Finite Group Databasebeta

← C7×#24049 #13201265

Subgroups — C7×#24049 #13201265

Order 1792.

1472 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#546C211yesmaximal, characteristic
2C2×C7×#480C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C7×#2487C211yesmaximal, characteristic, complemented
5C7×#2042C212yesmaximal, complemented
6C7×#2036C212yesmaximal, complemented
7C7×#1895C211yesmaximal, characteristic, complemented
8C7×#1870C211yesmaximal, characteristic, complemented
9C7×#1862C212yesmaximal, complemented
10C7×#1788C211yesmaximal, characteristic, complemented
11C7×#1786C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×D4C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic, complemented
15C7×#546—24no—
16C7×#546—24no—
17C2×C7×#124C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121—21no—
21C2×C7×#121C2211yescharacteristic
22C2×C7×#120—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic, complemented
26C22×C7×C22⋊C4—21no—
27C7×#480—24no—
28C2×C7×#107C2212yescomplemented
29C7×#417—24no—
30C2×C7×#103C2212yescomplemented
31C2×C7×#103C2212yescomplemented
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#102C2212yes—
36C2×C7×C22⋊C8C2212yes—
37C2×C7×C22⋊C8C2211yescharacteristic, complemented
38C7×#400—21no—
39C7×#399—21no—
40C7×#389C2212yescomplemented
41C7×#386—21no—
42C7×#385—21no—
43C7×#380C2212yes—
44C7×#380C2211yescharacteristic, complemented
45C7×#340C2212yescomplemented
46C7×#336C2212yescomplemented
47#24049C711yesmaximal, characteristic, direct product factor, Sylow
48C25×C7—21no—
49C2×C7×#41—22no—
50C2×C7×#41—24no—
51C2×C7×#41—42no—
52C2×C7×#41D414yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4—22no—
57C22×C7×D4—22no—
58C22×C7×D4—24no—
59C22×C7×D4—21no—
60C22×C7×D4—42no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C23×C4×C7C2311yescharacteristic
68C23×C4×C7C2311yescharacteristic
69C2×C7×M16—22no—
70C2×C7×M16—42no—
71C22×C7×C8C2312yes—
72C22×C7×C8C2312yes—
73C7×#124—22no—
74C7×#124—22no—
75C7×#123—28no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×#121—28no—
79C7×#121—22no—
80C7×#121D414yes—
81C7×#120—28no—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C4×C7×D4—22no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C22⋊C4—28no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4D411yescharacteristic
104C2×C7×C22⋊C4D411yescharacteristic
105C2×C42×C7C2311yescharacteristic
106C7×#107—24no—
107C7×#103—24no—
108C7×#103—42no—
109C7×#102—24no—
110C7×#102—24no—
111C7×#102—24no—
112C7×#102—42no—
113C7×C22⋊C8—22no—
114C7×C22⋊C8—42no—
115C7×#95—22no—
116C7×#95—21no—
117C7×#95—22no—
118C2×#546C2×C711yescharacteristic
119C2×#480C2×C711yescharacteristic, complemented
120C2×#417C2×C711yescharacteristic, complemented
121#2487C2×C711yescharacteristic, complemented
122#2042C2×C712yescomplemented
123#2036C2×C712yescomplemented
124#1895C2×C711yescharacteristic, complemented
125#1870C2×C711yescharacteristic, complemented
126#1862C2×C712yescomplemented
127#1788C2×C711yescharacteristic, complemented
128#1786C2×C712yescomplemented
129C24×C7—21no—
130C24×C7—24no—
131C24×C7—21no—
132C24×C7—44no—
133C24×C7—42no—
134C24×C7—42no—
135C24×C7—82no—
136C24×C7C2×D411yescharacteristic
137C24×C7C2×D411yescharacteristic
138C7×#41—28no—
139C7×#41—44no—
140C7×#41—48no—
141C2×C7×Q8—21no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8C2×D412yes—
145C2×C7×D4—24no—
146C2×C7×D4—22no—
147C2×C7×D4—24no—
148C2×C7×D4—22no—
149C2×C7×D4—24no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—41no—
156C2×C7×D4—44no—
157C2×C7×D4—48no—
158C2×C7×D4—44no—
159C2×C7×D4—44no—
160C2×C7×D4—44no—
161C2×C7×D4—82no—
162C2×C7×D4C2×D412yes—
163C2×C7×D4C2×D412yes—
164C2×C7×D4#4114yes—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—24no—
170C22×C4×C7—24no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—44no—
178C22×C4×C7—42no—
179C22×C4×C7—82no—
180C22×C4×C7—82no—
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D412yes—
