Finite Group Databasebeta

← C7×#16470 #13193686

Subgroups — C7×#16470 #13193686

Order 1792.

1438 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#397C211yesmaximal, characteristic
2C2×C7×#397C212yesmaximal
3C2×C7×#385C211yesmaximal, characteristic
4C7×#1442C212yesmaximal, complemented
5C7×#1442C212yesmaximal, complemented
6C7×#1441C212yesmaximal, complemented
7C7×#1441C212yesmaximal, complemented
8C7×#1439C211yesmaximal, characteristic
9C7×C22⋊#95C212yesmaximal
10C24×C4×C7C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C7×#397—28no—
20C7×#397—24no—
21C7×#385—24no—
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2211yescharacteristic
34#16470C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2311yescharacteristic
36C23×C4×C7—22no—
37C23×C4×C7—24no—
38C23×C4×C7—41no—
39C23×C4×C7—41no—
40C23×C4×C7—42no—
41C23×C4×C7C2312yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C2×C7×C4⋊C4—24no—
50C2×C7×C4⋊C4—24no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—42no—
55C2×C7×C4⋊C4—42no—
56C2×C7×C4⋊C4—41no—
57C2×C7×C4⋊C4—41no—
58C2×C7×C4⋊C4—41no—
59C2×C7×C4⋊C4—42no—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—41no—
68C2×C7×C22⋊C4—41no—
69C2×C7×C22⋊C4—42no—
70C2×C7×C22⋊C4D414yes—
71C2×C7×C22⋊C4D414yes—
72C7×#95—24no—
73C7×#95—24no—
74C7×#95—24no—
75C7×#95—22no—
76C7×#95—24no—
77C7×#95—24no—
78C7×#95—24no—
79C7×#95—24no—
80C7×#95—41no—
81C7×#95—42no—
82C7×#95—42no—
83C7×#95—42no—
84C2×#397C2×C711yescharacteristic
85C2×#397C2×C712yes—
86C2×#385C2×C711yescharacteristic
87#1442C2×C712yescomplemented
88#1442C2×C712yescomplemented
89#1441C2×C712yescomplemented
90#1441C2×C712yescomplemented
91#1439C2×C711yescharacteristic
92C22⋊#95C2×C712yes—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—21no—
96C24×C7—21no—
97C24×C7—22no—
98C24×C7—21no—
99C24×C7—21no—
100C24×C7—22no—
101C24×C7C2×D412yes—
102C24×C7C2×D412yes—
103C24×C7#4112yes—
104C24×C7C2411yescharacteristic
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
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—21no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
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—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
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—44no—
153C22×C4×C7—48no—
154C22×C4×C7—48no—
155C22×C4×C7—44no—
156C22×C4×C7—44no—
157C22×C4×C7C2×D414yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7#4114yes—
161C22×C4×C7#4114yes—
162C7×C4⋊C4—48no—
163C7×C4⋊C4—48no—
164C7×C4⋊C4—44no—
165C7×C4⋊C4—44no—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—44no—
168C7×C22⋊C4—48no—
169C7×C22⋊C4—48no—
170C24×C4C22×C711yescharacteristic
171C22×C4⋊C4C22×C712yes—
172C22×C4⋊C4C22×C712yes—
173C22×C4⋊C4C22×C711yescharacteristic
174C22×C4⋊C4C22×C711yescharacteristic
175C22×C22⋊C4C22×C712yes—
176C22×C22⋊C4C22×C712yes—
177C22×C22⋊C4C22×C711yescharacteristic
178C22×C22⋊C4C22×C711yescharacteristic
179#397—28no—
180#397—24no—
181#385—24no—
182C2×#95C22×C712yes—
183C2×#95C22×C712yes—
184C2×#95C22×C712yes—
185C2×#95C22×C712yes—
186C2×#95C22×C711yescharacteristic
187C2×#95C22×C712yes—
188C2×#95C22×C712yes—
189C2×#95C22×C712yes—
190C2×#95C22×C712yes—
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C2×#95C22×C711yescharacteristic
194C23×C7—24no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—22no—
198C23×C7—22no—
199C23×C7—22no—
200C23×C7—22no—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—41no—
211C23×C7—42no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—42no—
215C23×C7—41no—
216C23×C7#12114yes—
217C23×C7C22×D411yescharacteristic
218C23×C7C22×D411yescharacteristic
219C23×C7C2×#4111yescharacteristic
220C23×C7C2×#4112yes—
221C23×C7C2×#4112yes—
222C23×C7H(5,𝔽2)11yescharacteristic
223C23×C7H(5,𝔽2)11yescharacteristic
224C23×C7H(5,𝔽2)12yes—
225C23×C72−1+411yescharacteristic
226C23×C72−1+411yescharacteristic
227C23×C72−1+412yes—
228C2×C4×C7—24no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
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—416no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
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—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
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—
281C25C23×C711yescharacteristic
282C23×C4—22no—
283C23×C4—24no—
284C23×C4—41no—
285C23×C4—41no—
286C23×C4—42no—
287C23×C4C23×C712yes—
288C23×C4C23×C712yes—
