Finite Group Databasebeta

← C7×#24048 #13201264

Subgroups — C7×#24048 #13201264

Order 1792.

1492 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#546C211yesmaximal, characteristic
2C2×C7×#481C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C7×#2486C211yesmaximal, characteristic, complemented
5C7×#2042C212yesmaximal, complemented
6C7×#2035C212yesmaximal, complemented
7C7×#1895C211yesmaximal, characteristic, complemented
8C7×#1870C211yesmaximal, characteristic, complemented
9C7×#1864C212yesmaximal, complemented
10C7×#1788C211yesmaximal, characteristic, complemented
11C7×#1786C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×D4C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic, complemented
15C7×#546—24no—
16C7×#546—24no—
17C2×C7×#124C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×#123—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×#121—21no—
22C2×C7×#121C2211yescharacteristic
23C2×C7×#120—21no—
24C2×C4×C7×D4C2211yescharacteristic
25C2×C4×C7×D4C2211yescharacteristic
26C22×C7×C4⋊C4C2211yescharacteristic, complemented
27C22×C7×C22⋊C4—21no—
28C7×#481—24no—
29C2×C7×#106C2212yescomplemented
30C7×#417—24no—
31C2×C7×#103C2212yescomplemented
32C2×C7×#103C2212yescomplemented
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#102C2212yes—
36C2×C7×#102C2212yes—
37C2×C7×C22⋊C8C2212yes—
38C2×C7×C22⋊C8C2211yescharacteristic, complemented
39C7×#398—21no—
40C7×#397—21no—
41C7×#389C2212yescomplemented
42C7×#381—21no—
43C7×#380C2212yes—
44C7×#380C2211yescharacteristic, complemented
45C7×#340C2212yescomplemented
46C7×#336C2212yescomplemented
47#24048C711yesmaximal, characteristic, direct product factor, Sylow
48C25×C7—21no—
49C2×C7×#41—22no—
50C2×C7×#41—24no—
51C2×C7×#41—42no—
52C2×C7×#41D414yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4—22no—
57C22×C7×D4—22no—
58C22×C7×D4—24no—
59C22×C7×D4—21no—
60C22×C7×D4—42no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C23×C4×C7C2311yescharacteristic
68C23×C4×C7C2311yescharacteristic
69C2×C7×M16—22no—
70C2×C7×M16—42no—
71C22×C7×C8C2312yes—
72C22×C7×C8C2312yes—
73C7×#124—22no—
74C7×#124—22no—
75C7×#123—28no—
76C7×#123—44no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—22no—
79C7×#121—28no—
80C7×#121—22no—
81C7×#121D414yes—
82C7×#120—28no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2312yes—
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C22⋊C4—28no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4D411yescharacteristic
105C2×C7×C22⋊C4D411yescharacteristic
106C2×C42×C7C2311yescharacteristic
107C7×#106—24no—
108C7×#103—24no—
109C7×#103—42no—
110C7×#102—24no—
111C7×#102—24no—
112C7×#102—24no—
113C7×#102—42no—
114C7×C22⋊C8—22no—
115C7×C22⋊C8—42no—
116C7×#95—22no—
117C7×#95—21no—
118C7×#95—22no—
119C2×#546C2×C711yescharacteristic
120C2×#481C2×C711yescharacteristic, complemented
121C2×#417C2×C711yescharacteristic, complemented
122#2486C2×C711yescharacteristic, complemented
123#2042C2×C712yescomplemented
124#2035C2×C712yescomplemented
125#1895C2×C711yescharacteristic, complemented
126#1870C2×C711yescharacteristic, complemented
127#1864C2×C712yescomplemented
128#1788C2×C711yescharacteristic, complemented
129#1786C2×C712yescomplemented
130C24×C7—21no—
131C24×C7—24no—
132C24×C7—21no—
133C24×C7—44no—
134C24×C7—42no—
135C24×C7—42no—
136C24×C7—82no—
137C24×C7C2×D411yescharacteristic
138C24×C7C2×D411yescharacteristic
139C7×#41—28no—
140C7×#41—44no—
141C7×#41—48no—
142C2×C7×Q8—21no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8C2×D412yes—
146C2×C7×D4—24no—
147C2×C7×D4—22no—
148C2×C7×D4—24no—
149C2×C7×D4—22no—
150C2×C7×D4—24no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—41no—
157C2×C7×D4—44no—
158C2×C7×D4—48no—
159C2×C7×D4—44no—
160C2×C7×D4—44no—
161C2×C7×D4—44no—
162C2×C7×D4—82no—
163C2×C7×D4C2×D412yes—
164C2×C7×D4C2×D412yes—
165C2×C7×D4#4114yes—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—24no—
171C22×C4×C7—24no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—22no—
178C22×C4×C7—44no—
179C22×C4×C7—42no—
180C22×C4×C7—42no—
181C22×C4×C7—42no—
182C22×C4×C7—82no—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2×D412yes—
