Finite Group Databasebeta

← C7×#24011 #13201227

Subgroups — C7×#24011 #13201227

Order 1792.

1340 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C2×C7×#484C211yesmaximal, characteristic, complemented
3C2×C7×#480C211yesmaximal, characteristic, complemented
4C7×#2311C211yesmaximal, characteristic, complemented
5C7×#2036C212yesmaximal, complemented
6C7×#2033C212yesmaximal, complemented
7C7×#1875C211yesmaximal, characteristic, complemented
8C7×#1873C211yesmaximal, characteristic, complemented
9C7×#1852C212yesmaximal, complemented
10C7×#1788C212yesmaximal, complemented
11C7×#1780C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×#41C2211yescharacteristic
14C7×#537—24no—
15C7×#537—24no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8C2211yescharacteristic
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×#121—21no—
20C2×C7×#121—21no—
21C2×C7×#121—21no—
22C2×C7×#121C2211yescharacteristic
23C2×C7×#121C2211yescharacteristic
24C2×C7×#117C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic
26C22×C7×C4⋊C4C2211yescharacteristic
27C7×#484—24no—
28C7×#480—24no—
29C2×C7×#107C2212yes—
30C2×C7×#107C2212yes—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2211yescharacteristic
37C2×C7×C22⋊C8C2211yescharacteristic
38C2×C7×C22⋊C8C2212yes—
39C7×#391—21no—
40C7×#390—21no—
41C7×#386—21no—
42C7×#386—21no—
43C7×#380C2212yes—
44C7×#380C2212yes—
45C7×#340C2212yes—
46C7×#340C2212yes—
47#24011C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—24no—
53C2×C7×#41—24no—
54C2×C7×#41—42no—
55C2×C7×#41—42no—
56C22×C7×Q8D411yescharacteristic
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4—21no—
59C22×C7×D4—21no—
60C22×C7×D4—21no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4D411yescharacteristic
63C22×C7×D4D411yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C23×C4×C7—21no—
68C23×C4×C7C2311yescharacteristic
69C23×C4×C7C2311yescharacteristic
70C23×C4×C7C2311yescharacteristic
71C22×C7×C8C2312yes—
72C22×C7×C8C2312yes—
73C7×C22⋊Q8—28no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8D414yes—
78C7×#121—28no—
79C7×#121—28no—
80C7×#121—28no—
81C7×#121—22no—
82C7×#121—22no—
83C7×#121—22no—
84C7×#121D414yes—
85C7×#117—42no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4C2312yes—
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C4⋊C4C2311yescharacteristic
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—22no—
104C2×C7×C22⋊C4—22no—
105C2×C42×C7—21no—
106C7×#107—24no—
107C7×#107—24no—
108C7×#103—24no—
109C7×#103—42no—
110C7×#102—24no—
111C7×#102—42no—
112C7×C22⋊C8—22no—
113C7×C22⋊C8—22no—
114C7×C22⋊C8—24no—
115C7×C22⋊C8—42no—
116C7×#95—22no—
117C7×#95—22no—
118C2×#537C2×C711yescharacteristic
119C2×#484C2×C711yescharacteristic, complemented
120C2×#480C2×C711yescharacteristic, complemented
121#2311C2×C711yescharacteristic, complemented
122#2036C2×C712yescomplemented
123#2033C2×C712yescomplemented
124#1875C2×C711yescharacteristic, complemented
125#1873C2×C711yescharacteristic, complemented
126#1852C2×C712yescomplemented
127#1788C2×C712yescomplemented
128#1780C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C24×C7C2×D411yescharacteristic
133C7×#41—48no—
134C7×#41—48no—
135C7×#41—48no—
136C7×#41—48no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—21no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—41no—
143C2×C7×Q8C2×D412yes—
144C2×C7×D4—24no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—22no—
149C2×C7×D4—24no—
150C2×C7×D4—24no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—41no—
157C2×C7×D4—41no—
158C2×C7×D4—41no—
159C2×C7×D4—42no—
