Finite Group Databasebeta

← C127×#18331 #44582004

Subgroups — C127×#18331 #44582004

Order 32512.

998 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3460C211yesmaximal, characteristic
2C127×#3196C212yesmaximal, complemented
3C127×#2564C211yesmaximal, characteristic, complemented
4C127×#1618C212yesmaximal, complemented
5C127×#1617C211yesmaximal, characteristic, complemented
6C127×#1616C211yesmaximal, characteristic, complemented
7C127×#1606C212yesmaximal, complemented
8C127×#1605C211yesmaximal, characteristic, complemented
9C127×#1604C211yesmaximal, characteristic, complemented
10C127×#1504C212yesmaximal, complemented
11C127×#1465C211yesmaximal, characteristic, complemented
12#544×C127C2211yescharacteristic
13#542×C127—22no—
14#540×C127—22no—
15#538×C127—21no—
16#538×C127C2212yes—
17#537×C127C2211yescharacteristic
18#535×C127—21no—
19#532×C127—22no—
20#529×C127—22no—
21#525×C127—21no—
22#525×C127C2212yes—
23C2×C22⋊Q8×C127C2211yescharacteristic
24C4×#41×C127—21no—
25C2×C4×D4×C127C2211yescharacteristic
26#485×C127—22no—
27#484×C127C2212yes—
28#481×C127C2212yes—
29#470×C127—22no—
30#468×C127—22no—
31#452×C127—22no—
32#451×C127C2212yes—
33#448×C127C2212yes—
34#407×C127C2212yes—
35#400×C127C2211yescharacteristic
36#399×C127C2211yescharacteristic
37#395×C127C2211yescharacteristic
38#393×C127C2211yescharacteristic
39#387×C127—21no—
40#386×C127—21no—
41#385×C127—21no—
42#380×C127—21no—
43C2×#95×C127C2211yescharacteristic
44#328×C127C2212yes—
45#328×C127C2212yes—
46#327×C127C2212yes—
47#327×C127C2212yes—
48#324×C127C2211yescharacteristic
49#324×C127C2211yescharacteristic
50#323×C127C2212yes—
51C2×#41×C127—21no—
52C2×#41×C127—21no—
53C2×#41×C127D412yes—
54C22×Q8×C127C2311yescharacteristic
55C22×D4×C127D411yescharacteristic
56C23×C4×C127—21no—
57C23×C4×C127D411yescharacteristic
58C23×C4×C127C2311yescharacteristic
59C2×Q16×C127—22no—
60C2×SD16×C127—22no—
61C2×M16×C127—22no—
62C4⋊Q8×C127—22no—
63#127×C127—22no—
64#126×C127—22no—
65#126×C127—22no—
66#125×C127—22no—
67#124×C127—22no—
68#124×C127—22no—
69#124×C127—22no—
70#123×C127—22no—
71#123×C127—22no—
72#123×C127—22no—
73#123×C127—22no—
74C22⋊Q8×C127—21no—
75C22⋊Q8×C127—21no—
76C22⋊Q8×C127—22no—
77C22⋊Q8×C127—22no—
78C22⋊Q8×C127—22no—
79C22⋊Q8×C127D412yes—
80C22⋊Q8×C127C2312yes—
81#121×C127—21no—
82#121×C127—22no—
83#121×C127—22no—
84#121×C127—22no—
85#121×C127—22no—
86#121×C127—22no—
87#121×C127—21no—
88#121×C127D412yes—
89#121×C127C2312yes—
90#120×C127—22no—
91#120×C127—22no—
92C4×Q8×C127—21no—
93C4×Q8×C127—21no—
94C4×D4×C127—21no—
95C4×D4×C127—21no—
96C4×D4×C127—22no—
97C4×D4×C127—22no—
98C4×D4×C127—22no—
99C4×D4×C127—22no—
100C4×D4×C127—22no—
101C4×D4×C127—22no—
102#117×C127—22no—
103#117×C127—21no—
104#117×C127—21no—
105#117×C127D412yes—
106C2×C4⋊C4×C127—21no—
107C2×C4⋊C4×C127—21no—
108C2×C4⋊C4×C127—21no—
109C2×C4⋊C4×C127C2311yescharacteristic
110C2×C4⋊C4×C127C2311yescharacteristic
111C2×C22⋊C4×C127—21no—
112C2×C22⋊C4×C127—21no—
113C2×C22⋊C4×C127—21no—
114C2×C22⋊C4×C127D411yescharacteristic
115C2×C22⋊C4×C127C2311yescharacteristic
116C2×C42×C127—22no—
117C2×C42×C127D411yescharacteristic
118#107×C127—22no—
119#106×C127—22no—
120#103×C127—22no—
