Finite Group Databasebeta

← C7×#15961 #13193177

Subgroups — C7×#15961 #13193177

Order 1792.

2006 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×#386C212yesmaximal
2C2×C7×#385C212yesmaximal, complemented
3C2×C7×#385C212yesmaximal, complemented
4C7×#1445C211yesmaximal, characteristic, complemented
5C7×#1442C211yesmaximal, characteristic, complemented
6C7×#1441C212yesmaximal, complemented
7C7×#1439C212yesmaximal, complemented
8C7×#1439C212yesmaximal, complemented
9C7×#1438C211yesmaximal, characteristic
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2212yes—
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C7×#386—28no—
22C7×#385—28no—
23C7×#385—28no—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2211yescharacteristic
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2211yescharacteristic
33C2×C7×#95C2211yescharacteristic
34#15961C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2312yes—
36C22×C7×D4—24no—
37C22×C7×D4—24no—
38C22×C7×D4—41no—
39C22×C7×D4—41no—
40C22×C7×D4Q814yes—
41C23×C4×C7—24no—
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7—42no—
45C23×C4×C7—42no—
46C23×C4×C7—42no—
47C23×C4×C7—42no—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2312yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—41no—
61C2×C7×C4⋊C4—41no—
62C2×C7×C4⋊C4—41no—
63C2×C7×C4⋊C4—42no—
64C2×C7×C4⋊C4—42no—
65C2×C7×C4⋊C4—42no—
66C2×C7×C4⋊C4D414yescomplemented
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—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—24no—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—24no—
77C2×C7×C22⋊C4—42no—
78C2×C7×C22⋊C4—42no—
79C2×C7×C22⋊C4—42no—
80C2×C7×C22⋊C4—42no—
81C2×C7×C22⋊C4D418yes—
82C7×#95—24no—
83C7×#95—24no—
84C7×#95—24no—
85C7×#95—24no—
86C7×#95—24no—
87C7×#95—24no—
88C7×#95—22no—
89C7×#95—41no—
90C7×#95—41no—
91C7×#95—41no—
92C2×#386C2×C712yes—
93C2×#385C2×C712yescomplemented
94C2×#385C2×C712yescomplemented
95#1445C2×C711yescharacteristic, complemented
96#1442C2×C711yescharacteristic, complemented
97#1441C2×C712yescomplemented
98#1439C2×C712yescomplemented
99#1439C2×C712yescomplemented
100#1438C2×C711yescharacteristic
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×C7C2×D414yes—
114C24×C7C2×Q814yes—
115C24×C7#4114yes—
116C24×C7C2411yescharacteristic
117C2×C7×D4—216no—
118C2×C7×D4—44no—
119C2×C7×D4—44no—
120C2×C7×D4—48no—
121C2×C7×D4—44no—
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—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—21no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
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—21no—
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—48no—
166C22×C4×C7—48no—
167C22×C4×C7—48no—
168C22×C4×C7—48no—
169C22×C4×C7—48no—
170C22×C4×C7—48no—
171C22×C4×C7—48no—
172C22×C4×C7C2×D414yes—
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D414yes—
175C22×C4×C7C2×D414yes—
176C22×C4×C7C2×Q812yes—
177C22×C4×C7#4112yes—
178C22×C4×C7#4114yes—
179C7×C4⋊C4—44no—
180C7×C4⋊C4—44no—
181C7×C4⋊C4—48no—
182C7×C4⋊C4—48no—
183C7×C22⋊C4—216no—
184C7×C22⋊C4—216no—
185C7×C22⋊C4—48no—
186C7×C22⋊C4—48no—
187C7×C22⋊C4—48no—
188C23×D4C22×C711yescharacteristic
189C24×C4C22×C712yes—
190C22×C4⋊C4C22×C712yes—
191C22×C4⋊C4C22×C711yescharacteristic
