Finite Group Databasebeta

← C127×#21206 #44584879

Subgroups — C127×#21206 #44584879

Order 32512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#535×C127C211yesmaximal, characteristic
2C2×#417×C127C211yesmaximal, characteristic, complemented
3C2×#418×C127C211yesmaximal, characteristic, complemented
4C2×#416×C127C211yesmaximal, characteristic, complemented
5C22×#102×C127C211yesmaximal, characteristic, complemented
6C2×#405×C127C211yesmaximal, characteristic, complemented
7C127×#2317C211yesmaximal, characteristic, complemented
8C127×#1961C212yesmaximal, complemented
9C127×#1959C212yesmaximal, complemented
10C127×#1958C212yesmaximal, complemented
11C127×#1957C212yesmaximal, complemented
12C22×#41×C127C2211yescharacteristic
13C23×D4×C127C2211yescharacteristic
14C22×M16×C127C2211yescharacteristic
15C23×C8×C127C2211yescharacteristic
16#535×C127—24no—
17#535×C127C418yescomplemented
18C2×#127×C127C2212yes—
19C2×#124×C127C2212yes—
20C2×#121×C127—22no—
21C2×#120×C127—22no—
22C2×#117×C127C2211yescharacteristic
23C2×#117×C127C2211yescharacteristic
24C22×C4⋊C4×C127C2211yescharacteristic
25#418×C127—24no—
26#417×C127—24no—
27#416×C127—24no—
28C2×#103×C127C2211yescharacteristic, complemented
29C2×#103×C127C2211yescharacteristic, complemented
30C2×#103×C127C2212yescomplemented
31C2×#102×C127—24no—
32C2×#102×C127C2212yes—
33C2×#102×C127C2212yes—
34C2×#102×C127C2212yes—
35C2×#102×C127C2212yes—
36C2×#102×C127C2211yescharacteristic, complemented
37C2×#102×C127C2211yescharacteristic, complemented
38C2×#102×C127C2212yescomplemented
39#405×C127—24no—
40C2×#97×C127C2212yescomplemented
41C2×C4×C8×C127C2212yescomplemented
42#389×C127—22no—
43#385×C127—22no—
44#384×C127C2212yescomplemented
45#382×C127C2212yescomplemented
46C25×C127—21no—
47C2×#41×C127—22no—
48C2×#41×C127—22no—
49C2×#41×C127—22no—
50C2×#41×C127—22no—
51C2×#41×C127—22no—
52C2×#41×C127C2×C414yes—
53C22×Q8×C127C2311yescharacteristic
54C22×D4×C127—21no—
55C22×D4×C127—22no—
56C22×D4×C127—22no—
57C22×D4×C127—21no—
58C22×D4×C127—22no—
59C22×D4×C127—24no—
60C22×D4×C127—22no—
61C22×D4×C127—22no—
62C22×D4×C127C2×C414yes—
63C22×D4×C127C2312yes—
64C22×D4×C127C2311yescharacteristic
65C23×C4×C127—21no—
66C23×C4×C127—21no—
67C23×C4×C127C2311yescharacteristic
68C2×M16×C127—22no—
69C2×M16×C127—22no—
70C2×M16×C127D414yescomplemented
71C22×C8×C127—22no—
72C22×C8×C127—22no—
73C22×C8×C127D414yescomplemented
74C22×C8×C127C2311yescharacteristic
75C22×C8×C127C2311yescharacteristic
76C22×C8×C127C2312yes—
77#127×C127—24no—
78#127×C127C2×C418yescomplemented
79#124×C127—24no—
80#124×C127C2×C418yescomplemented
81#121×C127—216no—
82#120×C127—216no—
83#117×C127—22no—
84#117×C127—22no—
85#117×C127C2×C414yes—
86#117×C127D414yes—
87C2×C4⋊C4×C127—22no—
88C2×C4⋊C4×C127—22no—
89C2×C4⋊C4×C127—21no—
90C2×C4⋊C4×C127—24no—
91C2×C4⋊C4×C127D412yes—
92C2×C4⋊C4×C127C2312yes—
93C2×C4⋊C4×C127C2311yescharacteristic
94C2×C4⋊C4×C127C2311yescharacteristic
95C2×C22⋊C4×C127—24no—
96C2×C22⋊C4×C127—21no—
97C2×C22⋊C4×C127D412yes—
98C2×C42×C127—21no—
99C2×C42×C127C2312yes—