160C2×C7×D4—42no—
161C2×C7×D4—82no—
162C2×C7×D4—82no—
163C2×C7×D4C2×D412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—24no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—24no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—22no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—42no—
181C22×C4×C7—42no—
182C22×C4×C7—42no—
183C22×C4×C7—82no—
184C22×C4×C7—82no—
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2×D412yes—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7#4114yes—
191C22×C4×C7C2411yescharacteristic
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C7×C4⋊C4—24no—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—24no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—42no—
208C7×C4⋊C4C2×D412yes—
209C7×C4⋊C4C2×D412yes—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—24no—
212C7×C22⋊C4—24no—
213C7×C22⋊C4—24no—
214C7×C22⋊C4—42no—
215C7×C22⋊C4—82no—
216C7×C22⋊C4—82no—
217C42×C7—42no—
218C22×#41C22×C711yescharacteristic
219C22×#41C22×C711yescharacteristic
220#537—24no—
221#537—24no—
222C2×C22⋊Q8—21no—
223C2×C22⋊Q8C22×C711yescharacteristic
224C2×C22⋊Q8C22×C711yescharacteristic
225C2×#121—21no—
226C2×#121—21no—
227C2×#121—21no—
228C2×#121C22×C711yescharacteristic
229C2×#121C22×C711yescharacteristic
230C2×#117C22×C711yescharacteristic
231C22×C4⋊C4C22×C711yescharacteristic
232C22×C4⋊C4C22×C711yescharacteristic
233#484—24no—
234#480—24no—
235C2×#107C22×C712yes—
236C2×#107C22×C712yes—
237C2×#103C22×C712yes—
238C2×#103C22×C712yes—
239C2×#102C22×C712yes—
240C2×#102C22×C712yes—
241C2×C22⋊C8C22×C711yescharacteristic
242C2×C22⋊C8C22×C711yescharacteristic
243C2×C22⋊C8C22×C712yes—
244C2×C22⋊C8C22×C712yes—
245#391—21no—
246#390—21no—
247#386—21no—
248#386—21no—
249#380C22×C712yes—
250#380C22×C712yes—
251#340C22×C712yes—
252#340C22×C712yes—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—21no—
262C23×C7—22no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—42no—
270C23×C7—42no—
271C23×C7#12012yes—
272C23×C7#12312yes—
273C23×C7C22×D411yescharacteristic
274C7×Q8—24no—
275C7×Q8—44no—
276C7×Q8—42no—
277C7×D4—24no—
278C7×D4—44no—
279C7×D4—44no—
280C7×D4—44no—
281C7×D4—48no—
282C7×D4—42no—
283C7×D4—84no—
284C7×D4—84no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—24no—
290C2×C4×C7—24no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—48no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
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—42no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—84no—
313C2×C4×C7—84no—
314C2×C4×C7#12012yes—
315C2×C4×C7#12312yes—
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C2×#4112yes—
319C2×C4×C7H(5,𝔽2)12yes—
320C7×C8—42no—
321C7×C8—42no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41—24no—
327C2×#41—24no—
328C2×#41—42no—
329C2×#41—42no—
330C22×Q8C7×D411yescharacteristic
331C22×Q8C23×C711yescharacteristic
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4C7×D411yescharacteristic
336C22×D4C7×D411yescharacteristic
337C22×D4C7×D411yescharacteristic
338C22×D4C23×C711yescharacteristic
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4C23×C711yescharacteristic
343C23×C4C23×C711yescharacteristic
344C23×C4C23×C711yescharacteristic
345C22×C8C23×C712yes—
346C22×C8C23×C712yes—
347C22⋊Q8—28no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8C7×D414yes—
352#121—28no—
353#121—28no—
354#121—28no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121C7×D414yes—
359#117—42no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4C23×C712yes—
369C2×C4⋊C4C23×C712yes—
