Finite Group Databasebeta

← C7×#16481 #13193697

Subgroups — C7×#16481 #13193697

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#393C211yesmaximal, characteristic
2C2×C7×#385C211yesmaximal, characteristic
3C7×#1446C212yesmaximal, complemented
4C7×#1443C212yesmaximal, complemented
5C7×#1442C212yesmaximal, complemented
6C7×#1441C212yesmaximal, complemented
7C7×#1440C212yesmaximal
8C7×#1439C211yesmaximal, characteristic
9C7×C22⋊#95C212yesmaximal
10C23×C7×Q8C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2211yescharacteristic
18C7×#393—24no—
19C7×#385—24no—
20C2×C7×#95C2212yes—
21C2×C7×#95C2212yes—
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33#16481C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7C2311yescharacteristic
35C22×C7×Q8—22no—
36C22×C7×Q8—24no—
37C22×C7×Q8—42no—
38C22×C7×Q8—41no—
39C22×C7×Q8—41no—
40C23×C4×C7—22no—
41C23×C4×C7—24no—
42C23×C4×C7—41no—
43C23×C4×C7—41no—
44C23×C4×C7—42no—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2312yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—42no—
56C2×C7×C4⋊C4—41no—
57C2×C7×C4⋊C4—42no—
58C2×C7×C22⋊C4—22no—
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—41no—
64C2×C7×C22⋊C4—41no—
65C2×C7×C22⋊C4—42no—
66C2×C7×C22⋊C4—42no—
67C2×C7×C22⋊C4—42no—
68C2×C7×C22⋊C4D414yes—
69C2×C7×C22⋊C4Q814yescomplemented
70C7×#95—24no—
71C7×#95—24no—
72C7×#95—22no—
73C7×#95—24no—
74C7×#95—24no—
75C7×#95—42no—
76C7×#95—42no—
77C7×#95—41no—
78C7×#95—42no—
79C7×#95—42no—
80C7×#95—42no—
81C7×#95—42no—
82C7×#95—42no—
83C2×#393C2×C711yescharacteristic
84C2×#385C2×C711yescharacteristic
85#1446C2×C712yescomplemented
86#1443C2×C712yescomplemented
87#1442C2×C712yescomplemented
88#1441C2×C712yescomplemented
89#1440C2×C712yes—
90#1439C2×C711yescharacteristic
91C22⋊#95C2×C712yes—
92C24×C7—22no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—21no—
96C24×C7—21no—
97C24×C7—22no—
98C24×C7—21no—
99C24×C7—21no—
100C24×C7C2×D412yes—
101C24×C7C2×Q812yes—
102C24×C7#4112yes—
103C24×C7C2411yescharacteristic
104C2×C7×Q8—48no—
105C2×C7×Q8—44no—
106C2×C7×Q8—44no—
107C2×C7×Q8—48no—
108C2×C7×Q8—44no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
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—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—44no—
157C22×C4×C7—48no—
158C22×C4×C7—48no—
159C22×C4×C7—44no—
160C22×C4×C7—44no—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×Q814yes—
164C22×C4×C7#4114yes—
165C22×C4×C7#4114yes—
166C7×C4⋊C4—48no—
167C7×C4⋊C4—44no—
168C7×C22⋊C4—28no—
169C7×C22⋊C4—44no—
170C7×C22⋊C4—48no—
171C7×C22⋊C4—48no—
172C23×Q8C22×C711yescharacteristic
173C24×C4C22×C711yescharacteristic
174C22×C4⋊C4C22×C712yes—
175C22×C4⋊C4C22×C711yescharacteristic
176C22×C22⋊C4C22×C712yes—
177C22×C22⋊C4C22×C711yescharacteristic
178C22×C22⋊C4C22×C712yes—
179C22×C22⋊C4C22×C711yescharacteristic
180#393—24no—
181#385—24no—
182C2×#95C22×C712yes—
183C2×#95C22×C711yescharacteristic
184C2×#95C22×C712yes—
185C2×#95C22×C712yes—
186C2×#95C22×C712yes—
187C2×#95C22×C712yes—
188C2×#95C22×C712yes—
189C2×#95C22×C712yes—
190C2×#95C22×C711yescharacteristic
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C2×#95C22×C712yes—
195C23×C7—24no—
196C23×C7—22no—
197C23×C7—22no—
198C23×C7—24no—
199C23×C7—22no—
200C23×C7—22no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—41no—
212C23×C7—41no—
213C23×C7—42no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—42no—
217C23×C7C22⋊Q814yes—
218C23×C7C22×D411yescharacteristic
219C23×C7C22×Q811yescharacteristic
220C23×C7C2×#4111yescharacteristic
221C23×C7C2×#4112yes—
222C23×C7C2×#4112yes—
223C23×C7H(5,𝔽2)11yescharacteristic
224C23×C7H(5,𝔽2)11yescharacteristic
225C23×C7H(5,𝔽2)12yes—
226C23×C7H(5,𝔽2)12yes—
227C23×C72−1+411yescharacteristic
228C23×C72−1+411yescharacteristic
229C7×Q8—416no—
230C2×C4×C7—24no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—416no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C25C23×C711yescharacteristic
284C22×Q8—22no—
285C22×Q8—24no—
286C22×Q8—42no—
287C22×Q8—41no—
288C22×Q8—41no—
289C23×C4—22no—
290C23×C4—24no—
291C23×C4—41no—
292C23×C4—41no—
293C23×C4—42no—
294C23×C4C23×C712yes—
295C23×C4C23×C712yes—
296C23×C4C23×C712yes—
297C23×C4C23×C711yescharacteristic
298C23×C4C23×C711yescharacteristic
299C23×C4C23×C712yes—
300C23×C4C23×C712yes—