192C22×C4⋊C4C22×C712yes—
193C22×C4⋊C4C22×C711yescharacteristic
194C22×C22⋊C4C22×C712yes—
195C22×C22⋊C4C22×C712yes—
196C22×C22⋊C4C22×C712yes—
197C22×C22⋊C4C22×C712yes—
198C22×C22⋊C4C22×C712yes—
199#386—28no—
200#385—28no—
201#385—28no—
202C2×#95C22×C711yescharacteristic
203C2×#95C22×C712yes—
204C2×#95C22×C712yes—
205C2×#95C22×C712yes—
206C2×#95C22×C712yes—
207C2×#95C22×C712yes—
208C2×#95C22×C712yes—
209C2×#95C22×C711yescharacteristic
210C2×#95C22×C711yescharacteristic
211C2×#95C22×C711yescharacteristic
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—24no—
229C23×C7—24no—
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⋊Q818yes—
247C23×C7C22×D412yes—
248C23×C7C22×D411yescharacteristic
249C23×C7C22×Q811yescharacteristic
250C23×C7C2×#4112yes—
251C23×C7C2×#4112yes—
252C23×C7C2×#4111yescharacteristic
253C23×C7H(5,𝔽2)11yescharacteristic
254C23×C7H(5,𝔽2)12yes—
255C23×C7H(5,𝔽2)11yescharacteristic
256C23×C7H(5,𝔽2)11yescharacteristic
257C23×C72−1+411yescharacteristic
258C7×D4—416no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—22no—
268C2×C4×C7—24no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—432no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—41no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
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—
310C25C23×C712yes—
311C22×D4—24no—
312C22×D4—24no—
313C22×D4—41no—
314C22×D4—41no—
315C22×D4C7×Q814yes—
316C23×C4—24no—
317C23×C4—24no—
318C23×C4—24no—
319C23×C4—42no—
320C23×C4—42no—
321C23×C4—42no—
322C23×C4—42no—
323C23×C4C23×C712yes—
324C23×C4C23×C711yescharacteristic
325C23×C4C23×C712yes—
326C23×C4C23×C712yes—
327C23×C4C23×C712yes—
328C23×C4C23×C711yescharacteristic
329C23×C4C23×C712yes—
330C23×C4C23×C711yescharacteristic
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4—41no—
337C2×C4⋊C4—41no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4—42no—
341C2×C4⋊C4C7×D414yescomplemented
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4C7×D418yes—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—41no—
365#95—41no—
366#95—41no—
367C22×C7—28no—
368C22×C7—28no—
369C22×C7—28no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7—44no—
375C22×C7—44no—
376C22×C7—44no—
377C22×C7—44no—
378C22×C7—44no—
379C22×C7—44no—
380C22×C7—44no—
381C22×C7—44no—
382C22×C7C2×C22⋊Q812yes—
383C22×C7#52512yes—
384C22×C7#52512yes—
385C22×C7#53412yes—
386C22×C7#53411yescharacteristic
387C22×C7#53812yes—
388C22×C7#53912yes—
389C22×C7#53911yescharacteristic
390C22×C7#54311yescharacteristic
391C22×C7D4211yescharacteristic
392C22×C7D4212yes—
393C22×C7#54612yes—
394C22×C7#54712yes—
395C22×C7D4×Q811yescharacteristic
396C22×C7#55011yescharacteristic
397C22×C7#55111yescharacteristic
398C22×C7#55312yes—
399C22×C7#55312yes—
400C22×C7#55411yescharacteristic
401C22×C7#55412yes—
402C22×C7#55911yescharacteristic
403C22×C7H(3,𝔽4)11yescharacteristic
404C22×C7#56211yescharacteristic
405C4×C7—42no—
406C4×C7—44no—
407C4×C7—44no—
408C4×C7—42no—
409C4×C7—44no—
410C4×C7—42no—
411C4×C7—44no—
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—
425C24C2×C7×D414yes—
426C24C2×C7×Q814yes—
427C24C7×#4114yes—
428C24C24×C711yescharacteristic
429C2×D4—216no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—44no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
