Finite Group Databasebeta

← C7×#23982 #13201198

Subgroups — C7×#23982 #13201198

Order 1792.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic, complemented
2C2×C7×#450C211yesmaximal, characteristic, complemented
3C2×C7×#418C211yesmaximal, characteristic, complemented
4C7×#2408C211yesmaximal, characteristic
5C7×#2025C212yesmaximal, complemented
6C7×#2000C212yesmaximal, complemented
7C7×#1894C211yesmaximal, characteristic, complemented
8C7×#1880C212yesmaximal, complemented
9C7×#1874C212yesmaximal, complemented
10C7×#1870C211yesmaximal, characteristic, complemented
11C7×#1825C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C22×C7×M16C2211yescharacteristic
15C7×#552—24no—
16C7×#552—24no—
17C2×C7×#126—21no—
18C2×C7×#125—21no—
19C2×C7×#125C2211yescharacteristic, complemented
20C2×C7×#123—21no—
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C7×C22⋊Q8C2211yescharacteristic, complemented
23C2×C7×#121C2212yes—
24C2×C7×#120—21no—
25C2×C4×C7×Q8C2211yescharacteristic, complemented
26C2×C4×C7×D4C2211yescharacteristic
27C2×C7×#117—21no—
28C2×C7×#117C2211yescharacteristic
29C22×C7×C4⋊C4C2211yescharacteristic, complemented
30C7×#450—24no—
31C7×#418—24no—
32C2×C7×#103C2212yescomplemented
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#102C2212yes—
36C2×C7×C22⋊C8C2211yescharacteristic
37C2×C7×C22⋊C8C2212yescomplemented
38C7×#394—21no—
39C7×#392—21no—
40C7×#390C2212yes—
41C7×#380C2211yescharacteristic
42C7×#379—21no—
43C7×#340C2212yescomplemented
44C7×#340C2212yescomplemented
45C7×#340C2212yescomplemented
46C7×#336C2212yescomplemented
47#23982C711yesmaximal, characteristic, direct product factor, Sylow
48C25×C7—21no—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×D4—24no—
51C22×C7×D4—22no—
52C22×C7×D4—22no—
53C22×C7×D4—22no—
54C22×C7×D4—21no—
55C22×C7×D4—42no—
56C22×C7×D4D411yescharacteristic
57C22×C7×D4C2312yes—
58C23×C4×C7—21no—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C2×C7×SD16—24no—
62C2×C7×SD16—24no—
63C2×C7×SD16—42no—
64C2×C7×M16—22no—
65C2×C7×M16—42no—
66C22×C7×C8C2312yes—
67C22×C7×C8C2312yes—
68C7×#126—28no—
69C7×#125—28no—
70C7×#125—22no—
71C7×#125—22no—
72C7×#123—28no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8D414yescomplemented
77C7×#121—24no—
78C7×#121—42no—
79C7×#120—44no—
80C4×C7×Q8—22no—
81C4×C7×Q8—22no—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C7×#117—28no—
85C7×#117—22no—
86C7×#117D414yescomplemented
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—22no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4D411yescharacteristic
96C2×C7×C4⋊C4C2312yes—
97C2×C7×C4⋊C4C2311yescharacteristic
98C2×C7×C4⋊C4C2311yescharacteristic
99C2×C7×C4⋊C4C2311yescharacteristic
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—22no—
105C2×C7×C22⋊C4—22no—
106C2×C7×C22⋊C4—22no—
107C2×C7×C22⋊C4D411yescharacteristic
108C2×C7×C22⋊C4D411yescharacteristic
109C2×C42×C7—21no—
110C2×C42×C7—21no—
111C2×C42×C7C2311yescharacteristic
112C7×#103—42no—
113C7×#102—24no—
114C7×#102—24no—
115C7×#102—24no—
116C7×C22⋊C8—22no—
117C7×C22⋊C8—42no—
118C7×#95—21no—
119C2×#552C2×C711yescharacteristic, complemented
120C2×#450C2×C711yescharacteristic, complemented
121C2×#418C2×C711yescharacteristic, complemented
122#2408C2×C711yescharacteristic
123#2025C2×C712yescomplemented
124#2000C2×C712yescomplemented
125#1894C2×C711yescharacteristic, complemented
126#1880C2×C712yescomplemented
127#1874C2×C712yescomplemented
128#1870C2×C711yescharacteristic, complemented
129#1825C2×C711yescharacteristic, complemented
130C24×C7—22no—
131C24×C7—44no—
132C24×C7—42no—
133C24×C7—42no—
134C24×C7—42no—
135C24×C7—82no—
136C24×C7C2×D411yescharacteristic
137C24×C7C2×D411yescharacteristic
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8C2×D412yes—
142C2×C7×D4—28no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—22no—
149C2×C7×D4—22no—
150C2×C7×D4—48no—
151C2×C7×D4—41no—
152C2×C7×D4—44no—
153C2×C7×D4—44no—
154C2×C7×D4—44no—
155C2×C7×D4—44no—