370C2×C4⋊C4C23×C711yescharacteristic
371C2×C4⋊C4C23×C711yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C42—21no—
380#107—24no—
381#107—24no—
382#103—24no—
383#103—42no—
384#102—24no—
385#102—42no—
386C22⋊C8—22no—
387C22⋊C8—22no—
388C22⋊C8—24no—
389C22⋊C8—42no—
390#95—22no—
391#95—22no—
392C22×C7—24no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—41no—
401C22×C7—41no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—41no—
405C22×C7—42no—
406C22×C7—41no—
407C22×C7—42no—
408C22×C7—41no—
409C22×C7—42no—
410C22×C7—82no—
411C22×C7—82no—
412C22×C7C2×#12011yescharacteristic
413C22×C7C2×#12311yescharacteristic
414C22×C7#53411yescharacteristic
415C22×C7#57612yes—
416C22×C7#57812yes—
417C4×C7—22no—
418C4×C7—22no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—82no—
424C4×C7—82no—
425C4×C7#54612yes—
426C4×C7#54612yes—
427C24—21no—
428C24C2×C7×D411yescharacteristic
429C24C2×C7×D411yescharacteristic
430C24C2×C7×D411yescharacteristic
431#41—48no—
432#41—48no—
433#41—48no—
434#41—48no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—41no—
441C2×Q8C2×C7×D412yes—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—82no—
460C2×D4—82no—
461C2×D4C2×C7×D412yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C2×C7×D411yescharacteristic
484C22×C4C2×C7×D411yescharacteristic
485C22×C4C2×C7×D412yes—
486C22×C4C2×C7×D412yes—
487C22×C4C2×C7×D411yescharacteristic
488C22×C4C7×#4114yes—
489C22×C4C24×C711yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4C2×C7×D412yes—
507C4⋊C4C2×C7×D412yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—42no—
516C2×C7—42no—
517C2×C7—42no—
518C2×C7—81no—
519C2×C7—82no—
520C2×C7#240511yescharacteristic
521C2×C7#300811yescharacteristic
522C2×C7#301111yescharacteristic
523C2×C7#309812yes—
524C2×C7#320312yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23C7×#12012yes—
544C23C7×#12312yes—
545C23C22×C7×D411yescharacteristic
546Q8—24no—
547Q8—44no—
548Q8—42no—
549D4—24no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—48no—
554D4—42no—
555D4—84no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4C7×#12012yes—
587C2×C4C7×#12312yes—
588C2×C4C22×C7×D411yescharacteristic
589C2×C4C22×C7×D411yescharacteristic
590C2×C4C2×C7×#4112yes—
591C2×C4C7×H(5,𝔽2)12yes—
592C8—42no—
593C8—42no—
594C7#2401111yescharacteristic, direct product factor, Sylow
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22C2×C7×#12011yescharacteristic
616C22C2×C7×#12311yescharacteristic
617C22C7×#53411yescharacteristic
618C22C7×#57612yes—
619C22C7×#57812yes—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—82no—
628C4C7×#54612yes—
629C4C7×#54612yes—
630C2—42no—
631C2—42no—
632C2—81no—
633C2—82no—
634C2C7×#240511yescharacteristic
635C2C7×#300811yescharacteristic
636C2C7×#301111yescharacteristic
637C2C7×#309812yes—
638C2C7×#320312yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#484C2semidirect
3. C2×C7×#480C2semidirect
4. C7×#2311C2semidirect
5. C7×#2036C2semidirect
6. C7×#2033C2semidirect
7. C7×#1875C2semidirect
8. C7×#1873C2semidirect
9. C7×#1852C2semidirect
10. C7×#1788C2semidirect
11. C7×#1780C2semidirect
47. #24011C7direct (N × H)
119. C2×#484C2×C7semidirect
120. C2×#480C2×C7semidirect
121. #2311C2×C7semidirect
122. #2036C2×C7semidirect
123. #2033C2×C7semidirect
124. #1875C2×C7semidirect
125. #1873C2×C7semidirect
126. #1852C2×C7semidirect
127. #1788C2×C7semidirect
128. #1780C2×C7semidirect
594. C7#24011direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C2×C7×#4842
C2×C7×#4802
C7×#23112
C7×#20362
C7×#20332
C7×#18752
C7×#18732
C7×#18522
C7×#17882
C7×#17802
#240117