189C22×C4×C7C2411yescharacteristic
190C7×M16—44no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—44no—
205C7×C4⋊C4—42no—
206C7×C4⋊C4C2×D412yes—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—24no—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—24no—
214C7×C22⋊C4—42no—
215C7×C22⋊C4—44no—
216C42×C7—21no—
217C42×C7—21no—
218C22×#41C22×C711yescharacteristic
219C23×D4C22×C711yescharacteristic
220C22×M16C22×C711yescharacteristic, complemented
221#546—24no—
222#546—24no—
223C2×#124C22×C711yescharacteristic
224C2×#123—21no—
225C2×#123—21no—
226C2×C22⋊Q8C22×C711yescharacteristic
227C2×#121—21no—
228C2×#121C22×C711yescharacteristic
229C2×#120—21no—
230C2×C4×D4C22×C711yescharacteristic
231C2×C4×D4C22×C711yescharacteristic
232C22×C4⋊C4C22×C711yescharacteristic, complemented
233C22×C22⋊C4—21no—
234#481—24no—
235C2×#106C22×C712yescomplemented
236#417—24no—
237C2×#103C22×C712yescomplemented
238C2×#103C22×C712yescomplemented
239C2×#102C22×C712yes—
240C2×#102C22×C712yes—
241C2×#102C22×C712yes—
242C2×#102C22×C712yes—
243C2×C22⋊C8C22×C712yes—
244C2×C22⋊C8C22×C711yescharacteristic, complemented
245#398—21no—
246#397—21no—
247#389C22×C712yescomplemented
248#381—21no—
249#380C22×C712yes—
250#380C22×C711yescharacteristic, complemented
251#340C22×C712yescomplemented
252#336C22×C712yescomplemented
253C23×C7—21no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—22no—
257C23×C7—24no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—48no—
265C23×C7—48no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—42no—
269C23×C7—42no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—42no—
273C23×C7—88no—
274C23×C7—84no—
275C23×C7—88no—
276C23×C7#12012yes—
277C23×C7AGL(1,ℤ/8)12yes—
278C23×C7C22×D411yescharacteristic
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8—42no—
282C7×D4—22no—
283C7×D4—24no—
284C7×D4—24no—
285C7×D4—44no—
286C7×D4—48no—
287C7×D4—48no—
288C7×D4—41no—
289C7×D4—41no—
290C7×D4—84no—
291C7×D4#12314yescomplemented
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—24no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—22no—
305C2×C4×C7—48no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—42no—
316C2×C4×C7—42no—
317C2×C4×C7—84no—
318C2×C4×C7—84no—
319C2×C4×C7#12012yes—
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C22×D411yescharacteristic
322C2×C4×C7C2×#4112yes—
323C2×C4×C7H(5,𝔽2)12yes—
324C7×C8—42no—
325C7×C8—42no—
326C25—21no—
327C2×#41—22no—
328C2×#41—24no—
329C2×#41—42no—
330C2×#41C7×D414yes—
331C22×Q8C23×C711yescharacteristic
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—24no—
337C22×D4—21no—
338C22×D4—42no—
339C22×D4C7×D411yescharacteristic
340C22×D4C23×C711yescharacteristic
341C22×D4C23×C711yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4C23×C711yescharacteristic
346C23×C4C23×C711yescharacteristic
347C2×M16—22no—
348C2×M16—42no—
349C22×C8C23×C712yes—
350C22×C8C23×C712yes—
351#124—22no—
352#124—22no—
353#123—28no—
354#123—44no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357#121—28no—
358#121—22no—
359#121C7×D414yes—
360#120—28no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4C7×D411yescharacteristic
371C2×C4⋊C4C23×C712yes—
372C2×C4⋊C4C23×C712yes—
373C2×C4⋊C4C23×C711yescharacteristic
374C2×C22⋊C4—28no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4C7×D411yescharacteristic
383C2×C22⋊C4C7×D411yescharacteristic
384C2×C42C23×C711yescharacteristic
385#106—24no—
386#103—24no—
387#103—42no—
388#102—24no—
389#102—24no—
390#102—24no—
391#102—42no—
392C22⋊C8—22no—
393C22⋊C8—42no—
394#95—22no—
395#95—21no—
396#95—22no—
397C22×C7—22no—
398C22×C7—24no—
399C22×C7—22no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—24no—
403C22×C7—41no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—41no—
407C22×C7—42no—
408C22×C7—44no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—44no—
412C22×C7—42no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—816no—
416C22×C7—82no—
417C22×C7—82no—
418C22×C7C2×#12011yescharacteristic
419C22×C7#54612yes—
420C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
421C22×C7#57511yescharacteristic
