Finite Group Databasebeta

← C127×#18306 #44581979

Subgroups — C127×#18306 #44581979

Order 32512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3459C211yesmaximal, characteristic, complemented
2C127×#3211C212yesmaximal, complemented
3C127×#2631C211yesmaximal, characteristic, complemented
4C127×#1606C212yesmaximal, complemented
5C127×#1602C211yesmaximal, characteristic, complemented
6C127×#1601C211yesmaximal, characteristic, complemented
7C127×#1600C212yesmaximal, complemented
8C127×#1596C211yesmaximal, characteristic, complemented
9C127×#1595C211yesmaximal, characteristic, complemented
10C127×#1521C212yesmaximal, complemented
11C127×#1460C211yesmaximal, characteristic, complemented
12#543×C127C2211yescharacteristic
13#539×C127—22no—
14#539×C127—21no—
15#538×C127—22no—
16#538×C127C2212yes—
17#534×C127—21no—
18#534×C127C2211yescharacteristic
19#529×C127—22no—
20#529×C127C2212yes—
21C2×#124×C127C2211yescharacteristic, complemented
22#525×C127—21no—
23#525×C127—22no—
24C2×C4×D4×C127—21no—
25C2×C4×D4×C127C2211yescharacteristic
26#495×C127—22no—
27#488×C127—22no—
28#464×C127—22no—
29#463×C127—22no—
30#451×C127C2212yescomplemented
31#450×C127C2212yescomplemented
32#448×C127C2212yes—
33#447×C127C2212yes—
34#431×C127C2212yescomplemented
35#397×C127—21no—
36#393×C127C2211yescharacteristic
37#393×C127C2211yescharacteristic, complemented
38#392×C127—21no—
39#392×C127C2211yescharacteristic
40#392×C127C2211yescharacteristic, complemented
41#386×C127—21no—
42#386×C127—21no—
43#383×C127C2211yescharacteristic, complemented
44#328×C127C2212yescomplemented
45#328×C127C2212yescomplemented
46#327×C127C2212yes—
47#327×C127C2212yes—
48#323×C127C2211yescharacteristic
49#323×C127C2211yescharacteristic
50#323×C127C2212yescomplemented
51C2×#41×C127—21no—
52C22×Q8×C127C2311yescharacteristic
53C22×D4×C127—21no—
54C22×D4×C127—21no—
55C22×D4×C127—21no—
56C22×D4×C127D411yescharacteristic
57C22×D4×C127C2311yescharacteristic
58C23×C4×C127—21no—
59C23×C4×C127D411yescharacteristic
60C23×C4×C127D411yescharacteristic
61C2×Q16×C127—22no—
62C2×SD16×C127—22no—
63C2×SD16×C127—22no—
64C2×D8×C127—22no—
65#126×C127—22no—
66#126×C127—22no—
67#126×C127—22no—
68#125×C127—22no—
69#125×C127D412yescomplemented
70#124×C127—22no—
71#124×C127—22no—
72#124×C127—22no—
73#124×C127D412yescomplemented
74#123×C127—22no—
75#123×C127—22no—
76#123×C127—22no—
77#123×C127—22no—
78#123×C127—22no—
79#123×C127—22no—
80C22⋊Q8×C127—21no—
81C22⋊Q8×C127—22no—
82C22⋊Q8×C127—22no—
83C22⋊Q8×C127—21no—
84C22⋊Q8×C127—22no—
85C22⋊Q8×C127C2312yes—
86#121×C127—22no—
87#121×C127—21no—
88#121×C127—22no—
89#121×C127—22no—
90#121×C127—21no—
91#121×C127C2312yes—
92#120×C127—22no—
93#120×C127—22no—
94#120×C127—22no—
95C4×Q8×C127D412yescomplemented
96C4×D4×C127—22no—
97C4×D4×C127—22no—
98C4×D4×C127—22no—
99C4×D4×C127—22no—
100C4×D4×C127—22no—
101C4×D4×C127—22no—
102C4×D4×C127—22no—
103C4×D4×C127D412yescomplemented
104#117×C127—21no—
105#117×C127—22no—
106C2×C4⋊C4×C127—21no—
107C2×C4⋊C4×C127D411yescharacteristic
108C2×C22⋊C4×C127—21no—
