Finite Group Databasebeta

← C7×#24051 #13201267

Subgroups — C7×#24051 #13201267

Order 1792.

1352 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#548C211yesmaximal, characteristic
2C2×C7×#484C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C7×#2486C211yesmaximal, characteristic, complemented
5C7×#2042C212yesmaximal, complemented
6C7×#2033C212yesmaximal, complemented
7C7×#1894C211yesmaximal, characteristic, complemented
8C7×#1870C211yesmaximal, characteristic, complemented
9C7×#1863C212yesmaximal, complemented
10C7×#1788C211yesmaximal, characteristic, complemented
11C7×#1787C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C7×#548—24no—
16C7×#548—24no—
17C2×C7×#124—21no—
18C2×C7×#124C2211yescharacteristic
19C2×C7×#123—21no—
20C2×C7×C22⋊Q8—21no—
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C7×#121—21no—
23C2×C7×#121C2211yescharacteristic
24C2×C4×C7×Q8C2211yescharacteristic
25C2×C4×C7×D4—21no—
26C2×C4×C7×D4C2211yescharacteristic
27C22×C7×C4⋊C4C2211yescharacteristic
28C7×#484—24no—
29C2×C7×#107C2212yes—
30C7×#417—24no—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×#102C2212yes—
36C2×C7×#102C2212yes—
37C2×C7×C22⋊C8C2211yescharacteristic
38C2×C7×C22⋊C8C2212yes—
39C7×#398—21no—
40C7×#397—21no—
41C7×#389C2212yes—
42C7×#381—21no—
43C7×#380C2212yes—
44C7×#380C2211yescharacteristic
45C7×#340C2212yes—
46C7×#336C2212yes—
47#24051C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—24no—
50C2×C7×#41—42no—
51C2×C7×#41D414yes—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8—21no—
55C22×C7×Q8—24no—
56C22×C7×Q8—42no—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4—21no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C23×C4×C7C2311yescharacteristic
68C2×C7×M16—22no—
69C2×C7×M16—42no—
70C22×C7×C8C2312yes—
71C22×C7×C8C2312yes—
72C7×#124—28no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#123—44no—
76C7×C22⋊Q8—28no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8D414yes—
79C7×#121—28no—
80C7×#121—22no—
81C7×#121—22no—
82C4×C7×Q8—22no—
83C4×C7×Q8—22no—
84C4×C7×D4—28no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4D411yescharacteristic
94C2×C7×C4⋊C4C2312yes—
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C4⋊C4C2311yescharacteristic
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×C7—21no—
106C2×C42×C7C2311yescharacteristic
107C7×#107—24no—
108C7×#103—24no—
109C7×#103—24no—
110C7×#103—24no—
111C7×#103—42no—
112C7×#102—24no—
113C7×#102—42no—
114C7×C22⋊C8—22no—
115C7×C22⋊C8—42no—
116C7×#95—22no—
117C7×#95—21no—
118C7×#95—22no—
119C2×#548C2×C711yescharacteristic
120C2×#484C2×C711yescharacteristic, complemented
121C2×#417C2×C711yescharacteristic, complemented
122#2486C2×C711yescharacteristic, complemented
123#2042C2×C712yescomplemented
124#2033C2×C712yescomplemented
125#1894C2×C711yescharacteristic, complemented
126#1870C2×C711yescharacteristic, complemented
127#1863C2×C712yescomplemented
128#1788C2×C711yescharacteristic, complemented
129#1787C2×C712yescomplemented
130C24×C7—21no—
131C24×C7C2×D411yescharacteristic
132C24×C7C2×D411yescharacteristic
133C7×#41—28no—
134C7×#41—44no—
135C7×#41—48no—
136C2×C7×Q8—24no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—44no—
143C2×C7×Q8—48no—
144C2×C7×Q8—44no—
145C2×C7×Q8—44no—
146C2×C7×Q8C2×D412yes—
147C2×C7×Q8C2×D412yes—
148C2×C7×Q8#4114yes—
149C2×C7×D4—24no—
150C2×C7×D4—22no—
151C2×C7×D4—24no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—41no—
157C2×C7×D4—44no—
158C2×C7×D4—42no—
