Finite Group Databasebeta

← C7×#24217 #13201433

Subgroups — C7×#24217 #13201433

Order 1792.

1558 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#124C211yesmaximal, characteristic
2C2×C7×#464C212yesmaximal, complemented
3C2×C7×#451C211yesmaximal, characteristic, complemented
4C2×C7×#450C211yesmaximal, characteristic
5C4×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
6C7×#2042C212yesmaximal, complemented
7C7×#2015C212yesmaximal, complemented
8C7×#1983C212yesmaximal, complemented
9C7×#1797C211yesmaximal, characteristic, complemented
10C7×#1787C211yesmaximal, characteristic, complemented
11C7×#1786C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C23×C7×D4C2211yescharacteristic
14C22×C7×Q16C2212yescomplemented
15C22×C7×SD16C2212yes—
16C2×C7×#124—28no—
17C2×C7×#124—21no—
18C2×C7×#124—21no—
19C2×C7×#124C2212yes—
20C2×C7×#124C2212yes—
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic, complemented
23C22×C7×C22⋊C4—21no—
24C22×C7×C22⋊C4—21no—
25C22×C42×C7C2211yescharacteristic
26C7×#464—28no—
27C7×#451—24no—
28C7×#450—24no—
29C2×C7×C4⋊C8C2212yescomplemented
30C2×C7×#103C2212yescomplemented
31C2×C7×#103C2212yescomplemented
32C2×C7×#103C2212yescomplemented
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#102C2212yes—
36C2×C7×C22⋊C8C2211yescharacteristic
37C2×C7×C22⋊C8C2211yescharacteristic
38C7×#391C2211yescharacteristic, complemented
39C7×#389C2211yescharacteristic, complemented
40C7×#387—21no—
41C7×#385—21no—
42C7×#380C2211yescharacteristic
43C4×C7×C4⋊C4—21no—
44C4×C7×C4⋊C4C2211yescharacteristic, complemented
45C4×C7×C22⋊C4—21no—
46C7×#336C2212yescomplemented
47#24217C711yesmaximal, characteristic, direct product factor, Sylow
48C25×C7—21no—
49C22×C7×Q8—24no—
50C22×C7×Q8—21no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8—42no—
54C22×C7×Q8C2312yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—24no—
57C22×C7×D4—21no—
58C22×C7×D4—22no—
59C22×C7×D4—22no—
60C22×C7×D4—42no—
61C22×C7×D4C2312yes—
62C23×C4×C7—21no—
63C23×C4×C7D411yescharacteristic
64C23×C4×C7D411yescharacteristic
65C23×C4×C7C2311yescharacteristic
66C2×C7×Q16—24no—
67C2×C7×Q16—24no—
68C2×C7×Q16—42no—
69C2×C7×SD16—24no—
70C2×C7×SD16—24no—
71C2×C7×SD16—42no—
72C22×C7×C8C2312yes—
73C22×C7×C8C2312yes—
74C7×#124—28no—
75C7×#124—28no—
76C7×#124—24no—
77C7×#124—24no—
78C7×#124—24no—
79C7×#124D418yes—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8D414yes—
82C4×C7×Q8—24no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C22⋊C4—28no—
90C2×C7×C22⋊C4—28no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4—22no—
95C2×C7×C22⋊C4—22no—
96C2×C7×C22⋊C4—21no—
97C2×C42×C7—24no—
98C2×C42×C7—21no—
99C2×C42×C7—21no—
100C2×C42×C7D411yescharacteristic
101C2×C42×C7D411yescharacteristic
102C2×C42×C7C2311yescharacteristic
103C2×C42×C7C2311yescharacteristic
104C7×C4⋊C8—24no—
105C7×#103—24no—
106C7×#103—24no—
107C7×#103—42no—
108C7×#102—24no—
109C7×#102—24no—
110C7×#102—42no—
111C7×C22⋊C8—22no—
112C7×C22⋊C8—22no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95—21no—
116C7×#95—21no—
117C7×#95—21no—
118C22×#124C2×C711yescharacteristic
119C2×#464C2×C712yescomplemented
120C2×#451C2×C711yescharacteristic, complemented