301C23×C4C23×C712yes—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4—42no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4C7×D414yes—
318C2×C22⋊C4C7×Q814yescomplemented
319#95—24no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—42no—
325#95—42no—
326#95—41no—
327#95—42no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—42no—
332C22×C7—24no—
333C22×C7—28no—
334C22×C7—42no—
335C22×C7—44no—
336C22×C7—42no—
337C22×C7—42no—
338C22×C7—44no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—44no—
343C22×C7C2×C22⋊Q811yescharacteristic
344C22×C7#52912yes—
345C22×C7#53411yescharacteristic
346C22×C7#53611yescharacteristic
347C22×C7#53812yes—
348C22×C7#53812yes—
349C22×C7#53812yes—
350C22×C7#53911yescharacteristic
351C22×C7#53911yescharacteristic
352C22×C7#54212yes—
353C22×C7#54212yes—
354C22×C7#54411yescharacteristic
355C22×C7#54411yescharacteristic
356C22×C7#54612yes—
357C22×C7#54612yes—
358C22×C7#55111yescharacteristic
359C22×C7#55111yescharacteristic
360C22×C7#55412yes—
361C22×C7#55412yes—
362C22×C7#55612yes—
363C22×C7#56011yescharacteristic
364C22×C7H(3,𝔽4)11yescharacteristic
365C22×C7#56212yes—
366C4×C7—42no—
367C4×C7—44no—
368C4×C7—42no—
369C4×C7—44no—
370C4×C7—44no—
371C4×C7—82no—
372C4×C7—82no—
373C4×C7—82no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24C2×C7×D412yes—
383C24C2×C7×Q812yes—
384C24C7×#4112yes—
385C24C24×C711yescharacteristic
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—44no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4C2×C7×D412yes—
444C22×C4C2×C7×D412yes—
445C22×C4C2×C7×Q814yes—
446C22×C4C7×#4114yes—
447C22×C4C7×#4114yes—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C22⋊C4—28no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C2×C7—44no—
455C2×C7#245012yes—
456C2×C7#252311yescharacteristic
457C2×C7#262711yescharacteristic
458C2×C7#265312yes—
459C2×C7#268411yescharacteristic
460C2×C7#269711yescharacteristic
461C2×C7#270012yes—
462C2×C7#274211yescharacteristic
463C2×C7#276311yescharacteristic
464C2×C7#278812yes—
465C2×C7#282411yescharacteristic
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23C7×C22⋊Q814yes—
489C23C22×C7×D411yescharacteristic
490C23C22×C7×Q811yescharacteristic
491C23C2×C7×#4111yescharacteristic
492C23C2×C7×#4112yes—
493C23C2×C7×#4112yes—
494C23C7×H(5,𝔽2)11yescharacteristic
495C23C7×H(5,𝔽2)11yescharacteristic
496C23C7×H(5,𝔽2)12yes—
497C23C7×H(5,𝔽2)12yes—
498C23C7×2−1+411yescharacteristic
499C23C7×2−1+411yescharacteristic
500Q8—416no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—416no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C7#1648111yescharacteristic, direct product factor, Sylow
555C22—28no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22C2×C7×C22⋊Q811yescharacteristic
567C22C7×#52912yes—
568C22C7×#53411yescharacteristic
569C22C7×#53611yescharacteristic
570C22C7×#53812yes—
571C22C7×#53812yes—
572C22C7×#53812yes—
573C22C7×#53911yescharacteristic
574C22C7×#53911yescharacteristic
575C22C7×#54212yes—
576C22C7×#54212yes—
577C22C7×#54411yescharacteristic
578C22C7×#54411yescharacteristic
579C22C7×#54612yes—
580C22C7×#54612yes—
581C22C7×#55111yescharacteristic
582C22C7×#55111yescharacteristic
583C22C7×#55412yes—
584C22C7×#55412yes—
585C22C7×#55612yes—
586C22C7×#56011yescharacteristic
587C22C7×H(3,𝔽4)11yescharacteristic
588C22C7×#56212yes—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—44no—
593C4—44no—
594C4—82no—
595C4—82no—
596C4—82no—
597C2—44no—
598C2C7×#245012yes—
599C2C7×#252311yescharacteristic
600C2C7×#262711yescharacteristic
601C2C7×#265312yes—
602C2C7×#268411yescharacteristic
603C2C7×#269711yescharacteristic
604C2C7×#270012yes—
605C2C7×#274211yescharacteristic
606C2C7×#276311yescharacteristic
607C2C7×#278812yes—
608C2C7×#282411yescharacteristic
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
3. C7×#1446C2semidirect
4. C7×#1443C2semidirect
5. C7×#1442C2semidirect
6. C7×#1441C2semidirect
33. #16481C7direct (N × H)
69. C2×C7×C22⋊C4Q8semidirect
85. #1446C2×C7semidirect
86. #1443C2×C7semidirect
87. #1442C2×C7semidirect
88. #1441C2×C7semidirect
318. C2×C22⋊C4C7×Q8semidirect
554. C7#16481direct (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×#3852
C7×#14462
C7×#14432
C7×#14422
C7×#14412
C7×#14402
C7×#14392
C7×C22⋊#952
#164817