156C2×C7×D4—42no—
157C2×C7×D4—82no—
158C2×C7×D4#4114yes—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—24no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—42no—
172C22×C4×C7—42no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2411yescharacteristic
180C7×SD16—48no—
181C7×M16—44no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C2×C7×C8—22no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4C2×D412yes—
201C7×C4⋊C4C2×D412yes—
202C7×C4⋊C4C2×D412yes—
203C7×C22⋊C4—24no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—44no—
209C7×C22⋊C4—44no—
210C7×C22⋊C4—42no—
211C7×C22⋊C4—82no—
212C42×C7—24no—
213C42×C7—24no—
214C42×C7—21no—
215C42×C7—21no—
216C23×D4C22×C711yescharacteristic
217C22×SD16C22×C712yes—
218C22×M16C22×C711yescharacteristic
219#552—24no—
220#552—24no—
221C2×#126—21no—
222C2×#125—21no—
223C2×#125C22×C711yescharacteristic, complemented
224C2×#123—21no—
225C2×C22⋊Q8C22×C711yescharacteristic
226C2×C22⋊Q8C22×C711yescharacteristic, complemented
227C2×#121C22×C712yes—
228C2×#120—21no—
229C2×C4×Q8C22×C711yescharacteristic, complemented
230C2×C4×D4C22×C711yescharacteristic
231C2×#117—21no—
232C2×#117C22×C711yescharacteristic
233C22×C4⋊C4C22×C711yescharacteristic, complemented
234#450—24no—
235#418—24no—
236C2×#103C22×C712yescomplemented
237C2×#102C22×C712yes—
238C2×#102C22×C712yes—
239C2×#102C22×C712yes—
240C2×C22⋊C8C22×C711yescharacteristic
241C2×C22⋊C8C22×C712yescomplemented
242#394—21no—
243#392—21no—
244#390C22×C712yes—
245#380C22×C711yescharacteristic
246#379—21no—
247#340C22×C712yescomplemented
248#340C22×C712yescomplemented
249#340C22×C712yescomplemented
250#336C22×C712yescomplemented
251C23×C7—21no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—24no—
260C23×C7—48no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7—42no—
267C23×C7—84no—
268C23×C7—84no—
269C23×C7—88no—
270C23×C7—88no—
271C23×C7#12012yes—
272C23×C7AGL(1,ℤ/8)12yes—
273C23×C7C22×D411yescharacteristic
274C7×Q8—24no—
275C7×Q8—41no—
276C7×Q8—41no—
277C7×D4—24no—
278C7×D4—24no—
279C7×D4—48no—
280C7×D4—44no—
281C7×D4—42no—
282C7×D4—42no—
283C7×D4—88no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—24no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—48no—
300C2×C4×C7—42no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7#12012yes—
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C2×#4112yes—
316C2×C4×C7H(5,𝔽2)12yes—
317C7×C8—42no—
318C7×C8—42no—
319C25—21no—
320C22×Q8C23×C711yescharacteristic
321C22×D4—24no—
322C22×D4—22no—
323C22×D4—22no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—42no—
327C22×D4C7×D411yescharacteristic
328C22×D4C23×C712yes—
329C23×C4—21no—
330C23×C4C23×C711yescharacteristic
331C23×C4C23×C711yescharacteristic
332C2×SD16—24no—
333C2×SD16—24no—
334C2×SD16—42no—
335C2×M16—22no—
336C2×M16—42no—
337C22×C8C23×C712yes—
338C22×C8C23×C712yes—
339#126—28no—
340#125—28no—
341#125—22no—
342#125—22no—
343#123—28no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8C7×D414yescomplemented
348#121—24no—
349#121—42no—
350#120—44no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×D4—22no—
354C4×D4—22no—
355#117—28no—
356#117—22no—
357#117C7×D414yescomplemented
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4C7×D411yescharacteristic
367C2×C4⋊C4C23×C712yes—
368C2×C4⋊C4C23×C711yescharacteristic
369C2×C4⋊C4C23×C711yescharacteristic
370C2×C4⋊C4C23×C711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4C7×D411yescharacteristic
379C2×C22⋊C4C7×D411yescharacteristic
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C23×C711yescharacteristic
383#103—42no—
384#102—24no—
385#102—24no—
386#102—24no—
387C22⋊C8—22no—
388C22⋊C8—42no—
389#95—21no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—22no—
393C22×C7—22no—
394C22×C7—24no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—44no—
401C22×C7—44no—
402C22×C7—44no—
403C22×C7—816no—
404C22×C7—82no—
405C22×C7—82no—
406C22×C7—82no—
407C22×C7—82no—
408C22×C7C2×#12011yescharacteristic