100#103×C127—22no—
101#103×C127—22no—
102#103×C127—24no—
103#102×C127—24no—
104#102×C127—24no—
105#102×C127—24no—
106#102×C127—24no—
107#102×C127—22no—
108#102×C127—22no—
109#102×C127—24no—
110#97×C127—24no—
111C4×C8×C127—24no—
112#95×C127—24no—
113C24×C127—22no—
114C24×C127—21no—
115C24×C127—24no—
116C24×C127—42no—
117C24×C127—42no—
118C24×C127—82no—
119C24×C127—82no—
120C24×C127C2×D411yescharacteristic
121#41×C127—28no—
122#41×C127—28no—
123#41×C127—44no—
124#41×C127—44no—
125C2×Q8×C127—21no—
126C2×Q8×C127—21no—
127C2×Q8×C127—21no—
128C2×Q8×C127—21no—
129C2×Q8×C127—21no—
130C2×Q8×C127C22×C412yes—
131C2×D4×C127—24no—
132C2×D4×C127—24no—
133C2×D4×C127—216no—
134C2×D4×C127—28no—
135C2×D4×C127—28no—
136C2×D4×C127—22no—
137C2×D4×C127—22no—
138C2×D4×C127—21no—
139C2×D4×C127—21no—
140C2×D4×C127—22no—
141C2×D4×C127—21no—
142C2×D4×C127—22no—
143C2×D4×C127—22no—
144C2×D4×C127—21no—
145C2×D4×C127—21no—
146C2×D4×C127—42no—
147C2×D4×C127—42no—
148C2×D4×C127—44no—
149C2×D4×C127—44no—
150C2×D4×C127—42no—
151C2×D4×C127—42no—
152C2×D4×C127—84no—
153C2×D4×C127—84no—
154C2×D4×C127C22×C414yes—
155C2×D4×C127C22×C412yes—
156C22×C4×C127—21no—
157C22×C4×C127—24no—
158C22×C4×C127—21no—
159C22×C4×C127—22no—
160C22×C4×C127—21no—
161C22×C4×C127—42no—
162C22×C4×C127—42no—
163C22×C4×C127—42no—
164C22×C4×C127—82no—
165C22×C4×C127C22×C412yes—
166C22×C4×C127C2×D414yes—
167C22×C4×C127C2×D411yescharacteristic
168C22×C4×C127C2×D412yes—
169C22×C4×C127C2×D412yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127#4112yes—
172C22×C4×C127#4112yes—
173C22×C4×C127C2411yescharacteristic
174M16×C127—28no—
175C2×C8×C127—28no—
176C2×C8×C127—21no—
177C2×C8×C127—22no—
178C2×C8×C127—21no—
179C2×C8×C127—21no—
180C2×C8×C127—21no—
181C2×C8×C127—22no—
182C2×C8×C127C2×D412yes—
183C2×C8×C127C2×D412yes—
184C2×C8×C127C2×D414yes—
185C4⋊C4×C127—22no—
186C4⋊C4×C127—21no—
187C4⋊C4×C127—21no—
188C4⋊C4×C127—22no—
189C4⋊C4×C127—21no—
190C4⋊C4×C127—21no—
191C4⋊C4×C127—42no—
192C4⋊C4×C127—84no—
193C4⋊C4×C127C22⋊C418yes—
194C22⋊C4×C127—216no—
195C22⋊C4×C127—42no—
196C22⋊C4×C127C22⋊C418yes—
197C42×C127—42no—
198C42×C127C22×C414yes—
199C42×C127C2×D414yes—
200C23×C127—22no—
201C23×C127—22no—
202C23×C127—24no—
203C23×C127—22no—
204C23×C127—41no—
205C23×C127—42no—
206C23×C127—41no—
207C23×C127—42no—
208C23×C127—42no—
209C23×C127—44no—
210C23×C127—42no—
211C23×C127—44no—
212C23×C127—48no—
213C23×C127—88no—
214C23×C127—88no—
215C23×C127—84no—
216C23×C127—84no—
217C23×C127C2×C22⋊C412yes—
218C23×C127#12412yes—
219C23×C127#12712yes—
220C23×C127C22×D411yescharacteristic
221Q8×C127—22no—
222Q8×C127—22no—
223Q8×C127—41no—
224Q8×C127—41no—
225D4×C127—24no—
226D4×C127—24no—
227D4×C127—22no—
228D4×C127—22no—
229D4×C127—48no—
230D4×C127—42no—
231D4×C127—42no—
232D4×C127—41no—
233D4×C127—41no—
234D4×C127—48no—
235D4×C127—816no—
236C2×C4×C127—24no—
237C2×C4×C127—24no—
238C2×C4×C127—22no—
239C2×C4×C127—41no—
240C2×C4×C127—41no—
241C2×C4×C127—48no—
242C2×C4×C127—41no—
243C2×C4×C127—41no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—44no—