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—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
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—21no—
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—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4C2×C7×D414yes—
485C22×C4C2×C7×D412yes—
486C22×C4C2×C7×D414yes—
487C22×C4C2×C7×D414yes—
488C22×C4C2×C7×Q812yes—
489C22×C4C7×#4112yes—
490C22×C4C7×#4114yes—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C2×C7—48no—
501C2×C7#243711yescharacteristic
502C2×C7#244912yes—
503C2×C7#245511yescharacteristic
504C2×C7#245612yes—
505C2×C7#257912yes—
506C2×C7#260011yescharacteristic
507C2×C7#267911yescharacteristic
508C2×C7#270812yes—
509C2×C7#273211yescharacteristic
510C2×C7#278211yescharacteristic
511C2×C7#280411yescharacteristic
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
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—
546C23C7×C22⋊Q818yes—
547C23C22×C7×D412yes—
548C23C22×C7×D411yescharacteristic
549C23C22×C7×Q811yescharacteristic
550C23C2×C7×#4112yes—
551C23C2×C7×#4112yes—
552C23C2×C7×#4111yescharacteristic
553C23C7×H(5,𝔽2)11yescharacteristic
554C23C7×H(5,𝔽2)12yes—
555C23C7×H(5,𝔽2)11yescharacteristic
556C23C7×H(5,𝔽2)11yescharacteristic
557C23C7×2−1+411yescharacteristic
558D4—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
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×C4—42no—
610C7#1596111yescharacteristic, direct product factor, Sylow
611C22—28no—
612C22—28no—
613C22—28no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22C2×C7×C22⋊Q812yes—
627C22C7×#52512yes—
628C22C7×#52512yes—
629C22C7×#53412yes—
630C22C7×#53411yescharacteristic
631C22C7×#53812yes—
632C22C7×#53912yes—
633C22C7×#53911yescharacteristic
634C22C7×#54311yescharacteristic
635C22C7×D4211yescharacteristic
636C22C7×D4212yes—
637C22C7×#54612yes—
638C22C7×#54712yes—
639C22C7×D4×Q811yescharacteristic
640C22C7×#55011yescharacteristic
641C22C7×#55111yescharacteristic
642C22C7×#55312yes—
643C22C7×#55312yes—
644C22C7×#55412yes—
645C22C7×#55411yescharacteristic
646C22C7×#55911yescharacteristic
647C22C7×H(3,𝔽4)11yescharacteristic
648C22C7×#56211yescharacteristic
649C4—44no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—82no—
657C2—48no—
658C2C7×#243711yescharacteristic
659C2C7×#244912yes—
660C2C7×#245511yescharacteristic
661C2C7×#245612yes—
662C2C7×#257912yes—
663C2C7×#260011yescharacteristic
664C2C7×#267911yescharacteristic
665C2C7×#270812yes—
666C2C7×#273211yescharacteristic
667C2C7×#278211yescharacteristic
668C2C7×#280411yescharacteristic
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
2. C2×C7×#385C2semidirect
3. C2×C7×#385C2semidirect
4. C7×#1445C2semidirect
5. C7×#1442C2semidirect
6. C7×#1441C2semidirect
7. C7×#1439C2semidirect
8. C7×#1439C2semidirect
34. #15961C7direct (N × H)
66. C2×C7×C4⋊C4D4semidirect
93. C2×#385C2×C7semidirect
94. C2×#385C2×C7semidirect
95. #1445C2×C7semidirect
96. #1442C2×C7semidirect
97. #1441C2×C7semidirect
98. #1439C2×C7semidirect
99. #1439C2×C7semidirect
341. C2×C4⋊C4C7×D4semidirect
610. C7#15961direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3862
C2×C7×#3852
C7×#14452
C7×#14422
C7×#14412
C7×#14392
C7×#14382
#159617