Finite Group Databasebeta

← C7×#16596 #13193812

Subgroups — C7×#16596 #13193812

Order 1792.

1590 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C211yesmaximal, characteristic, complemented
2C2×C7×#394C212yesmaximal
3C2×C7×#390C211yesmaximal, characteristic
4C7×#1441C211yesmaximal, characteristic, complemented
5C7×#1440C212yesmaximal, complemented
6C7×#1440C212yesmaximal, complemented
7C7×#1439C212yesmaximal, complemented
8C7×#1439C212yesmaximal, complemented
9C7×#1436C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C7×#398—24no—
23C7×#394—28no—
24C7×#390—24no—
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#16596C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2312yes—
36C22×C7×D4—22no—
37C22×C7×D4—22no—
38C22×C7×D4—41no—
39C22×C7×D4—41no—
40C22×C7×D4—41no—
41C22×C7×D4—41no—
42C22×C7×D4D414yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2312yes—
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—41no—
56C2×C7×C4⋊C4—41no—
57C2×C7×C4⋊C4—41no—
58C2×C7×C4⋊C4—41no—
59C2×C7×C4⋊C4—42no—
60C2×C7×C4⋊C4—41no—
61C2×C7×C4⋊C4D414yes—
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—24no—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—42no—
73C2×C7×C22⋊C4—42no—
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⋊C4—42no—
80C2×C42×C7—22no—
81C2×C42×C7—41no—
82C2×C42×C7—41no—
83C7×#95—24no—
84C7×#95—22no—
85C7×#95—24no—
86C7×#95—24no—
87C7×#95—24no—
88C7×#95—42no—
89C7×#95—42no—
90C7×#95—42no—
91C7×#95—41no—
92C2×#398C2×C711yescharacteristic, complemented
93C2×#394C2×C712yes—
94C2×#390C2×C711yescharacteristic
95#1441C2×C711yescharacteristic, complemented
96#1440C2×C712yescomplemented
97#1440C2×C712yescomplemented
98#1439C2×C712yescomplemented
99#1439C2×C712yescomplemented
100#1436C2×C712yescomplemented
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7C2×D414yes—
116C24×C7C2411yescharacteristic
117C2×C7×D4—28no—
118C2×C7×D4—44no—
119C2×C7×D4—44no—
120C2×C7×D4—44no—
121C2×C7×D4—44no—
122C2×C7×D4—44no—
123C2×C7×D4—44no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
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—21no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
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—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—22no—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7#4114yes—
174C22×C4×C7#4114yes—
175C22×C4×C7#4112yes—
176C7×C4⋊C4—44no—
177C7×C4⋊C4—44no—
178C7×C4⋊C4—44no—
179C7×C4⋊C4—48no—
180C7×C22⋊C4—48no—
181C7×C22⋊C4—48no—
182C7×C22⋊C4—48no—
183C7×C22⋊C4—48no—
184C7×C22⋊C4—48no—
185C7×C22⋊C4—48no—
186C42×C7—44no—
187C23×D4C22×C711yescharacteristic
188C22×C4⋊C4C22×C712yes—
189C22×C4⋊C4C22×C711yescharacteristic
190C22×C4⋊C4C22×C711yescharacteristic
191C22×C4⋊C4C22×C711yescharacteristic
192C22×C22⋊C4C22×C712yes—
193C22×C22⋊C4C22×C712yes—
194C22×C22⋊C4C22×C712yes—
195C22×C22⋊C4C22×C712yes—
196C22×C22⋊C4C22×C712yes—
197C22×C22⋊C4C22×C712yes—
198C22×C42C22×C711yescharacteristic
199#398—24no—
200#394—28no—
201#390—24no—
202C2×#95C22×C712yes—
203C2×#95C22×C712yes—
204C2×#95C22×C711yescharacteristic
205C2×#95C22×C712yes—
206C2×#95C22×C712yes—
207C2×#95C22×C712yes—
208C2×#95C22×C712yes—
209C2×#95C22×C712yes—
210C2×#95C22×C711yescharacteristic
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7C22×D411yescharacteristic
247C23×C7C22×D411yescharacteristic
248C23×C7C2×#4111yescharacteristic
249C23×C7C2×#4112yes—
250C23×C7C2×#4112yes—
251C23×C7H(5,𝔽2)11yescharacteristic
252C23×C7H(5,𝔽2)11yescharacteristic
253C23×C7H(5,𝔽2)11yescharacteristic
254C23×C7H(5,𝔽2)11yescharacteristic
255C23×C7H(5,𝔽2)12yes—
256C23×C72−1+412yes—
257C7×D4—416no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
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—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
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#12114yes—
310C25C23×C712yes—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—41no—
314C22×D4—41no—
315C22×D4—41no—
316C22×D4—41no—
317C22×D4C7×D414yes—
318C23×C4C23×C712yes—
319C23×C4C23×C712yes—
320C23×C4C23×C712yes—
321C23×C4C23×C711yescharacteristic
322C23×C4C23×C711yescharacteristic
323C23×C4C23×C712yes—
324C23×C4C23×C712yes—
325C23×C4C23×C711yescharacteristic