121#103×C127—22no—
122#103×C127—22no—
123#102×C127—22no—
124C22⋊C8×C127—22no—
125C22⋊C8×C127C2312yes—
126C22⋊C8×C127C2312yes—
127#95×C127—21no—
128#95×C127—21no—
129#95×C127—21no—
130#95×C127—21no—
131#95×C127—21no—
132#95×C127C2311yescharacteristic
133#95×C127C2311yescharacteristic
134C24×C127C2×D411yescharacteristic
135C24×C127C2×D411yescharacteristic
136#41×C127—42no—
137#41×C127—42no—
138#41×C127—82no—
139#41×C127—82no—
140C2×Q8×C127—21no—
141C2×Q8×C127—21no—
142C2×Q8×C127—22no—
143C2×Q8×C127—81no—
144C2×Q8×C127C2×D412yes—
145C2×D4×C127—21no—
146C2×D4×C127—22no—
147C2×D4×C127—22no—
148C2×D4×C127—22no—
149C2×D4×C127—22no—
150C2×D4×C127—21no—
151C2×D4×C127—22no—
152C2×D4×C127—22no—
153C2×D4×C127—22no—
154C2×D4×C127—41no—
155C2×D4×C127C2×D412yes—
156C22×C4×C127—22no—
157C22×C4×C127—21no—
158C22×C4×C127—22no—
159C22×C4×C127—22no—
160C22×C4×C127—21no—
161C22×C4×C127—21no—
162C22×C4×C127—21no—
163C22×C4×C127—21no—
164C22×C4×C127—21no—
165C22×C4×C127—42no—
166C22×C4×C127—42no—
167C22×C4×C127—81no—
168C22×C4×C127—81no—
169C22×C4×C127—81no—
170C22×C4×C127C2×D411yescharacteristic
171C22×C4×C127C2×D411yescharacteristic
172C22×C4×C127C2×D411yescharacteristic
173C22×C4×C127C2×D411yescharacteristic
174C22×C4×C127C2×D412yes—
175C22×C4×C127C2×D412yes—
176C22×C4×C127C2411yescharacteristic
177Q16×C127—82no—
178SD16×C127—82no—
179M16×C127—82no—
180C2×C8×C127—22no—
181C2×C8×C127—22no—
182C4⋊C4×C127—21no—
183C4⋊C4×C127—21no—
184C4⋊C4×C127—22no—
185C4⋊C4×C127—21no—
186C4⋊C4×C127—21no—
187C4⋊C4×C127—22no—
188C4⋊C4×C127—22no—
189C4⋊C4×C127—22no—
190C4⋊C4×C127—22no—
191C4⋊C4×C127—22no—
192C4⋊C4×C127—42no—
193C4⋊C4×C127—41no—
194C4⋊C4×C127—41no—
195C4⋊C4×C127—81no—
196C4⋊C4×C127C2×D412yes—
197C4⋊C4×C127C2×D412yes—
198C22⋊C4×C127—22no—
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—42no—
210C22⋊C4×C127—42no—
211C22⋊C4×C127—81no—
212C42×C127—22no—
213C42×C127—22no—
214C42×C127—21no—
215C42×C127—22no—
216C42×C127—21no—
217C23×C127—21no—
218C23×C127—21no—
219C23×C127—21no—
220C23×C127—22no—
221C23×C127—21no—
222C23×C127—21no—
223C23×C127—41no—
224C23×C127—41no—
225C23×C127—41no—
226C23×C127—41no—
227C23×C127#12012yes—
228C23×C127C22×D411yescharacteristic
229Q8×C127—42no—
230Q8×C127—42no—
231Q8×C127—42no—
232Q8×C127—42no—
233Q8×C127—81no—
234D4×C127—44no—
235D4×C127—42no—
236D4×C127—42no—
237D4×C127—42no—
238D4×C127—42no—
239D4×C127—82no—
240D4×C127—82no—
241D4×C127—82no—
242D4×C127—81no—
243D4×C127—82no—
244C2×C4×C127—21no—
245C2×C4×C127—21no—
246C2×C4×C127—22no—
247C2×C4×C127—21no—
248C2×C4×C127—22no—
249C2×C4×C127—22no—
250C2×C4×C127—22no—
251C2×C4×C127—21no—
252C2×C4×C127—42no—
253C2×C4×C127—41no—
254C2×C4×C127—42no—
255C2×C4×C127—41no—
256C2×C4×C127—42no—
257C2×C4×C127—44no—
258C2×C4×C127—44no—
259C2×C4×C127—41no—
260C2×C4×C127—41no—
261C2×C4×C127—41no—
262C2×C4×C127—41no—
263C2×C4×C127—41no—
264C2×C4×C127—41no—
265C2×C4×C127—41no—
266C2×C4×C127—41no—
267C2×C4×C127—41no—
268C2×C4×C127—41no—
269C2×C4×C127—41no—
270C2×C4×C127—41no—
271C2×C4×C127—41no—
272C2×C4×C127—81no—
273C2×C4×C127—82no—
274C2×C4×C127—84no—
275C2×C4×C127#12011yescharacteristic
276C2×C4×C127#12011yescharacteristic
277C2×C4×C127#13712yes—