109C2×C22⋊C4×C127—21no—
110C2×C22⋊C4×C127—21no—
111C2×C22⋊C4×C127—21no—
112C2×C22⋊C4×C127—21no—
113C2×C22⋊C4×C127—21no—
114C2×C22⋊C4×C127—21no—
115C2×C22⋊C4×C127—21no—
116C2×C22⋊C4×C127C2311yescharacteristic
117C2×C22⋊C4×C127C2311yescharacteristic
118C2×C42×C127—21no—
119C2×C42×C127C2311yescharacteristic
120C4⋊C8×C127—22no—
121#103×C127—22no—
122#103×C127—22no—
123#102×C127—22no—
124#102×C127—22no—
125C22⋊C8×C127C2312yes—
126C22⋊C8×C127C2312yes—
127C4×C8×C127—22no—
128#95×C127—21no—
129#95×C127—21no—
130#95×C127—21no—
131#95×C127—21no—
132#95×C127C2311yescharacteristic
133#95×C127C2311yescharacteristic
134C24×C127—21no—
135C24×C127—21no—
136C24×C127C2×D411yescharacteristic
137C24×C127C2×D411yescharacteristic
138#41×C127—82no—
139C2×Q8×C127—21no—
140C2×Q8×C127—22no—
141C2×Q8×C127—81no—
142C2×Q8×C127C2×D412yes—
143C2×D4×C127—22no—
144C2×D4×C127—22no—
145C2×D4×C127—22no—
146C2×D4×C127—22no—
147C2×D4×C127—21no—
148C2×D4×C127—22no—
149C2×D4×C127—22no—
150C2×D4×C127—22no—
151C2×D4×C127—41no—
152C2×D4×C127—42no—
153C2×D4×C127—42no—
154C2×D4×C127—42no—
155C2×D4×C127—42no—
156C2×D4×C127—81no—
157C2×D4×C127—81no—
158C2×D4×C127—81no—
159C2×D4×C127—81no—
160C2×D4×C127—81no—
161C2×D4×C127C2×D412yes—
162C22×C4×C127—21no—
163C22×C4×C127—21no—
164C22×C4×C127—21no—
165C22×C4×C127—22no—
166C22×C4×C127—21no—
167C22×C4×C127—21no—
168C22×C4×C127—22no—
169C22×C4×C127—21no—
170C22×C4×C127—21no—
171C22×C4×C127—21no—
172C22×C4×C127—21no—
173C22×C4×C127—21no—
174C22×C4×C127—81no—
175C22×C4×C127—81no—
176C22×C4×C127—82no—
177C22×C4×C127C2×D411yescharacteristic
178C22×C4×C127C2×D411yescharacteristic
179C22×C4×C127C2×D411yescharacteristic
180C22×C4×C127C2×D411yescharacteristic
181C22×C4×C127C2411yescharacteristic
182Q16×C127—82no—
183SD16×C127—82no—
184SD16×C127—82no—
185D8×C127—82no—
186C2×C8×C127—22no—
187C2×C8×C127—22no—
188C4⋊C4×C127—22no—
189C4⋊C4×C127—21no—
190C4⋊C4×C127—21no—
191C4⋊C4×C127—22no—
192C4⋊C4×C127—22no—
193C4⋊C4×C127—22no—
194C4⋊C4×C127—22no—
195C4⋊C4×C127—22no—
196C4⋊C4×C127—22no—
197C4⋊C4×C127C2×D412yes—
198C4⋊C4×C127C2×D412yes—
199C22⋊C4×C127—22no—
200C22⋊C4×C127—22no—
201C22⋊C4×C127—22no—
202C22⋊C4×C127—22no—
203C22⋊C4×C127—22no—
204C22⋊C4×C127—22no—
205C22⋊C4×C127—22no—
206C22⋊C4×C127—22no—
207C22⋊C4×C127—22no—
208C22⋊C4×C127—22no—
209C22⋊C4×C127—22no—
210C22⋊C4×C127—22no—
211C22⋊C4×C127—41no—
212C22⋊C4×C127—42no—
213C22⋊C4×C127—41no—
214C22⋊C4×C127—81no—
215C22⋊C4×C127—81no—
216C22⋊C4×C127—81no—
217C22⋊C4×C127—81no—
218C42×C127—22no—
219C42×C127—22no—
220C42×C127C2×D412yes—
221C42×C127C2×D412yes—
222C23×C127—21no—
223C23×C127—21no—
224C23×C127—21no—
225C23×C127—22no—
226C23×C127—21no—
227C23×C127—21no—
228C23×C127—41no—
229C23×C127—41no—
230C23×C127—41no—
231C23×C127—41no—
232C23×C127—42no—
233C23×C127—41no—
234C23×C127—41no—
235C23×C127—41no—
236C23×C127—41no—
237C23×C127—41no—
238C23×C127—41no—
239C23×C127—41no—
240C23×C127—81no—
241C23×C127—81no—
242C23×C127#13512yes—
243C23×C127C22×D411yescharacteristic
244Q8×C127—42no—
245Q8×C127—42no—