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4C7×D414yes—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—42no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C42—22no—
356C2×C42—41no—
357C2×C42—41no—
358#95—24no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—41no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7—82no—
379C22×C7—82no—
380C22×C7—82no—
381C22×C7—82no—
382C22×C7C2×#12111yescharacteristic
383C22×C7#52912yes—
384C22×C7#53411yescharacteristic
385C22×C7#53411yescharacteristic
386C22×C7#53411yescharacteristic
387C22×C7#53511yescharacteristic
388C22×C7#53811yescharacteristic
389C22×C7#53811yescharacteristic
390C22×C7#53912yes—
391C22×C7#54012yes—
392C22×C7#54012yes—
393C22×C7#54212yes—
394C22×C7#54212yes—
395C22×C7#54612yes—
396C22×C7#54612yes—
397C22×C7#54712yes—
398C22×C7#54812yes—
399C22×C7#55312yes—
400C22×C7#55511yescharacteristic
401C22×C7#55911yescharacteristic
402C22×C7#56011yescharacteristic
403C22×C7#56212yes—
404C22×C7#56311yescharacteristic
405C4×C7—24no—
406C4×C7—44no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7—44no—
410C4×C7—82no—
411C4×C7—82no—
412C4×C7—82no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24C2×C7×D414yes—
428C24C24×C711yescharacteristic
429C2×D4—28no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
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—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4C2×C7×D412yes—
482C22×C4C2×C7×D412yes—
483C22×C4C2×C7×D412yes—
484C22×C4C2×C7×D412yes—
485C22×C4C7×#4114yes—
486C22×C4C7×#4114yes—
487C22×C4C7×#4112yes—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C42—44no—
499C2×C7—84no—
500C2×C7#242712yes—
501C2×C7#259311yescharacteristic
502C2×C7#261711yescharacteristic
503C2×C7#265112yes—
504C2×C7#269911yescharacteristic
505C2×C7#270112yes—
506C2×C7#270911yescharacteristic
507C2×C7#271112yes—
508C2×C7#272411yescharacteristic
509C2×C7#273111yescharacteristic
510C2×C7#283011yescharacteristic
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C22×C7×D411yescharacteristic
547C23C22×C7×D411yescharacteristic
548C23C2×C7×#4111yescharacteristic
549C23C2×C7×#4112yes—
550C23C2×C7×#4112yes—
551C23C7×H(5,𝔽2)11yescharacteristic
552C23C7×H(5,𝔽2)11yescharacteristic
553C23C7×H(5,𝔽2)11yescharacteristic
554C23C7×H(5,𝔽2)11yescharacteristic
555C23C7×H(5,𝔽2)12yes—
556C23C7×2−1+412yes—
557D4—416no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
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—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
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×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C7×#12114yes—
610C7#1659611yescharacteristic, direct product factor, Sylow
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×C7×#12111yescharacteristic
627C22C7×#52912yes—
628C22C7×#53411yescharacteristic
629C22C7×#53411yescharacteristic
630C22C7×#53411yescharacteristic
631C22C7×#53511yescharacteristic
632C22C7×#53811yescharacteristic
633C22C7×#53811yescharacteristic
634C22C7×#53912yes—
635C22C7×#54012yes—
636C22C7×#54012yes—
637C22C7×#54212yes—
638C22C7×#54212yes—
639C22C7×#54612yes—
640C22C7×#54612yes—
641C22C7×#54712yes—
642C22C7×#54812yes—
643C22C7×#55312yes—
644C22C7×#55511yescharacteristic
645C22C7×#55911yescharacteristic
646C22C7×#56011yescharacteristic
647C22C7×#56212yes—
648C22C7×#56311yescharacteristic
649C4—24no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C2—84no—
658C2C7×#242712yes—
659C2C7×#259311yescharacteristic
660C2C7×#261711yescharacteristic
661C2C7×#265112yes—
662C2C7×#269911yescharacteristic
663C2C7×#270112yes—
664C2C7×#270911yescharacteristic
665C2C7×#271112yes—
666C2C7×#272411yescharacteristic
667C2C7×#273111yescharacteristic
668C2C7×#283011yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#398C2semidirect
4. C7×#1441C2semidirect
5. C7×#1440C2semidirect
6. C7×#1440C2semidirect
7. C7×#1439C2semidirect
8. C7×#1439C2semidirect
9. C7×#1436C2semidirect
34. #16596C7direct (N × H)
92. C2×#398C2×C7semidirect
95. #1441C2×C7semidirect
96. #1440C2×C7semidirect
97. #1440C2×C7semidirect
98. #1439C2×C7semidirect
99. #1439C2×C7semidirect
100. #1436C2×C7semidirect
610. C7#16596direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3982
C2×C7×#3942
C2×C7×#3902
C7×#14412
C7×#14402
C7×#14392
C7×#14362
#165967