278C2×C4×C127C22×D412yes—
279C8×C127—42no—
280C8×C127—42no—
281C22×C127—21no—
282C22×C127—21no—
283C22×C127—22no—
284C22×C127—22no—
285C22×C127—22no—
286C22×C127—42no—
287C22×C127—42no—
288C22×C127—42no—
289C22×C127—42no—
290C22×C127—41no—
291C22×C127—42no—
292C22×C127—41no—
293C22×C127—41no—
294C22×C127—42no—
295C22×C127—81no—
296C22×C127C2×#12011yescharacteristic
297C22×C127C2×#13711yescharacteristic
298C22×C127#57511yescharacteristic
299C4×C127—22no—
300C4×C127—22no—
301C4×C127—21no—
302C4×C127—21no—
303C4×C127—42no—
304C4×C127—42no—
305C4×C127—41no—
306C4×C127—42no—
307C4×C127—41no—
308C4×C127—42no—
309C4×C127—81no—
310C4×C127—81no—
311C4×C127—81no—
312#18331C12711yesmaximal, characteristic, direct product factor, Sylow
313C2×C127—21no—
314C2×C127—21no—
315C2×C127—42no—
316C2×C127—82no—
317C2×C127—81no—
318C2×C127#300711yescharacteristic
319C2×C127#301811yescharacteristic
320C2×C127#302611yescharacteristic
321#3460C2×C12711yescharacteristic
322#3196C2×C12712yescomplemented
323#2564C2×C12711yescharacteristic, complemented
324#1618C2×C12712yescomplemented
325#1617C2×C12711yescharacteristic, complemented
326#1616C2×C12711yescharacteristic, complemented
327#1606C2×C12712yescomplemented
328#1605C2×C12711yescharacteristic, complemented
329#1604C2×C12711yescharacteristic, complemented
330#1504C2×C12712yescomplemented
331#1465C2×C12711yescharacteristic, complemented
332C127#1833111yescharacteristic, direct product factor, Sylow
333#544C22×C12711yescharacteristic
334#542—22no—
335#540—22no—
336#538—21no—
337#538C22×C12712yes—
338#537C22×C12711yescharacteristic
339#535—21no—
340#532—22no—
341#529—22no—
342#525—21no—
343#525C22×C12712yes—
344C2×C22⋊Q8C22×C12711yescharacteristic
345C4×#41—21no—
346C2×C4×D4C22×C12711yescharacteristic
347#485—22no—
348#484C22×C12712yes—
349#481C22×C12712yes—
350#470—22no—
351#468—22no—
352#452—22no—
353#451C22×C12712yes—
354#448C22×C12712yes—
355#407C22×C12712yes—
356#400C22×C12711yescharacteristic
357#399C22×C12711yescharacteristic
358#395C22×C12711yescharacteristic
359#393C22×C12711yescharacteristic
360#387—21no—
361#386—21no—
362#385—21no—
363#380—21no—
364C2×#95C22×C12711yescharacteristic
365#328C22×C12712yes—
366#328C22×C12712yes—
367#327C22×C12712yes—
368#327C22×C12712yes—
369#324C22×C12711yescharacteristic
370#324C22×C12711yescharacteristic
371#323C22×C12712yes—
372C2×#41—21no—
373C2×#41—21no—
374C2×#41D4×C12712yes—
375C22×Q8C23×C12711yescharacteristic
376C22×D4D4×C12711yescharacteristic
377C23×C4—21no—
378C23×C4D4×C12711yescharacteristic
379C23×C4C23×C12711yescharacteristic
380C2×Q16—22no—
381C2×SD16—22no—
382C2×M16—22no—
383C4⋊Q8—22no—
384#127—22no—
385#126—22no—
386#126—22no—
387#125—22no—
388#124—22no—
389#124—22no—
390#124—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—22no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8D4×C12712yes—
401C22⋊Q8C23×C12712yes—
402#121—21no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—22no—
408#121—21no—
409#121D4×C12712yes—
410#121C23×C12712yes—
411#120—22no—
412#120—22no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423#117—22no—
424#117—21no—
425#117—21no—