246Q8×C127—42no—
247Q8×C127—81no—
248D4×C127—44no—
249D4×C127—42no—
250D4×C127—42no—
251D4×C127—42no—
252D4×C127—44no—
253D4×C127—42no—
254D4×C127—42no—
255D4×C127—82no—
256D4×C127—81no—
257D4×C127—82no—
258D4×C127—84no—
259D4×C127—84no—
260C2×C4×C127—21no—
261C2×C4×C127—21no—
262C2×C4×C127—22no—
263C2×C4×C127—22no—
264C2×C4×C127—21no—
265C2×C4×C127—21no—
266C2×C4×C127—22no—
267C2×C4×C127—41no—
268C2×C4×C127—41no—
269C2×C4×C127—42no—
270C2×C4×C127—42no—
271C2×C4×C127—44no—
272C2×C4×C127—42no—
273C2×C4×C127—41no—
274C2×C4×C127—41no—
275C2×C4×C127—41no—
276C2×C4×C127—41no—
277C2×C4×C127—41no—
278C2×C4×C127—41no—
279C2×C4×C127—41no—
280C2×C4×C127—41no—
281C2×C4×C127—41no—
282C2×C4×C127—41no—
283C2×C4×C127—41no—
284C2×C4×C127—81no—
285C2×C4×C127—81no—
286C2×C4×C127—81no—
287C2×C4×C127—82no—
288C2×C4×C127—81no—
289C2×C4×C127#12011yescharacteristic
290C2×C4×C127#12012yes—
291C2×C4×C127#12011yescharacteristic
292C2×C4×C127C22×D412yes—
293C8×C127—42no—
294C8×C127—42no—
295C22×C127—21no—
296C22×C127—21no—
297C22×C127—22no—
298C22×C127—42no—
299C22×C127—42no—
300C22×C127—41no—
301C22×C127—41no—
302C22×C127—41no—
303C22×C127—42no—
304C22×C127—41no—
305C22×C127—42no—
306C22×C127—42no—
307C22×C127—41no—
308C22×C127—81no—
309C22×C127—81no—
310C22×C127—81no—
311C22×C127—81no—
312C22×C127—81no—
313C22×C127—81no—
314C22×C127—84no—
315C22×C127—81no—
316C22×C127—81no—
317C22×C127—81no—
318C22×C127C2×#12011yescharacteristic
319C22×C127C2×#13511yescharacteristic
320C22×C127#57511yescharacteristic
321C4×C127—22no—
322C4×C127—21no—
323C4×C127—21no—
324C4×C127—22no—
325C4×C127—42no—
326C4×C127—41no—
327C4×C127—41no—
328C4×C127—42no—
329C4×C127—42no—
330C4×C127—81no—
331C4×C127—81no—
332C4×C127—81no—
333#18306C12711yesmaximal, characteristic, direct product factor, Sylow
334C2×C127—21no—
335C2×C127—21no—
336C2×C127—42no—
337C2×C127—81no—
338C2×C127—82no—
339C2×C127—161no—
340C2×C127#300211yescharacteristic
341C2×C127#301911yescharacteristic
342C2×C127#302711yescharacteristic
343#3459C2×C12711yescharacteristic, complemented
344#3211C2×C12712yescomplemented
345#2631C2×C12711yescharacteristic, complemented
346#1606C2×C12712yescomplemented
347#1602C2×C12711yescharacteristic, complemented
348#1601C2×C12711yescharacteristic, complemented
349#1600C2×C12712yescomplemented
350#1596C2×C12711yescharacteristic, complemented
351#1595C2×C12711yescharacteristic, complemented
352#1521C2×C12712yescomplemented
353#1460C2×C12711yescharacteristic, complemented
354C127#1830611yescharacteristic, direct product factor, Sylow
355#543C22×C12711yescharacteristic
356#539—22no—
357#539—21no—
358#538—22no—
359#538C22×C12712yes—
360#534—21no—
361#534C22×C12711yescharacteristic
362#529—22no—
363#529C22×C12712yes—
364C2×#124C22×C12711yescharacteristic, complemented
365#525—21no—
366#525—22no—
367C2×C4×D4—21no—
368C2×C4×D4C22×C12711yescharacteristic
369#495—22no—
370#488—22no—
371#464—22no—
372#463—22no—
373#451C22×C12712yescomplemented
374#450C22×C12712yescomplemented
375#448C22×C12712yes—
376#447C22×C12712yes—