159C2×C7×D4—42no—
160C2×C7×D4C2×D412yes—
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—24no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—24no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—44no—
175C22×C4×C7—44no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7—42no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D412yes—
184C22×C4×C7C2×D412yes—
185C22×C4×C7C2411yescharacteristic
186C7×M16—44no—
187C2×C7×C8—22no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—44no—
202C7×C4⋊C4—42no—
203C7×C4⋊C4C2×D412yes—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—42no—
211C7×C22⋊C4—44no—
212C42×C7—24no—
213C42×C7—21no—
214C42×C7—21no—
215C22×#41C22×C711yescharacteristic
216C23×Q8C22×C711yescharacteristic
217C22×M16C22×C711yescharacteristic
218#548—24no—
219#548—24no—
220C2×#124—21no—
221C2×#124C22×C711yescharacteristic
222C2×#123—21no—
223C2×C22⋊Q8—21no—
224C2×C22⋊Q8C22×C711yescharacteristic
225C2×#121—21no—
226C2×#121C22×C711yescharacteristic
227C2×C4×Q8C22×C711yescharacteristic
228C2×C4×D4—21no—
229C2×C4×D4C22×C711yescharacteristic
230C22×C4⋊C4C22×C711yescharacteristic
231#484—24no—
232C2×#107C22×C712yes—
233#417—24no—
234C2×#103C22×C712yes—
235C2×#103C22×C712yes—
236C2×#103C22×C712yes—
237C2×#103C22×C712yes—
238C2×#102C22×C712yes—
239C2×#102C22×C712yes—
240C2×C22⋊C8C22×C711yescharacteristic
241C2×C22⋊C8C22×C712yes—
242#398—21no—
243#397—21no—
244#389C22×C712yes—
245#381—21no—
246#380C22×C711yescharacteristic
247#380C22×C712yes—
248#340C22×C712yes—
249#336C22×C712yes—
250C23×C7—21no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—42no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—42no—
263C23×C7#12012yes—
264C23×C7AGL(1,ℤ/8)12yes—
265C23×C7C22×D411yescharacteristic
266C7×Q8—22no—
267C7×Q8—24no—
268C7×Q8—24no—
269C7×Q8—48no—
270C7×Q8—41no—
271C7×Q8—41no—
272C7×Q8#12314yescomplemented
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—48no—
276C7×D4—44no—
277C7×D4—48no—
278C7×D4—42no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—48no—
295C2×C4×C7—48no—
296C2×C4×C7—42no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—84no—
309C2×C4×C7#12012yes—
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4112yes—
313C2×C4×C7H(5,𝔽2)12yes—
314C7×C8—42no—
315C7×C8—42no—
316C2×#41—22no—
317C2×#41—24no—
318C2×#41—42no—
319C2×#41C7×D414yes—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—21no—
323C22×Q8—24no—
324C22×Q8—42no—
325C22×Q8C23×C711yescharacteristic
326C22×Q8C23×C711yescharacteristic
327C22×D4—21no—
328C22×D4—21no—
329C22×D4C7×D411yescharacteristic
330C22×D4C23×C711yescharacteristic
331C23×C4—21no—
332C23×C4—21no—
333C23×C4—21no—
334C23×C4C23×C711yescharacteristic
335C23×C4C23×C711yescharacteristic
336C2×M16—22no—
337C2×M16—42no—
338C22×C8C23×C712yes—
339C22×C8C23×C712yes—
340#124—28no—
341#124—22no—
342#124—22no—
343#123—44no—
344C22⋊Q8—28no—
345C22⋊Q8—22no—
346C22⋊Q8C7×D414yes—
347#121—28no—
348#121—22no—
349#121—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×D4—28no—
353C4×D4—22no—
354C4×D4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C7×D411yescharacteristic
362C2×C4⋊C4C23×C712yes—
363C2×C4⋊C4C23×C712yes—
364C2×C4⋊C4C23×C711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C7×D411yescharacteristic
372C2×C22⋊C4C7×D411yescharacteristic
373C2×C42—21no—
374C2×C42C23×C711yescharacteristic
375#107—24no—
376#103—24no—
377#103—24no—
378#103—24no—
379#103—42no—
380#102—24no—
381#102—42no—
382C22⋊C8—22no—
383C22⋊C8—42no—
384#95—22no—