426#117D4×C12712yes—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4C23×C12711yescharacteristic
431C2×C4⋊C4C23×C12711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4D4×C12711yescharacteristic
436C2×C22⋊C4C23×C12711yescharacteristic
437C2×C42—22no—
438C2×C42D4×C12711yescharacteristic
439#107—22no—
440#106—22no—
441#103—22no—
442#103—22no—
443#103—22no—
444#102—22no—
445C22⋊C8—22no—
446C22⋊C8C23×C12712yes—
447C22⋊C8C23×C12712yes—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95C23×C12711yescharacteristic
454#95C23×C12711yescharacteristic
455C24C2×D4×C12711yescharacteristic
456C24C2×D4×C12711yescharacteristic
457#41—42no—
458#41—42no—
459#41—82no—
460#41—82no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—81no—
465C2×Q8C2×D4×C12712yes—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—41no—
476C2×D4C2×D4×C12712yes—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4C2×D4×C12711yescharacteristic
492C22×C4C2×D4×C12711yescharacteristic
493C22×C4C2×D4×C12711yescharacteristic
494C22×C4C2×D4×C12711yescharacteristic
495C22×C4C2×D4×C12712yes—
496C22×C4C2×D4×C12712yes—
497C22×C4C24×C12711yescharacteristic
498Q16—82no—
499SD16—82no—
500M16—82no—
501C2×C8—22no—
502C2×C8—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—81no—
517C4⋊C4C2×D4×C12712yes—
518C4⋊C4C2×D4×C12712yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—81no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—22no—
537C42—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#120×C12712yes—
549C23C22×D4×C12711yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—81no—
555D4—44no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—82no—
561D4—82no—
562D4—81no—
563D4—82no—
564D4—82no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4#120×C12711yescharacteristic
597C2×C4#120×C12711yescharacteristic
598C2×C4#137×C12712yes—
599C2×C4C22×D4×C12712yes—
600C8—42no—
601C8—42no—
602C22—21no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—81no—
617C22C2×#120×C12711yescharacteristic
618C22C2×#137×C12711yescharacteristic
619C22#575×C12711yescharacteristic
620C4—22no—
621C4—22no—
622C4—21no—
623C4—21no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—21no—
634C2—21no—
635C2—42no—
636C2—81no—
637C2—82no—
638C2C127×#300711yescharacteristic
639C2C127×#301811yescharacteristic
640C2C127×#302611yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3460C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3196C2semidirect
3. C127×#2564C2semidirect
4. C127×#1618C2semidirect
5. C127×#1617C2semidirect
6. C127×#1616C2semidirect
7. C127×#1606C2semidirect
8. C127×#1605C2semidirect
9. C127×#1604C2semidirect
10. C127×#1504C2semidirect
11. C127×#1465C2semidirect
312. #18331C127direct (N × H)
322. #3196C2×C127semidirect
323. #2564C2×C127semidirect
324. #1618C2×C127semidirect
325. #1617C2×C127semidirect
326. #1616C2×C127semidirect
327. #1606C2×C127semidirect
328. #1605C2×C127semidirect
329. #1604C2×C127semidirect
330. #1504C2×C127semidirect
331. #1465C2×C127semidirect
332. C127#18331direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34602
C127×#31962
C127×#25642
C127×#16182
C127×#16172
C127×#16162
C127×#16062
C127×#16052
C127×#16042
C127×#15042
C127×#14652
#18331127