422C22×C7#57712yes—
423C4×C7—22no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7—41no—
427C4×C7—41no—
428C4×C7—42no—
429C4×C7—82no—
430C4×C7—82no—
431C4×C7C2×#12311yescharacteristic
432C4×C7#53411yescharacteristic
433C4×C7#54612yes—
434C24—21no—
435C24—24no—
436C24—21no—
437C24—44no—
438C24—42no—
439C24—42no—
440C24—82no—
441C24C2×C7×D411yescharacteristic
442C24C2×C7×D411yescharacteristic
443#41—28no—
444#41—44no—
445#41—48no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8C2×C7×D412yes—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—41no—
461C2×D4—44no—
462C2×D4—48no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—82no—
467C2×D4C2×C7×D412yes—
468C2×D4C2×C7×D412yes—
469C2×D4C7×#4114yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—82no—
487C22×C4C2×C7×D411yescharacteristic
488C22×C4C2×C7×D411yescharacteristic
489C22×C4C2×C7×D411yescharacteristic
490C22×C4C2×C7×D411yescharacteristic
491C22×C4C2×C7×D412yes—
492C22×C4C2×C7×D412yes—
493C22×C4C24×C711yescharacteristic
494M16—44no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4C2×C7×D412yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C42—21no—
521C42—21no—
522C2×C7—42no—
523C2×C7—42no—
524C2×C7—44no—
525C2×C7—82no—
526C2×C7#240511yescharacteristic
527C2×C7#300611yescharacteristic
528C2×C7#300911yescharacteristic
529C2×C7#330612yes—
530C2×C7#332312yes—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—48no—
543C23—48no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—88no—
552C23—84no—
553C23—88no—
554C23C7×#12012yes—
555C23C7×AGL(1,ℤ/8)12yes—
556C23C22×C7×D411yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—42no—
560D4—22no—
561D4—24no—
562D4—24no—
563D4—44no—
564D4—48no—
565D4—48no—
566D4—41no—
567D4—41no—
568D4—84no—
569D4C7×#12314yescomplemented
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4C7×#12012yes—
598C2×C4C22×C7×D411yescharacteristic
599C2×C4C22×C7×D411yescharacteristic
600C2×C4C2×C7×#4112yes—
601C2×C4C7×H(5,𝔽2)12yes—
602C8—42no—
603C8—42no—
604C7#2404811yescharacteristic, direct product factor, Sylow
605C22—22no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—24no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—816no—
624C22—82no—
625C22—82no—
626C22C2×C7×#12011yescharacteristic
627C22C7×#54612yes—
628C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
629C22C7×#57511yescharacteristic
630C22C7×#57712yes—
631C4—22no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4C2×C7×#12311yescharacteristic
640C4C7×#53411yescharacteristic
641C4C7×#54612yes—
642C2—42no—
643C2—42no—
644C2—44no—
645C2—82no—
646C2C7×#240511yescharacteristic
647C2C7×#300611yescharacteristic
648C2C7×#300911yescharacteristic
649C2C7×#330612yes—
650C2C7×#332312yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#481C2semidirect
3. C2×C7×#417C2semidirect
4. C7×#2486C2semidirect
5. C7×#2042C2semidirect
6. C7×#2035C2semidirect
7. C7×#1895C2semidirect
8. C7×#1870C2semidirect
9. C7×#1864C2semidirect
10. C7×#1788C2semidirect
11. C7×#1786C2semidirect
14. C22×C7×M16C22semidirect
26. C22×C7×C4⋊C4C22semidirect
29. C2×C7×#106C22semidirect
31. C2×C7×#103C22semidirect
32. C2×C7×#103C22semidirect
38. C2×C7×C22⋊C8C22semidirect
41. C7×#389C22semidirect
44. C7×#380C22semidirect
45. C7×#340C22semidirect
46. C7×#336C22semidirect
47. #24048C7direct (N × H)
120. C2×#481C2×C7semidirect
121. C2×#417C2×C7semidirect
122. #2486C2×C7semidirect
123. #2042C2×C7semidirect
124. #2035C2×C7semidirect
125. #1895C2×C7semidirect
126. #1870C2×C7semidirect
127. #1864C2×C7semidirect
128. #1788C2×C7semidirect
129. #1786C2×C7semidirect
220. C22×M16C22×C7semidirect
232. C22×C4⋊C4C22×C7semidirect
235. C2×#106C22×C7semidirect
237. C2×#103C22×C7semidirect
238. C2×#103C22×C7semidirect
244. C2×C22⋊C8C22×C7semidirect
247. #389C22×C7semidirect
250. #380C22×C7semidirect
251. #340C22×C7semidirect
252. #336C22×C7semidirect
291. C7×D4#123semidirect
569. D4C7×#123semidirect
604. C7#24048direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5462
C2×C7×#4812
C2×C7×#4172
C7×#24862
C7×#20422
C7×#20352
C7×#18952
C7×#18702
C7×#18642
C7×#17882
C7×#17862
#240487