Finite Group Databasebeta

← C127×#19774 #44583447

Subgroups — C127×#19774 #44583447

Order 32512.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3380C211yesmaximal, characteristic, complemented
2C127×#3331C211yesmaximal, characteristic, complemented
3C127×#3295C211yesmaximal, characteristic, complemented
4C127×#3294C211yesmaximal, characteristic
5C127×#1719C211yesmaximal, characteristic, complemented
6C127×#1697C211yesmaximal, characteristic, complemented
7C127×#1683C211yesmaximal, characteristic, complemented
8C127×#1682C211yesmaximal, characteristic, complemented
9C127×#1675C211yesmaximal, characteristic, complemented
10C127×#1644C211yesmaximal, characteristic, complemented
11C127×#1635C211yesmaximal, characteristic, complemented
12C127×#1629C211yesmaximal, characteristic, complemented
13C127×#1622C211yesmaximal, characteristic, complemented
14C127×#1581C211yesmaximal, characteristic, complemented
15C127×SD16⋊C8C211yesmaximal, characteristic, complemented
16#554×C127C2211yescharacteristic
17#554×C127C2211yescharacteristic, complemented
18#547×C127C2211yescharacteristic
19D42×C127C2211yescharacteristic
20#500×C127C2211yescharacteristic, complemented
21#499×C127—21no—
22#492×C127C2211yescharacteristic
23#487×C127C2211yescharacteristic, complemented
24#487×C127C2211yescharacteristic, complemented
25#486×C127C2211yescharacteristic
26#486×C127C2211yescharacteristic
27#483×C127—21no—
28#481×C127—21no—
29#480×C127—21no—
30#478×C127—21no—
31#478×C127—21no—
32#476×C127C2211yescharacteristic
33#476×C127C2211yescharacteristic
34#476×C127C2211yescharacteristic
35#476×C127C2211yescharacteristic, complemented
36#474×C127C2211yescharacteristic
37#470×C127—21no—
38#466×C127—21no—
39#465×C127—21no—
40#462×C127C2211yescharacteristic, complemented
41#460×C127C2211yescharacteristic
42#459×C127C2211yescharacteristic
43#452×C127—21no—
44#450×C127—21no—
45#447×C127—21no—
46C8×Q8×C127C2211yescharacteristic, complemented
47C4×SD16×C127C2211yescharacteristic
48C4×D8×C127—21no—
49C4×D8×C127C2211yescharacteristic
50#436×C127C2211yescharacteristic, complemented
51C8×D4×C127C2211yescharacteristic
52C8×D4×C127C2211yescharacteristic
53#417×C127—21no—
54C2×#102×C127—21no—
55C2×#102×C127—21no—
56#335×C127C2211yescharacteristic, complemented
57#335×C127C2211yescharacteristic, complemented
58#333×C127C2211yescharacteristic, complemented
59#333×C127C2211yescharacteristic, complemented
60#332×C127C2211yescharacteristic, complemented
61#331×C127C2211yescharacteristic
62Q8⋊C8×C127C2211yescharacteristic, complemented
63D4⋊C8×C127C2211yescharacteristic
64D4⋊C8×C127C2211yescharacteristic
65D4⋊C8×C127C2211yescharacteristic, complemented
66#322×C127C2211yescharacteristic, complemented
67C2×#41×C127—21no—
68C22×D4×C127—21no—
69C22×D4×C127—21no—
70C2×Q16×C127—21no—
71C2×SD16×C127—21no—
72C2×SD16×C127—41no—
73C2×SD16×C127D411yescharacteristic
74C2×D8×C127—21no—
75C2×D8×C127D411yescharacteristic
76C2×M16×C127—21no—
77C22×C8×C127—21no—
78C22×C8×C127—21no—
79C4⋊Q8×C127C2311yescharacteristic
80C4⋊Q8×C127C2311yescharacteristic
81C4⋊Q8×C127C2311yescharacteristic
82#127×C127C2311yescharacteristic