121C2×#450C2×C711yescharacteristic
122C4×C22⋊Q8C2×C711yescharacteristic, complemented
123#2042C2×C712yescomplemented
124#2015C2×C712yescomplemented
125#1983C2×C712yescomplemented
126#1797C2×C711yescharacteristic, complemented
127#1787C2×C711yescharacteristic, complemented
128#1786C2×C711yescharacteristic
129C24×C7—24no—
130C24×C7—22no—
131C24×C7—42no—
132C24×C7—42no—
133C24×C7—42no—
134C24×C7—42no—
135C24×C7—42no—
136C24×C7—42no—
137C24×C7C2×D411yescharacteristic
138C2×C7×Q8—24no—
139C2×C7×Q8—22no—
140C2×C7×Q8—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8—22no—
146C2×C7×Q8—22no—
147C2×C7×Q8—48no—
148C2×C7×Q8—48no—
149C2×C7×Q8—42no—
150C2×C7×Q8—42no—
151C2×C7×Q8C2×D414yes—
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—24no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—48no—
160C2×C7×D4—48no—
161C2×C7×D4—42no—
162C2×C7×D4—42no—
163C2×C7×D4C2×D414yes—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—24no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—41no—
177C22×C4×C7—41no—
178C22×C4×C7—42no—
179C22×C4×C7—42no—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D412yes—
186C22×C4×C7#4112yes—
187C22×C4×C7C2411yescharacteristic
188C7×Q16—48no—
189C7×SD16—48no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C7×C4⋊C4—24no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—42no—
201C7×C4⋊C4—42no—
202C7×C4⋊C4C2×D412yes—
203C7×C22⋊C4—28no—
204C7×C22⋊C4—28no—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—28no—
207C7×C22⋊C4—28no—
208C7×C22⋊C4—42no—
209C42×C7—22no—
210C42×C7—22no—
211C42×C7—22no—
212C42×C7—42no—
213C42×C7—42no—
214C42×C7C2×D414yes—
215C23×Q8C22×C711yescharacteristic
216C23×D4C22×C711yescharacteristic
217C22×Q16C22×C712yescomplemented
218C22×SD16C22×C712yes—
219C2×#124—28no—
220C2×#124—21no—
221C2×#124—21no—
222C2×#124C22×C712yes—
223C2×#124C22×C712yes—
224C2×C22⋊Q8C22×C711yescharacteristic
225C2×C4×Q8C22×C711yescharacteristic, complemented
226C22×C22⋊C4—21no—
227C22×C22⋊C4—21no—
228C22×C42C22×C711yescharacteristic
229#464—28no—
230#451—24no—
231#450—24no—
232C2×C4⋊C8C22×C712yescomplemented
233C2×#103C22×C712yescomplemented
234C2×#103C22×C712yescomplemented
235C2×#103C22×C712yescomplemented
236C2×#102C22×C712yes—
237C2×#102C22×C712yes—
238C2×#102C22×C712yes—
239C2×C22⋊C8C22×C711yescharacteristic
240C2×C22⋊C8C22×C711yescharacteristic
241#391C22×C711yescharacteristic, complemented
242#389C22×C711yescharacteristic, complemented
243#387—21no—
244#385—21no—
245#380C22×C711yescharacteristic
246C4×C4⋊C4—21no—
247C4×C4⋊C4C22×C711yescharacteristic, complemented
248C4×C22⋊C4—21no—
249#336C22×C712yescomplemented
250C23×C7—21no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—24no—
255C23×C7—48no—
256C23×C7—48no—
257C23×C7—48no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—84no—
265C23×C7—84no—
266C23×C7—84no—
267C23×C7—84no—
268C23×C7#12112yes—
269C23×C7#13512yes—
270C23×C7#13712yes—
271C23×C7C22×D411yescharacteristic
272C7×Q8—24no—
273C7×Q8—24no—
274C7×Q8—24no—
275C7×Q8—48no—
276C7×Q8—42no—
277C7×Q8—42no—
278C7×Q8—42no—
279C7×D4—24no—
280C7×D4—24no—
281C7×D4—48no—
282C7×D4—42no—
283C7×D4—42no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—24no—
