Finite Group Databasebeta

← C7×#16566 #13193782

Subgroups — C7×#16566 #13193782

Order 1792.

1454 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#393C211yesmaximal, characteristic
2C2×C7×#391C211yesmaximal, characteristic, complemented
3C2×C7×#388C211yesmaximal, characteristic
4C2×C7×#382C211yesmaximal, characteristic, complemented
5C7×#1446C211yesmaximal, characteristic, complemented
6C7×#1444C211yesmaximal, characteristic, complemented
7C7×#1442C212yesmaximal, complemented
8C7×#1441C212yesmaximal, complemented
9C7×#1439C212yesmaximal
10C7×#1439C211yesmaximal, characteristic
11C7×#1439C212yesmaximal
12C23×C7×Q8C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2211yescharacteristic
21C22×C7×C22⋊C4C2212yes—
22C22×C42×C7C2211yescharacteristic
23C7×#393—24no—
24C7×#391—24no—
25C7×#388—24no—
26C7×#382—24no—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2211yescharacteristic
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2212yes—
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37C2×C7×#95C2212yes—
38C2×C7×#95C2211yescharacteristic
39#16566C711yesmaximal, characteristic, direct product factor, Sylow
40C25×C7C2311yescharacteristic
41C22×C7×Q8—22no—
42C22×C7×Q8—22no—
43C22×C7×Q8—24no—
44C22×C7×Q8—22no—
45C22×C7×Q8—42no—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2312yes—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—42no—
61C2×C7×C4⋊C4—42no—
62C2×C7×C4⋊C4—42no—
63C2×C7×C4⋊C4—42no—
64C2×C7×C4⋊C4—42no—
65C2×C7×C4⋊C4—42no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—42no—
75C2×C7×C22⋊C4—42no—
76C2×C7×C22⋊C4—42no—
77C2×C7×C22⋊C4—42no—
78C2×C7×C22⋊C4—42no—
79C2×C7×C22⋊C4Q814yescomplemented
80C2×C42×C7—22no—
81C2×C42×C7—22no—
82C2×C42×C7D414yes—
83C7×#95—24no—
84C7×#95—22no—
85C7×#95—22no—
86C7×#95—22no—
87C7×#95—24no—
88C7×#95—24no—
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—42no—
92C7×#95—42no—
93C7×#95—42no—
94C7×#95D414yes—
95C2×#393C2×C711yescharacteristic
96C2×#391C2×C711yescharacteristic, complemented
97C2×#388C2×C711yescharacteristic
98C2×#382C2×C711yescharacteristic, complemented
99#1446C2×C711yescharacteristic, complemented
100#1444C2×C711yescharacteristic, complemented
101#1442C2×C712yescomplemented
102#1441C2×C712yescomplemented
103#1439C2×C711yescharacteristic
104#1439C2×C712yes—
105#1439C2×C712yes—
106C24×C7—21no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—21no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—21no—
114C24×C7—22no—
115C24×C7—21no—
116C24×C7C2×Q812yes—
117C24×C7C2411yescharacteristic
118C2×C7×Q8—28no—
119C2×C7×Q8—48no—
120C2×C7×Q8—48no—
121C2×C7×Q8—44no—
122C2×C7×Q8—44no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—21no—
172C22×C4×C7—22no—
173C22×C4×C7C2×D414yes—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D412yes—
176C22×C4×C7C2×D412yes—
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×Q812yes—
179C22×C4×C7C2×Q812yes—
180C22×C4×C7#4114yes—
181C22×C4×C7#4112yes—
182C22×C4×C7#4112yes—
183C7×C4⋊C4—48no—
184C7×C4⋊C4—44no—
185C7×C4⋊C4—44no—
186C7×C4⋊C4—48no—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—48no—
189C7×C22⋊C4—44no—
190C7×C22⋊C4—48no—
191C7×C22⋊C4—44no—
192C42×C7—28no—
193C23×Q8C22×C711yescharacteristic
194C22×C4⋊C4C22×C712yes—