83#125×C127—21no—
84#125×C127—21no—
85#123×C127—41no—
86C22⋊Q8×C127—21no—
87C22⋊Q8×C127—21no—
88C22⋊Q8×C127—21no—
89C22⋊Q8×C127—41no—
90C22⋊Q8×C127—41no—
91#121×C127—21no—
92#121×C127—21no—
93#121×C127—21no—
94#120×C127—41no—
95C4×Q8×C127C2311yescharacteristic
96C4×D4×C127—21no—
97C4×D4×C127—21no—
98C4×D4×C127C2311yescharacteristic
99C4×D4×C127C2311yescharacteristic
100C4×D4×C127C2311yescharacteristic
101C2×C4⋊C4×C127—21no—
102C2×C4⋊C4×C127—21no—
103C2×C4⋊C4×C127—21no—
104#107×C127—21no—
105#107×C127—21no—
106#107×C127—21no—
107#107×C127D411yescharacteristic
108#106×C127—21no—
109#106×C127—21no—
110#106×C127—21no—
111#106×C127—21no—
112#106×C127D411yescharacteristic
113C4⋊C8×C127—21no—
114C4⋊C8×C127C2311yescharacteristic
115C4⋊C8×C127C2311yescharacteristic
116C4⋊C8×C127C2311yescharacteristic
117C4⋊C8×C127C2311yescharacteristic
118#103×C127—21no—
119#103×C127—21no—
120#103×C127—21no—
121#103×C127—41no—
122#103×C127D411yescharacteristic
123#102×C127—21no—
124#102×C127—21no—
125#102×C127—21no—
126#102×C127—21no—
127#102×C127—21no—
128#102×C127—21no—
129#102×C127—21no—
130#102×C127—21no—
131#102×C127—21no—
132#102×C127—21no—
133#102×C127—21no—
134#102×C127D411yescharacteristic
135#102×C127D411yescharacteristic
136#102×C127D411yescharacteristic
137C22⋊C8×C127—21no—
138C22⋊C8×C127—21no—
139C22⋊C8×C127—21no—
140C4×C8×C127—21no—
141C4×C8×C127C2311yescharacteristic
142C4×C8×C127C2311yescharacteristic
143C4×C8×C127C2311yescharacteristic
144C24×C127—41no—
145#41×C127—42no—
146#41×C127—81no—
147C2×Q8×C127—21no—
148C2×Q8×C127—21no—
149C2×Q8×C127—21no—
150C2×Q8×C127C2×D411yescharacteristic
151C2×D4×C127—21no—
152C2×D4×C127—21no—
153C2×D4×C127—21no—
154C2×D4×C127—41no—
155C2×D4×C127—41no—
156C2×D4×C127—42no—
157C2×D4×C127—42no—
158C2×D4×C127—41no—
159C2×D4×C127—81no—
160C2×D4×C127—81no—
161C2×D4×C127C2×D411yescharacteristic
162C2×D4×C127C2×D411yescharacteristic
163C2×D4×C127C2×D411yescharacteristic
164C22×C4×C127—21no—
165C22×C4×C127—21no—
166C22×C4×C127—21no—
167C22×C4×C127—41no—
168C22×C4×C127—41no—
169C22×C4×C127—41no—
170Q16×C127—81no—
171SD16×C127—22no—
172SD16×C127—81no—
173SD16×C127—82no—
174D8×C127—22no—
175D8×C127—81no—
176M16×C127—81no—
177C2×C8×C127—21no—
178C2×C8×C127—21no—
179C2×C8×C127—21no—
180C2×C8×C127—21no—
181C2×C8×C127—21no—
182C2×C8×C127—81no—
183C2×C8×C127—81no—
184C2×C8×C127C2×D411yescharacteristic
185C2×C8×C127C2×D411yescharacteristic
186C2×C8×C127C2×D411yescharacteristic
187C2×C8×C127C2×D411yescharacteristic
188C4⋊C4×C127—21no—
189C4⋊C4×C127—21no—
190C4⋊C4×C127—21no—
191C4⋊C4×C127—21no—
192C4⋊C4×C127—21no—
193C4⋊C4×C127—21no—
194C4⋊C4×C127—21no—
195C4⋊C4×C127—41no—
196C4⋊C4×C127—41no—
197C4⋊C4×C127—41no—
198C4⋊C4×C127—41no—
199C4⋊C4×C127—41no—
200C4⋊C4×C127C2×D411yescharacteristic
201C4⋊C4×C127C2×D411yescharacteristic
202C4⋊C4×C127C2×D411yescharacteristic
203C4⋊C4×C127C2×D411yescharacteristic
204C22⋊C4×C127—21no—
205C22⋊C4×C127—21no—
206C22⋊C4×C127—21no—
207C22⋊C4×C127—41no—
208C22⋊C4×C127—41no—
209C22⋊C4×C127—41no—
210C22⋊C4×C127—41no—
211C42×C127—21no—
212C42×C127C2411yescharacteristic
213C23×C127—21no—
214C23×C127—21no—
215C23×C127—21no—