377#431C22×C12712yescomplemented
378#397—21no—
379#393C22×C12711yescharacteristic
380#393C22×C12711yescharacteristic, complemented
381#392—21no—
382#392C22×C12711yescharacteristic
383#392C22×C12711yescharacteristic, complemented
384#386—21no—
385#386—21no—
386#383C22×C12711yescharacteristic, complemented
387#328C22×C12712yescomplemented
388#328C22×C12712yescomplemented
389#327C22×C12712yes—
390#327C22×C12712yes—
391#323C22×C12711yescharacteristic
392#323C22×C12711yescharacteristic
393#323C22×C12712yescomplemented
394C2×#41—21no—
395C22×Q8C23×C12711yescharacteristic
396C22×D4—21no—
397C22×D4—21no—
398C22×D4—21no—
399C22×D4D4×C12711yescharacteristic
400C22×D4C23×C12711yescharacteristic
401C23×C4—21no—
402C23×C4D4×C12711yescharacteristic
403C23×C4D4×C12711yescharacteristic
404C2×Q16—22no—
405C2×SD16—22no—
406C2×SD16—22no—
407C2×D8—22no—
408#126—22no—
409#126—22no—
410#126—22no—
411#125—22no—
412#125D4×C12712yescomplemented
413#124—22no—
414#124—22no—
415#124—22no—
416#124D4×C12712yescomplemented
417#123—22no—
418#123—22no—
419#123—22no—
420#123—22no—
421#123—22no—
422#123—22no—
423C22⋊Q8—21no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—22no—
428C22⋊Q8C23×C12712yes—
429#121—22no—
430#121—21no—
431#121—22no—
432#121—22no—
433#121—21no—
434#121C23×C12712yes—
435#120—22no—
436#120—22no—
437#120—22no—
438C4×Q8D4×C12712yescomplemented
439C4×D4—22no—
440C4×D4—22no—
441C4×D4—22no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—22no—
446C4×D4D4×C12712yescomplemented
447#117—21no—
448#117—22no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4D4×C12711yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—21no—
459C2×C22⋊C4C23×C12711yescharacteristic
460C2×C22⋊C4C23×C12711yescharacteristic
461C2×C42—21no—
462C2×C42C23×C12711yescharacteristic
463C4⋊C8—22no—
464#103—22no—
465#103—22no—
466#102—22no—
467#102—22no—
468C22⋊C8C23×C12712yes—
469C22⋊C8C23×C12712yes—
470C4×C8—22no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95C23×C12711yescharacteristic
476#95C23×C12711yescharacteristic
477C24—21no—
478C24—21no—
479C24C2×D4×C12711yescharacteristic
480C24C2×D4×C12711yescharacteristic
481#41—82no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—81no—
485C2×Q8C2×D4×C12712yes—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—41no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—81no—
500C2×D4—81no—
501C2×D4—81no—
502C2×D4—81no—
503C2×D4—81no—
504C2×D4C2×D4×C12712yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—81no—
518C22×C4—81no—
519C22×C4—82no—
520C22×C4C2×D4×C12711yescharacteristic
521C22×C4C2×D4×C12711yescharacteristic
522C22×C4C2×D4×C12711yescharacteristic
523C22×C4C2×D4×C12711yescharacteristic
524C22×C4C24×C12711yescharacteristic
525Q16—82no—
526SD16—82no—
527SD16—82no—
528D8—82no—
529C2×C8—22no—
530C2×C8—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4C2×D4×C12712yes—
541C4⋊C4C2×D4×C12712yes—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—41no—
555C22⋊C4—42no—
556C22⋊C4—41no—
557C22⋊C4—81no—
558C22⋊C4—81no—