409C22×C7#54612yes—
410C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C7#57511yescharacteristic
412C22×C7#57712yes—
413C4×C7—22no—
414C4×C7—41no—
415C4×C7—42no—
416C4×C7—41no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7—82no—
422C4×C7#52511yescharacteristic
423C4×C7#53511yescharacteristic
424C4×C7#54612yes—
425C24—22no—
426C24—44no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—82no—
431C24C2×C7×D411yescharacteristic
432C24C2×C7×D411yescharacteristic
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8C2×C7×D412yes—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—48no—
446C2×D4—41no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—82no—
453C2×D4C7×#4114yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C2×C7×D412yes—
473C22×C4C2×C7×D412yes—
474C22×C4C24×C711yescharacteristic
475SD16—48no—
476M16—44no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4C2×C7×D412yes—
496C4⋊C4C2×C7×D412yes—
497C4⋊C4C2×C7×D412yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—82no—
507C42—24no—
508C42—24no—
509C42—21no—
510C42—21no—
511C2×C7—42no—
512C2×C7—42no—
513C2×C7—84no—
514C2×C7#240411yescharacteristic
515C2×C7#300611yescharacteristic
516C2×C7#301011yescharacteristic
517C2×C7#331012yes—
518C2×C7#332312yes—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—84no—
536C23—84no—
537C23—88no—
538C23—88no—
539C23C7×#12012yes—
540C23C7×AGL(1,ℤ/8)12yes—
541C23C22×C7×D411yescharacteristic
542Q8—24no—
543Q8—41no—
544Q8—41no—
545D4—24no—
546D4—24no—
547D4—48no—
548D4—44no—
549D4—42no—
550D4—42no—
551D4—88no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4C7×#12012yes—
581C2×C4C22×C7×D411yescharacteristic
582C2×C4C22×C7×D411yescharacteristic
583C2×C4C2×C7×#4112yes—
584C2×C4C7×H(5,𝔽2)12yes—
585C8—42no—
586C8—42no—
587C7#2398211yescharacteristic, direct product factor, Sylow
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—816no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C2×C7×#12011yescharacteristic
607C22C7×#54612yes—
608C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
609C22C7×#57511yescharacteristic
610C22C7×#57712yes—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—82no—
620C4C7×#52511yescharacteristic
621C4C7×#53511yescharacteristic
622C4C7×#54612yes—
623C2—42no—
624C2—42no—
625C2—84no—
626C2C7×#240411yescharacteristic
627C2C7×#300611yescharacteristic
628C2C7×#301011yescharacteristic
629C2C7×#331012yes—
630C2C7×#332312yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#2408C2
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
1. C2×C7×#552C2semidirect
2. C2×C7×#450C2semidirect
3. C2×C7×#418C2semidirect
5. C7×#2025C2semidirect
6. C7×#2000C2semidirect
7. C7×#1894C2semidirect
8. C7×#1880C2semidirect
9. C7×#1874C2semidirect
10. C7×#1870C2semidirect
11. C7×#1825C2semidirect
19. C2×C7×#125C22semidirect
22. C2×C7×C22⋊Q8C22semidirect
25. C2×C4×C7×Q8C22semidirect
29. C22×C7×C4⋊C4C22semidirect
32. C2×C7×#103C22semidirect
37. C2×C7×C22⋊C8C22semidirect
43. C7×#340C22semidirect
44. C7×#340C22semidirect
45. C7×#340C22semidirect
46. C7×#336C22semidirect
47. #23982C7direct (N × H)
76. C7×C22⋊Q8D4semidirect
86. C7×#117D4semidirect
119. C2×#552C2×C7semidirect
120. C2×#450C2×C7semidirect
121. C2×#418C2×C7semidirect
123. #2025C2×C7semidirect
124. #2000C2×C7semidirect
125. #1894C2×C7semidirect
126. #1880C2×C7semidirect
127. #1874C2×C7semidirect
128. #1870C2×C7semidirect
129. #1825C2×C7semidirect
223. C2×#125C22×C7semidirect
226. C2×C22⋊Q8C22×C7semidirect
229. C2×C4×Q8C22×C7semidirect
233. C22×C4⋊C4C22×C7semidirect
236. C2×#103C22×C7semidirect
241. C2×C22⋊C8C22×C7semidirect
247. #340C22×C7semidirect
248. #340C22×C7semidirect
249. #340C22×C7semidirect
250. #336C22×C7semidirect
347. C22⋊Q8C7×D4semidirect
357. #117C7×D4semidirect
587. C7#23982direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×C7×#4502
C2×C7×#4182
C7×#24082
C7×#20252
C7×#20002
C7×#18942
C7×#18802
C7×#18742
C7×#18702
C7×#18252
#239827