216C23×C127—41no—
217C23×C127—82no—
218C23×C127—82no—
219C23×C127—161no—
220Q8×C127—21no—
221Q8×C127—21no—
222Q8×C127—41no—
223Q8×C127—41no—
224Q8×C127—81no—
225Q8×C127—81no—
226D4×C127—21no—
227D4×C127—21no—
228D4×C127—21no—
229D4×C127—21no—
230D4×C127—41no—
231D4×C127—41no—
232D4×C127—41no—
233D4×C127—41no—
234D4×C127—81no—
235D4×C127—82no—
236D4×C127—82no—
237D4×C127—82no—
238D4×C127#13512yes—
239D4×C127#13712yes—
240C2×C4×C127—21no—
241C2×C4×C127—21no—
242C2×C4×C127—21no—
243C2×C4×C127—21no—
244C2×C4×C127—21no—
245C2×C4×C127—41no—
246C2×C4×C127—41no—
247C2×C4×C127—41no—
248C2×C4×C127—41no—
249C2×C4×C127—41no—
250C2×C4×C127—41no—
251C2×C4×C127—41no—
252C2×C4×C127—41no—
253C2×C4×C127—41no—
254C2×C4×C127—82no—
255C2×C4×C127—161no—
256C2×C4×C127—161no—
257C2×C4×C127C22×D411yescharacteristic
258C2×C4×C127C22×D411yescharacteristic
259C2×C4×C127H(5,𝔽2)11yescharacteristic
260C8×C127—21no—
261C8×C127—21no—
262C8×C127—21no—
263C8×C127—21no—
264C8×C127—41no—
265C8×C127—41no—
266C8×C127—41no—
267C8×C127—41no—
268C8×C127—41no—
269C22×C127—22no—
270C22×C127—22no—
271C22×C127—41no—
272C22×C127—41no—
273C22×C127—41no—
274C22×C127—41no—
275C22×C127—41no—
276C22×C127—41no—
277C22×C127—41no—
278C22×C127—81no—
279C22×C127—81no—
280C22×C127—81no—
281C22×C127—164no—
282C22×C127D4211yescharacteristic
283C4×C127—21no—
284C4×C127—41no—
285C4×C127—41no—
286C4×C127—41no—
287C4×C127—41no—
288C4×C127—41no—
289C4×C127—81no—
290C4×C127—81no—
291C4×C127—81no—
292C4×C127C2×#13511yescharacteristic
293C4×C127C2×#13711yescharacteristic
294C4×C127#57611yescharacteristic
295C4×C127#57711yescharacteristic
296#19774C12711yesmaximal, characteristic, direct product factor, Sylow
297C2×C127—42no—
298C2×C127—42no—
299C2×C127—81no—
300C2×C127—161no—
301C2×C127#326611yescharacteristic
302C2×C127#328011yescharacteristic
303C2×C127#328811yescharacteristic
304#3380C2×C12711yescharacteristic, complemented
305#3331C2×C12711yescharacteristic, complemented
306#3295C2×C12711yescharacteristic, complemented
307#3294C2×C12711yescharacteristic
308#1719C2×C12711yescharacteristic, complemented
309#1697C2×C12711yescharacteristic, complemented
310#1683C2×C12711yescharacteristic, complemented
311#1682C2×C12711yescharacteristic, complemented
312#1675C2×C12711yescharacteristic, complemented
313#1644C2×C12711yescharacteristic, complemented
314#1635C2×C12711yescharacteristic, complemented
315#1629C2×C12711yescharacteristic, complemented
316#1622C2×C12711yescharacteristic, complemented
317#1581C2×C12711yescharacteristic, complemented
318SD16⋊C8C2×C12711yescharacteristic, complemented
319C127#1977411yescharacteristic, direct product factor, Sylow
320#554C22×C12711yescharacteristic
321#554C22×C12711yescharacteristic, complemented
322#547C22×C12711yescharacteristic
323D42C22×C12711yescharacteristic
324#500C22×C12711yescharacteristic, complemented
325#499—21no—
326#492C22×C12711yescharacteristic
327#487C22×C12711yescharacteristic, complemented
328#487C22×C12711yescharacteristic, complemented
329#486C22×C12711yescharacteristic
330#486C22×C12711yescharacteristic
331#483—21no—
332#481—21no—
333#480—21no—
334#478—21no—