247C2×C4×C127—42no—
248C2×C4×C127—84no—
249C2×C4×C127C2×C22⋊C418yes—
250C2×C4×C127C2×C22⋊C412yes—
251C2×C4×C127C4×D414yes—
252C2×C4×C127C4×D414yes—
253C2×C4×C127#12114yes—
254C2×C4×C127#12114yes—
255C2×C4×C127#12412yes—
256C2×C4×C127#12712yes—
257C2×C4×C127C23×C411yescharacteristic
258C2×C4×C127C22×D411yescharacteristic
259C2×C4×C127C22×D411yescharacteristic
260C2×C4×C127C22×D411yescharacteristic
261C2×C4×C127C2×#4111yescharacteristic
262C2×C4×C127C2×#4111yescharacteristic
263C8×C127—22no—
264C8×C127—22no—
265C8×C127—24no—
266C22×C127—24no—
267C22×C127—24no—
268C22×C127—22no—
269C22×C127—22no—
270C22×C127—44no—
271C22×C127—48no—
272C22×C127—42no—
273C22×C127—44no—
274C22×C127—816no—
275C22×C127—82no—
276C22×C127—84no—
277C22×C127—81no—
278C22×C127—82no—
279C22×C127—81no—
280C22×C127—82no—
281C22×C127#39014yes—
282C22×C127C22×C22⋊C411yescharacteristic
283C22×C127C2×#12411yescharacteristic
284C22×C127C2×#12711yescharacteristic
285C22×C127#57612yes—
286C22×C127#57712yes—
287C4×C127—28no—
288C4×C127—42no—
289C4×C127—81no—
290C4×C127—81no—
291C4×C127—82no—
292C4×C127#39014yes—
293C4×C127C2×C4×D411yescharacteristic
294C4×C127C2×C4×D411yescharacteristic
295C4×C127C2×#12111yescharacteristic
296C4×C127C2×#12111yescharacteristic
297#21206C12711yesmaximal, characteristic, direct product factor, Sylow
298C2×C127—24no—
299C2×C127—82no—
300C2×C127—84no—
301C2×C127C2×#39011yescharacteristic
302C2×C127#289712yes—
303C2×C127#313612yes—
304C2×C127#315411yescharacteristic
305C2×C127#315511yescharacteristic
306C2×#535C2×C12711yescharacteristic
307C2×#417C2×C12711yescharacteristic, complemented
308C2×#418C2×C12711yescharacteristic, complemented
309C2×#416C2×C12711yescharacteristic, complemented
310C22×#102C2×C12711yescharacteristic, complemented
311C2×#405C2×C12711yescharacteristic, complemented
312#2317C2×C12711yescharacteristic, complemented
313#1961C2×C12712yescomplemented
314#1959C2×C12712yescomplemented
315#1958C2×C12712yescomplemented
316#1957C2×C12712yescomplemented
317C127#2120611yescharacteristic, direct product factor, Sylow
318C22×#41C22×C12711yescharacteristic
319C23×D4C22×C12711yescharacteristic
320C22×M16C22×C12711yescharacteristic
321C23×C8C22×C12711yescharacteristic
322#535—24no—
323#535C4×C12718yescomplemented
324C2×#127C22×C12712yes—
325C2×#124C22×C12712yes—
326C2×#121—22no—
327C2×#120—22no—
328C2×#117C22×C12711yescharacteristic
329C2×#117C22×C12711yescharacteristic
330C22×C4⋊C4C22×C12711yescharacteristic
331#418—24no—
332#417—24no—
333#416—24no—
334C2×#103C22×C12711yescharacteristic, complemented
335C2×#103C22×C12711yescharacteristic, complemented
336C2×#103C22×C12712yescomplemented
337C2×#102—24no—
338C2×#102C22×C12712yes—
339C2×#102C22×C12712yes—
340C2×#102C22×C12712yes—
341C2×#102C22×C12712yes—
342C2×#102C22×C12711yescharacteristic, complemented
343C2×#102C22×C12711yescharacteristic, complemented
344C2×#102C22×C12712yescomplemented
345#405—24no—
346C2×#97C22×C12712yescomplemented
347C2×C4×C8C22×C12712yescomplemented
348#389—22no—
349#385—22no—