291C2×C4×C7—24no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—24no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—44no—
302C2×C4×C7—44no—
303C2×C4×C7—48no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7#12012yes—
317C2×C4×C7#12012yes—
318C2×C4×C7#12112yes—
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C22×D412yes—
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7H(5,𝔽2)12yes—
323C7×C8—42no—
324C7×C8—42no—
325C25—21no—
326C22×Q8—24no—
327C22×Q8—21no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—42no—
331C22×Q8C23×C711yescharacteristic
332C22×Q8C23×C712yes—
333C22×D4—24no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—42no—
338C22×D4C23×C712yes—
339C23×C4—21no—
340C23×C4C7×D411yescharacteristic
341C23×C4C7×D411yescharacteristic
342C23×C4C23×C711yescharacteristic
343C2×Q16—24no—
344C2×Q16—24no—
345C2×Q16—42no—
346C2×SD16—24no—
347C2×SD16—24no—
348C2×SD16—42no—
349C22×C8C23×C712yes—
350C22×C8C23×C712yes—
351#124—28no—
352#124—28no—
353#124—24no—
354#124—24no—
355#124—24no—
356#124C7×D418yes—
357C22⋊Q8—22no—
358C22⋊Q8C7×D414yes—
359C4×Q8—24no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C23×C711yescharacteristic
364C2×C4⋊C4C23×C711yescharacteristic
365C2×C4⋊C4C23×C711yescharacteristic
366C2×C22⋊C4—28no—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C42—24no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C7×D411yescharacteristic
378C2×C42C7×D411yescharacteristic
379C2×C42C23×C711yescharacteristic
380C2×C42C23×C711yescharacteristic
381C4⋊C8—24no—
382#103—24no—
383#103—24no—
384#103—42no—
385#102—24no—
386#102—24no—
387#102—42no—
388C22⋊C8—22no—
389C22⋊C8—22no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—22no—
398C22×C7—22no—
399C22×C7—22no—
400C22×C7—22no—
401C22×C7—44no—
402C22×C7—44no—
403C22×C7—44no—
404C22×C7—816no—
405C22×C7—82no—
406C22×C7—82no—
407C22×C7—82no—
408C22×C7—82no—
409C22×C7#46414yescomplemented
410C22×C7C2×#12011yescharacteristic
411C22×C7C2×#12111yescharacteristic
412C22×C7#53511yescharacteristic
413C22×C7C2×#13511yescharacteristic
414C22×C7C2×#13711yescharacteristic
415C22×C7#57511yescharacteristic
416C22×C7#57511yescharacteristic
417C4×C7—22no—
418C4×C7—22no—
419C4×C7—22no—
420C4×C7—44no—
421C4×C7—42no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7—42no—
425C4×C7—81no—
426C4×C7—81no—
427C4×C7D4212yes—
428C4×C7#54612yes—
429C24—24no—
430C24—22no—
431C24—42no—
432C24—42no—
433C24—42no—
434C24—42no—
435C24—42no—
436C24—42no—
437C24C2×C7×D411yescharacteristic
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—48no—
448C2×Q8—48no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8C2×C7×D414yes—
452C2×Q8C2×C7×D412yes—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—48no—
460C2×D4—48no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4C2×C7×D414yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C2×C7×D411yescharacteristic
481C22×C4C2×C7×D411yescharacteristic
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C2×C7×D411yescharacteristic
484C22×C4C2×C7×D411yescharacteristic
485C22×C4C2×C7×D412yes—
486C22×C4C7×#4112yes—
487C22×C4C24×C711yescharacteristic
488Q16—48no—