335#478—21no—
336#476C22×C12711yescharacteristic
337#476C22×C12711yescharacteristic
338#476C22×C12711yescharacteristic
339#476C22×C12711yescharacteristic, complemented
340#474C22×C12711yescharacteristic
341#470—21no—
342#466—21no—
343#465—21no—
344#462C22×C12711yescharacteristic, complemented
345#460C22×C12711yescharacteristic
346#459C22×C12711yescharacteristic
347#452—21no—
348#450—21no—
349#447—21no—
350C8×Q8C22×C12711yescharacteristic, complemented
351C4×SD16C22×C12711yescharacteristic
352C4×D8—21no—
353C4×D8C22×C12711yescharacteristic
354#436C22×C12711yescharacteristic, complemented
355C8×D4C22×C12711yescharacteristic
356C8×D4C22×C12711yescharacteristic
357#417—21no—
358C2×#102—21no—
359C2×#102—21no—
360#335C22×C12711yescharacteristic, complemented
361#335C22×C12711yescharacteristic, complemented
362#333C22×C12711yescharacteristic, complemented
363#333C22×C12711yescharacteristic, complemented
364#332C22×C12711yescharacteristic, complemented
365#331C22×C12711yescharacteristic
366Q8⋊C8C22×C12711yescharacteristic, complemented
367D4⋊C8C22×C12711yescharacteristic
368D4⋊C8C22×C12711yescharacteristic
369D4⋊C8C22×C12711yescharacteristic, complemented
370#322C22×C12711yescharacteristic, complemented
371C2×#41—21no—
372C22×D4—21no—
373C22×D4—21no—
374C2×Q16—21no—
375C2×SD16—21no—
376C2×SD16—41no—
377C2×SD16D4×C12711yescharacteristic
378C2×D8—21no—
379C2×D8D4×C12711yescharacteristic
380C2×M16—21no—
381C22×C8—21no—
382C22×C8—21no—
383C4⋊Q8C23×C12711yescharacteristic
384C4⋊Q8C23×C12711yescharacteristic
385C4⋊Q8C23×C12711yescharacteristic
386#127C23×C12711yescharacteristic
387#125—21no—
388#125—21no—
389#123—41no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—41no—
394C22⋊Q8—41no—
395#121—21no—
396#121—21no—
397#121—21no—
398#120—41no—
399C4×Q8C23×C12711yescharacteristic
400C4×D4—21no—
401C4×D4—21no—
402C4×D4C23×C12711yescharacteristic
403C4×D4C23×C12711yescharacteristic
404C4×D4C23×C12711yescharacteristic
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408#107—21no—
409#107—21no—
410#107—21no—
411#107D4×C12711yescharacteristic
412#106—21no—
413#106—21no—
414#106—21no—
415#106—21no—
416#106D4×C12711yescharacteristic
417C4⋊C8—21no—
418C4⋊C8C23×C12711yescharacteristic
419C4⋊C8C23×C12711yescharacteristic
420C4⋊C8C23×C12711yescharacteristic
421C4⋊C8C23×C12711yescharacteristic
422#103—21no—
423#103—21no—
424#103—21no—
425#103—41no—
426#103D4×C12711yescharacteristic
427#102—21no—
428#102—21no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102D4×C12711yescharacteristic
439#102D4×C12711yescharacteristic
440#102D4×C12711yescharacteristic
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—21no—
444C4×C8—21no—
445C4×C8C23×C12711yescharacteristic
446C4×C8C23×C12711yescharacteristic
447C4×C8C23×C12711yescharacteristic
448C24—41no—
449#41—42no—
450#41—81no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C2×D4×C12711yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—41no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4C2×D4×C12711yescharacteristic
466C2×D4C2×D4×C12711yescharacteristic
467C2×D4C2×D4×C12711yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474Q16—81no—
