157