489SD16—48no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4C2×C7×D412yes—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—42no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—42no—
513C42—42no—
514C42C2×C7×D414yes—
515C2×C7—24no—
516C2×C7—84no—
517C2×C7#240811yescharacteristic
518C2×C7#300211yescharacteristic
519C2×C7#300711yescharacteristic
520C2×C7C2×#46411yescharacteristic
521C2×C7#304011yescharacteristic
522C2×C7#321111yescharacteristic
523C2×C7#321511yescharacteristic
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23C7×#12112yes—
543C23C7×#13512yes—
544C23C7×#13712yes—
545C23C22×C7×D411yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—24no—
549Q8—48no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553D4—24no—
554D4—24no—
555D4—48no—
556D4—42no—
557D4—42no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4C7×#12012yes—
591C2×C4C7×#12012yes—
592C2×C4C7×#12112yes—
593C2×C4C22×C7×D411yescharacteristic
594C2×C4C22×C7×D412yes—
595C2×C4C2×C7×#4111yescharacteristic
596C2×C4C7×H(5,𝔽2)12yes—
597C8—42no—
598C8—42no—
599C7#2421711yescharacteristic, direct product factor, Sylow
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—816no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22C7×#46414yescomplemented
615C22C2×C7×#12011yescharacteristic
616C22C2×C7×#12111yescharacteristic
617C22C7×#53511yescharacteristic
618C22C2×C7×#13511yescharacteristic
619C22C2×C7×#13711yescharacteristic
620C22C7×#57511yescharacteristic
621C22C7×#57511yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—44no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4C7×D4212yes—
633C4C7×#54612yes—
634C2—24no—
635C2—84no—
636C2C7×#240811yescharacteristic
637C2C7×#300211yescharacteristic
638C2C7×#300711yescharacteristic
639C2C2×C7×#46411yescharacteristic
640C2C7×#304011yescharacteristic
641C2C7×#321111yescharacteristic
642C2C7×#321511yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#464C2semidirect
3. C2×C7×#451C2semidirect
5. C4×C7×C22⋊Q8C2semidirect
6. C7×#2042C2semidirect
7. C7×#2015C2semidirect
8. C7×#1983C2semidirect
9. C7×#1797C2semidirect
10. C7×#1787C2semidirect
14. C22×C7×Q16C22semidirect
22. C2×C4×C7×Q8C22semidirect
29. C2×C7×C4⋊C8C22semidirect
30. C2×C7×#103C22semidirect
31. C2×C7×#103C22semidirect
32. C2×C7×#103C22semidirect
38. C7×#391C22semidirect
39. C7×#389C22semidirect
44. C4×C7×C4⋊C4C22semidirect
46. C7×#336C22semidirect
47. #24217C7direct (N × H)
119. C2×#464C2×C7semidirect
120. C2×#451C2×C7semidirect
122. C4×C22⋊Q8C2×C7semidirect
123. #2042C2×C7semidirect
124. #2015C2×C7semidirect
125. #1983C2×C7semidirect
126. #1797C2×C7semidirect
127. #1787C2×C7semidirect
217. C22×Q16C22×C7semidirect
225. C2×C4×Q8C22×C7semidirect
232. C2×C4⋊C8C22×C7semidirect
233. C2×#103C22×C7semidirect
234. C2×#103C22×C7semidirect
235. C2×#103C22×C7semidirect
241. #391C22×C7semidirect
242. #389C22×C7semidirect
247. C4×C4⋊C4C22×C7semidirect
249. #336C22×C7semidirect
409. C22×C7#464semidirect
599. C7#24217direct (N × H)
614. C22C7×#464semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1242
C2×C7×#4642
C2×C7×#4512
C2×C7×#4502
C4×C7×C22⋊Q82
C7×#20422
C7×#20152
C7×#19832
C7×#17972
C7×#17872
C7×#17862
#242177