559C22⋊C4—81no—
560C22⋊C4—81no—
561C42—22no—
562C42—22no—
563C42C2×D4×C12712yes—
564C42C2×D4×C12712yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—21no—
570C23—21no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—81no—
584C23—81no—
585C23#135×C12712yes—
586C23C22×D4×C12711yescharacteristic
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—81no—
591D4—44no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—44no—
596D4—42no—
597D4—42no—
598D4—81no—
599D4—82no—
600D4—82no—
601D4—84no—
602D4—84no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4#120×C12711yescharacteristic
633C2×C4#120×C12712yes—
634C2×C4#120×C12711yescharacteristic
635C2×C4C22×D4×C12712yes—
636C8—42no—
637C8—42no—
638C22—21no—
639C22—21no—
640C22—22no—
641C22—42no—
642C22—42no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—41no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—84no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×#120×C12711yescharacteristic
662C22C2×#135×C12711yescharacteristic
663C22#575×C12711yescharacteristic
664C4—22no—
665C4—21no—
666C4—21no—
667C4—22no—
668C4—42no—
669C4—41no—
670C4—41no—
671C4—42no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—81no—
676C2—21no—
677C2—21no—
678C2—42no—
679C2—81no—
680C2—82no—
681C2—161no—
682C2C127×#300211yescharacteristic
683C2C127×#301911yescharacteristic
684C2C127×#302711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3459C2semidirect
2. C127×#3211C2semidirect
3. C127×#2631C2semidirect
4. C127×#1606C2semidirect
5. C127×#1602C2semidirect
6. C127×#1601C2semidirect
7. C127×#1600C2semidirect
8. C127×#1596C2semidirect
9. C127×#1595C2semidirect
10. C127×#1521C2semidirect
11. C127×#1460C2semidirect
21. C2×#124×C127C22semidirect
30. #451×C127C22semidirect
31. #450×C127C22semidirect
34. #431×C127C22semidirect
37. #393×C127C22semidirect
40. #392×C127C22semidirect
43. #383×C127C22semidirect
44. #328×C127C22semidirect
45. #328×C127C22semidirect
50. #323×C127C22semidirect
69. #125×C127D4semidirect
73. #124×C127D4semidirect
95. C4×Q8×C127D4semidirect
103. C4×D4×C127D4semidirect
333. #18306C127direct (N × H)
343. #3459C2×C127semidirect
344. #3211C2×C127semidirect
345. #2631C2×C127semidirect
346. #1606C2×C127semidirect
347. #1602C2×C127semidirect
348. #1601C2×C127semidirect
349. #1600C2×C127semidirect
350. #1596C2×C127semidirect
351. #1595C2×C127semidirect
352. #1521C2×C127semidirect
353. #1460C2×C127semidirect
354. C127#18306direct (N × H)
364. C2×#124C22×C127semidirect
373. #451C22×C127semidirect
374. #450C22×C127semidirect
377. #431C22×C127semidirect
380. #393C22×C127semidirect
383. #392C22×C127semidirect
386. #383C22×C127semidirect
387. #328C22×C127semidirect
388. #328C22×C127semidirect
393. #323C22×C127semidirect
412. #125D4×C127semidirect
416. #124D4×C127semidirect
438. C4×Q8D4×C127semidirect
446. C4×D4D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34592
C127×#32112
C127×#26312
C127×#16062
C127×#16022
C127×#16012
C127×#16002
C127×#15962
C127×#15952
C127×#15212
C127×#14602
#18306127