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2411yescharacteristic
188C7×M16—44no—
189C2×C7×C8—22no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C7×C4⋊C4—24no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—42no—
203C7×C4⋊C4—82no—
204C7×C4⋊C4C2×D412yes—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—24no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—24no—
212C7×C22⋊C4—42no—
213C7×C22⋊C4—82no—
214C42×C7—21no—
215C42×C7—21no—
216C22×#41C22×C711yescharacteristic
217C23×D4C22×C711yescharacteristic
218C22×M16C22×C711yescharacteristic, complemented
219#546—24no—
220#546—24no—
221C2×#124C22×C711yescharacteristic
222C2×#123—21no—
223C2×C22⋊Q8C22×C711yescharacteristic
224C2×#121—21no—
225C2×#121C22×C711yescharacteristic
226C2×#120—21no—
227C2×C4×D4C22×C711yescharacteristic
228C2×C4×D4C22×C711yescharacteristic
229C22×C4⋊C4C22×C711yescharacteristic, complemented
230C22×C22⋊C4—21no—
231#480—24no—
232C2×#107C22×C712yescomplemented
233#417—24no—
234C2×#103C22×C712yescomplemented
235C2×#103C22×C712yescomplemented
236C2×#102C22×C712yes—
237C2×#102C22×C712yes—
238C2×#102C22×C712yes—
239C2×#102C22×C712yes—
240C2×C22⋊C8C22×C712yes—
241C2×C22⋊C8C22×C711yescharacteristic, complemented
242#400—21no—
243#399—21no—
244#389C22×C712yescomplemented
245#386—21no—
246#385—21no—
247#380C22×C712yes—
248#380C22×C711yescharacteristic, complemented
249#340C22×C712yescomplemented
250#336C22×C712yescomplemented
251C23×C7—21no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—22no—
255C23×C7—24no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—22no—
262C23×C7—48no—
263C23×C7—48no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—42no—
267C23×C7—42no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—42no—
271C23×C7—88no—
272C23×C7—84no—
273C23×C7—88no—
274C23×C7#12012yes—
275C23×C7#13712yes—
276C23×C7C22×D411yescharacteristic
277C7×Q8—22no—
278C7×Q8—22no—
279C7×Q8—42no—
280C7×D4—22no—
281C7×D4—24no—
282C7×D4—24no—
283C7×D4—44no—
284C7×D4—48no—
285C7×D4—48no—
286C7×D4—41no—
287C7×D4—41no—
288C7×D4—84no—
289C7×D4#12314yes—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—24no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—48no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—84no—
316C2×C4×C7—84no—
317C2×C4×C7#12012yes—
318C2×C4×C7C22×D411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4112yes—
321C2×C4×C7H(5,𝔽2)12yes—
322C7×C8—42no—
323C7×C8—42no—
324C25—21no—
325C2×#41—22no—
326C2×#41—24no—
327C2×#41—42no—
328C2×#41C7×D414yes—
329C22×Q8C23×C711yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—22no—
333C22×D4—22no—
334C22×D4—24no—
335C22×D4—21no—
336C22×D4—42no—
337C22×D4C7×D411yescharacteristic
338C22×D4C23×C711yescharacteristic
339C22×D4C23×C711yescharacteristic
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—21no—
343C23×C4C23×C711yescharacteristic
344C23×C4C23×C711yescharacteristic
345C2×M16—22no—
346C2×M16—42no—
347C22×C8C23×C712yes—
348C22×C8C23×C712yes—
349#124—22no—
350#124—22no—
351#123—28no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354#121—28no—
355#121—22no—
356#121C7×D414yes—
357#120—28no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4C7×D411yescharacteristic
368C2×C4⋊C4C23×C712yes—
369C2×C4⋊C4C23×C712yes—
370C2×C4⋊C4C23×C711yescharacteristic
371C2×C22⋊C4—28no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4C7×D411yescharacteristic
380C2×C22⋊C4C7×D411yescharacteristic
381C2×C42C23×C711yescharacteristic
382#107—24no—
383#103—24no—
384#103—42no—
385#102—24no—
386#102—24no—
387#102—24no—
388#102—42no—
389C22⋊C8—22no—
390C22⋊C8—42no—
391#95—22no—
392#95—21no—
393#95—22no—
394C22×C7—22no—
395C22×C7—24no—
396C22×C7—22no—
397C22×C7—22no—
398C22×C7—22no—
399C22×C7—24no—
400C22×C7—41no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—41no—
404C22×C7—42no—
405C22×C7—44no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—44no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—816no—
413C22×C7—82no—