385#95—21no—
386#95—22no—
387C22×C7—22no—
388C22×C7—22no—
389C22×C7—22no—
390C22×C7—22no—
391C22×C7—24no—
392C22×C7—41no—
393C22×C7—41no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—82no—
401C22×C7—82no—
402C22×C7C2×#12011yescharacteristic
403C22×C7#54612yes—
404C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C7#57511yescharacteristic
406C22×C7#57812yes—
407C4×C7—22no—
408C4×C7—24no—
409C4×C7—41no—
410C4×C7—42no—
411C4×C7—41no—
412C4×C7—42no—
413C4×C7—41no—
414C4×C7—82no—
415C4×C7—82no—
416C4×C7C2×#12311yescharacteristic
417C4×C7#53411yescharacteristic
418C4×C7#54612yes—
419C24—21no—
420C24C2×C7×D411yescharacteristic
421C24C2×C7×D411yescharacteristic
422#41—28no—
423#41—44no—
424#41—48no—
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8C2×C7×D412yes—
436C2×Q8C2×C7×D412yes—
437C2×Q8C7×#4114yes—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4C2×C7×D412yes—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C2×C7×D412yes—
473C22×C4C2×C7×D412yes—
474C22×C4C24×C711yescharacteristic
475M16—44no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4C2×C7×D412yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C42—24no—
502C42—21no—
503C42—21no—
504C2×C7—42no—
505C2×C7—42no—
506C2×C7—82no—
507C2×C7#240511yescharacteristic
508C2×C7#300611yescharacteristic
509C2×C7#301111yescharacteristic
510C2×C7#330812yes—
511C2×C7#332712yes—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23C7×#12012yes—
526C23C7×AGL(1,ℤ/8)12yes—
527C23C22×C7×D411yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8—24no—
531Q8—48no—
532Q8—41no—
533Q8—41no—
534Q8C7×#12314yescomplemented
535D4—22no—
536D4—22no—
537D4—48no—
538D4—44no—
539D4—48no—
540D4—42no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4C7×#12012yes—
572C2×C4C22×C7×D411yescharacteristic
573C2×C4C22×C7×D411yescharacteristic
574C2×C4C2×C7×#4112yes—
575C2×C4C7×H(5,𝔽2)12yes—
576C8—42no—
577C8—42no—
578C7#2405111yescharacteristic, direct product factor, Sylow
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—24no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22C2×C7×#12011yescharacteristic
595C22C7×#54612yes—
596C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
597C22C7×#57511yescharacteristic
598C22C7×#57812yes—
599C4—22no—
600C4—24no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—82no—
607C4—82no—
608C4C2×C7×#12311yescharacteristic
609C4C7×#53411yescharacteristic
610C4C7×#54612yes—
611C2—42no—
612C2—42no—
613C2—82no—
614C2C7×#240511yescharacteristic
615C2C7×#300611yescharacteristic
616C2C7×#301111yescharacteristic
617C2C7×#330812yes—
618C2C7×#332712yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#548C2
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×#484C2semidirect
3. C2×C7×#417C2semidirect
4. C7×#2486C2semidirect
5. C7×#2042C2semidirect
6. C7×#2033C2semidirect
7. C7×#1894C2semidirect
8. C7×#1870C2semidirect
9. C7×#1863C2semidirect
10. C7×#1788C2semidirect
11. C7×#1787C2semidirect
47. #24051C7direct (N × H)
120. C2×#484C2×C7semidirect
121. C2×#417C2×C7semidirect
122. #2486C2×C7semidirect
123. #2042C2×C7semidirect
124. #2033C2×C7semidirect
125. #1894C2×C7semidirect
126. #1870C2×C7semidirect
127. #1863C2×C7semidirect
128. #1788C2×C7semidirect
129. #1787C2×C7semidirect
272. C7×Q8#123semidirect
534. Q8C7×#123semidirect
578. C7#24051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5482
C2×C7×#4842
C2×C7×#4172
C7×#24862
C7×#20422
C7×#20332
C7×#18942
C7×#18702
C7×#18632
C7×#17882
C7×#17872
#240517