195C22×C4⋊C4C22×C711yescharacteristic
196C22×C4⋊C4C22×C711yescharacteristic
197C22×C4⋊C4C22×C712yes—
198C22×C22⋊C4C22×C711yescharacteristic
199C22×C22⋊C4C22×C712yes—
200C22×C22⋊C4C22×C711yescharacteristic
201C22×C22⋊C4C22×C711yescharacteristic
202C22×C22⋊C4C22×C712yes—
203C22×C42C22×C711yescharacteristic
204#393—24no—
205#391—24no—
206#388—24no—
207#382—24no—
208C2×#95C22×C712yes—
209C2×#95C22×C712yes—
210C2×#95C22×C712yes—
211C2×#95C22×C712yes—
212C2×#95C22×C712yes—
213C2×#95C22×C711yescharacteristic
214C2×#95C22×C711yescharacteristic
215C2×#95C22×C711yescharacteristic
216C2×#95C22×C712yes—
217C2×#95C22×C712yes—
218C2×#95C22×C712yes—
219C2×#95C22×C711yescharacteristic
220C23×C7—22no—
221C23×C7—24no—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—22no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—41no—
238C23×C7—42no—
239C23×C7—41no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7C22×D411yescharacteristic
244C23×C7C22×D411yescharacteristic
245C23×C7C22×Q811yescharacteristic
246C23×C7C2×#4111yescharacteristic
247C23×C7C2×#4111yescharacteristic
248C23×C7C2×#4112yes—
249C23×C7H(5,𝔽2)12yes—
250C23×C7H(5,𝔽2)12yes—
251C23×C7H(5,𝔽2)11yescharacteristic
252C23×C72−1+412yes—
253C23×C72−1+411yescharacteristic
254C7×Q8—416no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—42no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
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—42no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7#12114yes—
314C2×C4×C7C22⋊Q814yes—
315C25C23×C711yescharacteristic
316C22×Q8—22no—
317C22×Q8—22no—
318C22×Q8—24no—
319C22×Q8—22no—
320C22×Q8—42no—
321C23×C4C23×C711yescharacteristic
322C23×C4C23×C712yes—
323C23×C4C23×C711yescharacteristic
324C23×C4C23×C712yes—
325C23×C4C23×C712yes—
326C23×C4C23×C711yescharacteristic
327C23×C4C23×C711yescharacteristic
328C23×C4C23×C712yes—
329C23×C4C23×C711yescharacteristic
330C23×C4C23×C711yescharacteristic
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4—42no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4C7×Q814yescomplemented
355C2×C42—22no—
356C2×C42—22no—
357C2×C42C7×D414yes—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—42no—
367#95—42no—
368#95—42no—
369#95C7×D414yes—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—44no—
374C22×C7—44no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—82no—
380C22×C7—82no—
381C22×C7C2×#12111yescharacteristic
382C22×C7C2×C22⋊Q811yescharacteristic
383C22×C7#52511yescharacteristic
384C22×C7#53511yescharacteristic
385C22×C7#53511yescharacteristic
386C22×C7#53611yescharacteristic
387C22×C7#53711yescharacteristic
388C22×C7#53712yes—
389C22×C7#54112yes—
390C22×C7#54212yes—
391C22×C7#54212yes—
392C22×C7#54212yes—
393C22×C7#54412yes—
394C22×C7#54612yes—
395C22×C7#54611yescharacteristic
396C22×C7D4×Q811yescharacteristic
397C22×C7#55112yes—
398C22×C7#55211yescharacteristic
399C22×C7#55412yes—
400C22×C7#55411yescharacteristic
401C22×C7#55912yes—
402C22×C7H(3,𝔽4)12yes—
403C22×C7#56212yes—
404C4×C7—24no—
405C4×C7—24no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—44no—
409C4×C7—44no—
410C4×C7—81no—
411C4×C7—82no—
412C4×C7—81no—
413C4×C7—82no—
414C24—21no—
415C24—21no—
416C24—21no—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—21no—