414C22×C7—82no—
415C22×C7C2×#12011yescharacteristic
416C22×C7#54612yes—
417C22×C7C2×#13711yescharacteristic
418C22×C7#57511yescharacteristic
419C22×C7#57612yes—
420C4×C7—22no—
421C4×C7—41no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7—82no—
427C4×C7—82no—
428C4×C7C2×#12311yescharacteristic
429C4×C7#53411yescharacteristic
430C4×C7#54612yes—
431C24—21no—
432C24—24no—
433C24—21no—
434C24—44no—
435C24—42no—
436C24—42no—
437C24—82no—
438C24C2×C7×D411yescharacteristic
439C24C2×C7×D411yescharacteristic
440#41—28no—
441#41—44no—
442#41—48no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8C2×C7×D412yes—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—41no—
458C2×D4—44no—
459C2×D4—48no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—82no—
464C2×D4C2×C7×D412yes—
465C2×D4C2×C7×D412yes—
466C2×D4C7×#4114yes—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C2×C7×D411yescharacteristic
484C22×C4C2×C7×D411yescharacteristic
485C22×C4C2×C7×D411yescharacteristic
486C22×C4C2×C7×D411yescharacteristic
487C22×C4C2×C7×D412yes—
488C22×C4C2×C7×D412yes—
489C22×C4C24×C711yescharacteristic
490M16—44no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4C2×C7×D412yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—82no—
516C42—21no—
517C42—21no—
518C2×C7—42no—
519C2×C7—42no—
520C2×C7—44no—
521C2×C7—82no—
522C2×C7#240511yescharacteristic
523C2×C7#300711yescharacteristic
524C2×C7#300811yescharacteristic
525C2×C7#331412yes—
526C2×C7#332212yes—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—88no—
548C23—84no—
549C23—88no—
550C23C7×#12012yes—
551C23C7×#13712yes—
552C23C22×C7×D411yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—42no—
556D4—22no—
557D4—24no—
558D4—24no—
559D4—44no—
560D4—48no—
561D4—48no—
562D4—41no—
563D4—41no—
564D4—84no—
565D4C7×#12314yes—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4C7×#12012yes—
594C2×C4C22×C7×D411yescharacteristic
595C2×C4C22×C7×D411yescharacteristic
596C2×C4C2×C7×#4112yes—
597C2×C4C7×H(5,𝔽2)12yes—
598C8—42no—
599C8—42no—
600C7#2404911yescharacteristic, direct product factor, Sylow
601C22—22no—
602C22—24no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22C2×C7×#12011yescharacteristic
623C22C7×#54612yes—
624C22C2×C7×#13711yescharacteristic
625C22C7×#57511yescharacteristic
626C22C7×#57612yes—
627C4—22no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4C2×C7×#12311yescharacteristic
636C4C7×#53411yescharacteristic
637C4C7×#54612yes—
638C2—42no—
639C2—42no—
640C2—44no—
641C2—82no—
642C2C7×#240511yescharacteristic
643C2C7×#300711yescharacteristic
644C2C7×#300811yescharacteristic
645C2C7×#331412yes—
646C2C7×#332212yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#480C2semidirect
3. C2×C7×#417C2semidirect
4. C7×#2487C2semidirect
5. C7×#2042C2semidirect
6. C7×#2036C2semidirect
7. C7×#1895C2semidirect
8. C7×#1870C2semidirect
9. C7×#1862C2semidirect
10. C7×#1788C2semidirect
11. C7×#1786C2semidirect
14. C22×C7×M16C22semidirect
25. C22×C7×C4⋊C4C22semidirect
28. C2×C7×#107C22semidirect
30. C2×C7×#103C22semidirect
31. C2×C7×#103C22semidirect
37. C2×C7×C22⋊C8C22semidirect
40. C7×#389C22semidirect
44. C7×#380C22semidirect
45. C7×#340C22semidirect
46. C7×#336C22semidirect
47. #24049C7direct (N × H)
119. C2×#480C2×C7semidirect
120. C2×#417C2×C7semidirect
121. #2487C2×C7semidirect
122. #2042C2×C7semidirect
123. #2036C2×C7semidirect
124. #1895C2×C7semidirect
125. #1870C2×C7semidirect
126. #1862C2×C7semidirect
127. #1788C2×C7semidirect
128. #1786C2×C7semidirect
218. C22×M16C22×C7semidirect
229. C22×C4⋊C4C22×C7semidirect
232. C2×#107C22×C7semidirect
234. C2×#103C22×C7semidirect
235. C2×#103C22×C7semidirect
241. C2×C22⋊C8C22×C7semidirect
244. #389C22×C7semidirect
248. #380C22×C7semidirect
249. #340C22×C7semidirect
250. #336C22×C7semidirect
600. C7#24049direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5462
C2×C7×#4802
C2×C7×#4172
C7×#24872
C7×#20422
C7×#20362
C7×#18952
C7×#18702
C7×#18622
C7×#17882
C7×#17862
#240497