350#384C22×C12712yescomplemented
351#382C22×C12712yescomplemented
352C25—21no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41C2×C4×C12714yes—
359C22×Q8C23×C12711yescharacteristic
360C22×D4—21no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—21no—
364C22×D4—22no—
365C22×D4—24no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4C2×C4×C12714yes—
369C22×D4C23×C12712yes—
370C22×D4C23×C12711yescharacteristic
371C23×C4—21no—
372C23×C4—21no—
373C23×C4C23×C12711yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16D4×C12714yescomplemented
377C22×C8—22no—
378C22×C8—22no—
379C22×C8D4×C12714yescomplemented
380C22×C8C23×C12711yescharacteristic
381C22×C8C23×C12711yescharacteristic
382C22×C8C23×C12712yes—
383#127—24no—
384#127C2×C4×C12718yescomplemented
385#124—24no—
386#124C2×C4×C12718yescomplemented
387#121—216no—
388#120—216no—
389#117—22no—
390#117—22no—
391#117C2×C4×C12714yes—
392#117D4×C12714yes—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4D4×C12712yes—
398C2×C4⋊C4C23×C12712yes—
399C2×C4⋊C4C23×C12711yescharacteristic
400C2×C4⋊C4C23×C12711yescharacteristic
401C2×C22⋊C4—24no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4D4×C12712yes—
404C2×C42—21no—
405C2×C42C23×C12712yes—
406#103—22no—
407#103—22no—
408#103—24no—
409#102—24no—
410#102—24no—
411#102—24no—
412#102—24no—
413#102—22no—
414#102—22no—
415#102—24no—
416#97—24no—
417C4×C8—24no—
418#95—24no—
419C24—22no—
420C24—21no—
421C24—24no—
422C24—42no—
423C24—42no—
424C24—82no—
425C24—82no—
426C24C2×D4×C12711yescharacteristic
427#41—28no—
428#41—28no—
429#41—44no—
430#41—44no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C22×C4×C12712yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—216no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4C22×C4×C12714yes—
461C2×D4C22×C4×C12712yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—82no—
471C22×C4C22×C4×C12712yes—
472C22×C4C2×D4×C12714yes—
473C22×C4C2×D4×C12711yescharacteristic
474C22×C4C2×D4×C12712yes—
475C22×C4C2×D4×C12712yes—
476C22×C4C2×D4×C12712yes—
477C22×C4#41×C12712yes—
478C22×C4#41×C12712yes—
479C22×C4C24×C12711yescharacteristic
480M16—28no—
481C2×C8—28no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8C2×D4×C12712yes—
489C2×C8C2×D4×C12712yes—
490C2×C8C2×D4×C12714yes—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—84no—
499C4⋊C4C22⋊C4×C12718yes—
500C22⋊C4—216no—
501C22⋊C4—42no—
502C22⋊C4C22⋊C4×C12718yes—
503C42—42no—
504C42C22×C4×C12714yes—
505C42C2×D4×C12714yes—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—48no—
519C23—88no—
520C23—88no—
521C23—84no—
522C23—84no—
523C23C2×C22⋊C4×C12712yes—
524C23#124×C12712yes—
525C23#127×C12712yes—
526C23C22×D4×C12711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—24no—
532D4—24no—
533D4—22no—
534D4—22no—
535D4—48no—
536D4—42no—
537D4—42no—
538D4—41no—
539D4—41no—
540D4—48no—
541D4—816no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—84no—
555C2×C4C2×C22⋊C4×C12718yes—
556C2×C4C2×C22⋊C4×C12712yes—
557C2×C4C4×D4×C12714yes—
558C2×C4C4×D4×C12714yes—