289C23×C4C23×C712yes—
290C23×C4C23×C711yescharacteristic
291C23×C4C23×C712yes—
292C23×C4C23×C711yescharacteristic
293C23×C4C23×C712yes—
294C23×C4C23×C712yes—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—41no—
303C2×C4⋊C4—41no—
304C2×C4⋊C4—41no—
305C2×C4⋊C4—42no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4C7×D414yes—
317C2×C22⋊C4C7×D414yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—41no—
327#95—42no—
328#95—42no—
329#95—42no—
330C22×C7—24no—
331C22×C7—28no—
332C22×C7—42no—
333C22×C7—44no—
334C22×C7—42no—
335C22×C7—44no—
336C22×C7—44no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7C2×#12111yescharacteristic
342C22×C7#52912yes—
343C22×C7#53411yescharacteristic
344C22×C7#53611yescharacteristic
345C22×C7#53711yescharacteristic
346C22×C7#53711yescharacteristic
347C22×C7#53812yes—
348C22×C7#53911yescharacteristic
349C22×C7#53912yes—
350C22×C7#53911yescharacteristic
351C22×C7#53912yes—
352C22×C7#54112yes—
353C22×C7#54212yes—
354C22×C7#54612yes—
355C22×C7#54612yes—
356C22×C7#54712yes—
357C22×C7#54712yes—
358C22×C7#55111yescharacteristic
359C22×C7#55212yes—
360C22×C7#55911yescharacteristic
361C22×C7H(3,𝔽4)11yescharacteristic
362C22×C7#56312yes—
363C22×C7#56311yescharacteristic
364C4×C7—44no—
365C4×C7—42no—
366C4×C7—44no—
367C4×C7—44no—
368C4×C7—42no—
369C4×C7—82no—
370C4×C7—82no—
371C4×C7—82no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24C2×C7×D412yes—
381C24C2×C7×D412yes—
382C24C7×#4112yes—
383C24C24×C711yescharacteristic
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
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—22no—
412C22×C4—22no—
413C22×C4—22no—
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—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4C2×C7×D414yes—
437C22×C4C2×C7×D412yes—
438C22×C4C2×C7×D412yes—
439C22×C4C7×#4114yes—
440C22×C4C7×#4114yes—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C22⋊C4—28no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C2×C7—44no—
450C2×C7#242512yes—
451C2×C7#251711yescharacteristic
452C2×C7#261811yescharacteristic
453C2×C7#265212yes—
454C2×C7#268411yescharacteristic
455C2×C7#269512yes—
456C2×C7#269812yes—
457C2×C7#271211yescharacteristic
458C2×C7#272211yescharacteristic
459C2×C7#272411yescharacteristic
460C2×C7#283511yescharacteristic
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23C7×#12114yes—
484C23C22×C7×D411yescharacteristic
485C23C22×C7×D411yescharacteristic
486C23C2×C7×#4111yescharacteristic
487C23C2×C7×#4112yes—
488C23C2×C7×#4112yes—
489C23C7×H(5,𝔽2)11yescharacteristic
490C23C7×H(5,𝔽2)11yescharacteristic
491C23C7×H(5,𝔽2)12yes—
492C23C7×2−1+411yescharacteristic
493C23C7×2−1+411yescharacteristic
494C23C7×2−1+412yes—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
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—
548C7#1647011yescharacteristic, direct product factor, Sylow
549C22—28no—
550C22—24no—
551C22—44no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22C2×C7×#12111yescharacteristic
561C22C7×#52912yes—
562C22C7×#53411yescharacteristic
563C22C7×#53611yescharacteristic
564C22C7×#53711yescharacteristic
565C22C7×#53711yescharacteristic
566C22C7×#53812yes—
567C22C7×#53911yescharacteristic
568C22C7×#53912yes—
569C22C7×#53912yes—
570C22C7×#53911yescharacteristic
571C22C7×#54112yes—
572C22C7×#54212yes—
573C22C7×#54612yes—
574C22C7×#54612yes—
575C22C7×#54712yes—
576C22C7×#54712yes—
577C22C7×#55111yescharacteristic
578C22C7×#55212yes—
579C22C7×#55911yescharacteristic
580C22C7×H(3,𝔽4)11yescharacteristic
581C22C7×#56312yes—
582C22C7×#56311yescharacteristic
583C4—44no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—82no—
590C4—82no—
591C2—44no—
592C2C7×#242512yes—
593C2C7×#251711yescharacteristic
594C2C7×#261811yescharacteristic
595C2C7×#265212yes—
596C2C7×#268411yescharacteristic
597C2C7×#269512yes—
598C2C7×#269812yes—
599C2C7×#271211yescharacteristic
600C2C7×#272211yescharacteristic
601C2C7×#272411yescharacteristic
602C2C7×#283511yescharacteristic
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
4. C7×#1442C2semidirect
5. C7×#1442C2semidirect
6. C7×#1441C2semidirect
7. C7×#1441C2semidirect
34. #16470C7direct (N × H)
87. #1442C2×C7semidirect
88. #1442C2×C7semidirect
89. #1441C2×C7semidirect
90. #1441C2×C7semidirect
548. C7#16470direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3972
C2×C7×#3852
C7×#14422
C7×#14412
C7×#14392
C7×C22⋊#952
#164707