422C24—22no—
423C24—21no—
424C24C2×C7×Q812yes—
425C24C24×C711yescharacteristic
426C2×Q8—28no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—44no—
430C2×Q8—44no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4C2×C7×D414yes—
482C22×C4C2×C7×D412yes—
483C22×C4C2×C7×D412yes—
484C22×C4C2×C7×D412yes—
485C22×C4C2×C7×D412yes—
486C22×C4C2×C7×Q812yes—
487C22×C4C2×C7×Q812yes—
488C22×C4C7×#4114yes—
489C22×C4C7×#4112yes—
490C22×C4C7×#4112yes—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C22⋊C4—28no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C42—28no—
501C2×C7—82no—
502C2×C7#241111yescharacteristic
503C2×C7#241711yescharacteristic
504C2×C7#258011yescharacteristic
505C2×C7#262011yescharacteristic
506C2×C7#262811yescharacteristic
507C2×C7#263711yescharacteristic
508C2×C7#270412yes—
509C2×C7#271212yes—
510C2×C7#275412yes—
511C2×C7#276312yes—
512C2×C7#281911yescharacteristic
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C22×C7×D411yescharacteristic
537C23C22×C7×D411yescharacteristic
538C23C22×C7×Q811yescharacteristic
539C23C2×C7×#4111yescharacteristic
540C23C2×C7×#4111yescharacteristic
541C23C2×C7×#4112yes—
542C23C7×H(5,𝔽2)12yes—
543C23C7×H(5,𝔽2)12yes—
544C23C7×H(5,𝔽2)11yescharacteristic
545C23C7×2−1+412yes—
546C23C7×2−1+411yescharacteristic
547Q8—416no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
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—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4C7×#12114yes—
607C2×C4C7×C22⋊Q814yes—
608C7#1656611yescharacteristic, direct product factor, Sylow
609C22—42no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C2×C7×#12111yescharacteristic
621C22C2×C7×C22⋊Q811yescharacteristic
622C22C7×#52511yescharacteristic
623C22C7×#53511yescharacteristic
624C22C7×#53511yescharacteristic
625C22C7×#53611yescharacteristic
626C22C7×#53711yescharacteristic
627C22C7×#53712yes—
628C22C7×#54112yes—
629C22C7×#54212yes—
630C22C7×#54212yes—
631C22C7×#54212yes—
632C22C7×#54412yes—
633C22C7×#54612yes—
634C22C7×#54611yescharacteristic
635C22C7×D4×Q811yescharacteristic
636C22C7×#55112yes—
637C22C7×#55211yescharacteristic
638C22C7×#55411yescharacteristic
639C22C7×#55412yes—
640C22C7×#55912yes—
641C22C7×H(3,𝔽4)12yes—
642C22C7×#56212yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—81no—
653C2—82no—
654C2C7×#241111yescharacteristic
655C2C7×#241711yescharacteristic
656C2C7×#258011yescharacteristic
657C2C7×#262011yescharacteristic
658C2C7×#262811yescharacteristic
659C2C7×#263711yescharacteristic
660C2C7×#270412yes—
661C2C7×#271212yes—
662C2C7×#275412yes—
663C2C7×#276312yes—
664C2C7×#281911yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#391C2semidirect
4. C2×C7×#382C2semidirect
5. C7×#1446C2semidirect
6. C7×#1444C2semidirect
7. C7×#1442C2semidirect
8. C7×#1441C2semidirect
39. #16566C7direct (N × H)
79. C2×C7×C22⋊C4Q8semidirect
96. C2×#391C2×C7semidirect
98. C2×#382C2×C7semidirect
99. #1446C2×C7semidirect
100. #1444C2×C7semidirect
101. #1442C2×C7semidirect
102. #1441C2×C7semidirect
354. C2×C22⋊C4C7×Q8semidirect
608. C7#16566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C2×C7×#3912
C2×C7×#3882
C2×C7×#3822
C7×#14462
C7×#14442
C7×#14422
C7×#14412
C7×#14392
#165667