559C2×C4#121×C12714yes—
560C2×C4#121×C12714yes—
561C2×C4#124×C12712yes—
562C2×C4#127×C12712yes—
563C2×C4C23×C4×C12711yescharacteristic
564C2×C4C22×D4×C12711yescharacteristic
565C2×C4C22×D4×C12711yescharacteristic
566C2×C4C22×D4×C12711yescharacteristic
567C2×C4C2×#41×C12711yescharacteristic
568C2×C4C2×#41×C12711yescharacteristic
569C8—22no—
570C8—22no—
571C8—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—44no—
577C22—48no—
578C22—42no—
579C22—44no—
580C22—816no—
581C22—84no—
582C22—82no—
583C22—82no—
584C22—81no—
585C22—82no—
586C22—81no—
587C22#390×C12714yes—
588C22C22×C22⋊C4×C12711yescharacteristic
589C22C2×#124×C12711yescharacteristic
590C22C2×#127×C12711yescharacteristic
591C22#576×C12712yes—
592C22#577×C12712yes—
593C4—28no—
594C4—42no—
595C4—81no—
596C4—82no—
597C4—81no—
598C4#390×C12714yes—
599C4C2×C4×D4×C12711yescharacteristic
600C4C2×C4×D4×C12711yescharacteristic
601C4C2×#121×C12711yescharacteristic
602C4C2×#121×C12711yescharacteristic
603C2—24no—
604C2—84no—
605C2—82no—
606C2C2×#390×C12711yescharacteristic
607C2C127×#289712yes—
608C2C127×#313612yes—
609C2C127×#315411yescharacteristic
610C2C127×#315511yescharacteristic
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×#535×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×#117×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#417×C127C2semidirect
3. C2×#418×C127C2semidirect
4. C2×#416×C127C2semidirect
5. C22×#102×C127C2semidirect
6. C2×#405×C127C2semidirect
7. C127×#2317C2semidirect
8. C127×#1961C2semidirect
9. C127×#1959C2semidirect
10. C127×#1958C2semidirect
11. C127×#1957C2semidirect
17. #535×C127C4semidirect
28. C2×#103×C127C22semidirect
29. C2×#103×C127C22semidirect
30. C2×#103×C127C22semidirect
36. C2×#102×C127C22semidirect
37. C2×#102×C127C22semidirect
38. C2×#102×C127C22semidirect
40. C2×#97×C127C22semidirect
41. C2×C4×C8×C127C22semidirect
44. #384×C127C22semidirect
45. #382×C127C22semidirect
70. C2×M16×C127D4semidirect
73. C22×C8×C127D4semidirect
78. #127×C127C2×C4semidirect
80. #124×C127C2×C4semidirect
297. #21206C127direct (N × H)
307. C2×#417C2×C127semidirect
308. C2×#418C2×C127semidirect
309. C2×#416C2×C127semidirect
310. C22×#102C2×C127semidirect
311. C2×#405C2×C127semidirect
312. #2317C2×C127semidirect
313. #1961C2×C127semidirect
314. #1959C2×C127semidirect
315. #1958C2×C127semidirect
316. #1957C2×C127semidirect
317. C127#21206direct (N × H)
323. #535C4×C127semidirect
334. C2×#103C22×C127semidirect
335. C2×#103C22×C127semidirect
336. C2×#103C22×C127semidirect
342. C2×#102C22×C127semidirect
343. C2×#102C22×C127semidirect
344. C2×#102C22×C127semidirect
346. C2×#97C22×C127semidirect
347. C2×C4×C8C22×C127semidirect
350. #384C22×C127semidirect
351. #382C22×C127semidirect
376. C2×M16D4×C127semidirect
379. C22×C8D4×C127semidirect
384. #127C2×C4×C127semidirect
386. #124C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#535×C1272
C2×#417×C1272
C2×#418×C1272
C2×#416×C1272
C22×#102×C1272
C2×#405×C1272
C127×#23172
C127×#19612
C127×#19592
C127×#19582
C127×#19572
#21206127