475SD16—22no—
476SD16—81no—
477SD16—82no—
478D8—22no—
479D8—81no—
480M16—81no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—81no—
487C2×C8—81no—
488C2×C8C2×D4×C12711yescharacteristic
489C2×C8C2×D4×C12711yescharacteristic
490C2×C8C2×D4×C12711yescharacteristic
491C2×C8C2×D4×C12711yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4C2×D4×C12711yescharacteristic
505C4⋊C4C2×D4×C12711yescharacteristic
506C4⋊C4C2×D4×C12711yescharacteristic
507C4⋊C4C2×D4×C12711yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—21no—
516C42C24×C12711yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—161no—
524Q8—21no—
525Q8—21no—
526Q8—41no—
527Q8—41no—
528Q8—81no—
529Q8—81no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4#135×C12712yes—
543D4#137×C12712yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—82no—
559C2×C4—161no—
560C2×C4—161no—
561C2×C4C22×D4×C12711yescharacteristic
562C2×C4C22×D4×C12711yescharacteristic
563C2×C4H(5,𝔽2)×C12711yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—41no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—164no—
586C22D42×C12711yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4C2×#135×C12711yescharacteristic
597C4C2×#137×C12711yescharacteristic
598C4#576×C12711yescharacteristic
599C4#577×C12711yescharacteristic
600C2—42no—
601C2—42no—
602C2—81no—
603C2—161no—
604C2C127×#326611yescharacteristic
605C2C127×#328011yescharacteristic
606C2C127×#328811yescharacteristic
subgroupG/H
center, socleC22×C127D42
commutator, Frattini, cosocleC42C24×C127
prime radicalC127×#3294C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3380C2semidirect
2. C127×#3331C2semidirect
3. C127×#3295C2semidirect
5. C127×#1719C2semidirect
6. C127×#1697C2semidirect
7. C127×#1683C2semidirect
8. C127×#1682C2semidirect
9. C127×#1675C2semidirect
10. C127×#1644C2semidirect
11. C127×#1635C2semidirect
12. C127×#1629C2semidirect
13. C127×#1622C2semidirect
14. C127×#1581C2semidirect
15. C127×SD16⋊C8C2semidirect
17. #554×C127C22semidirect
20. #500×C127C22semidirect
23. #487×C127C22semidirect
24. #487×C127C22semidirect
35. #476×C127C22semidirect
40. #462×C127C22semidirect
46. C8×Q8×C127C22semidirect
50. #436×C127C22semidirect
56. #335×C127C22semidirect
57. #335×C127C22semidirect
58. #333×C127C22semidirect
59. #333×C127C22semidirect
60. #332×C127C22semidirect
62. Q8⋊C8×C127C22semidirect
65. D4⋊C8×C127C22semidirect
66. #322×C127C22semidirect
296. #19774C127direct (N × H)
304. #3380C2×C127semidirect
305. #3331C2×C127semidirect
306. #3295C2×C127semidirect
308. #1719C2×C127semidirect
309. #1697C2×C127semidirect
310. #1683C2×C127semidirect
311. #1682C2×C127semidirect
312. #1675C2×C127semidirect
313. #1644C2×C127semidirect
314. #1635C2×C127semidirect
315. #1629C2×C127semidirect
316. #1622C2×C127semidirect
317. #1581C2×C127semidirect
318. SD16⋊C8C2×C127semidirect
319. C127#19774direct (N × H)
321. #554C22×C127semidirect
324. #500C22×C127semidirect
327. #487C22×C127semidirect
328. #487C22×C127semidirect
339. #476C22×C127semidirect
344. #462C22×C127semidirect
350. C8×Q8C22×C127semidirect
354. #436C22×C127semidirect
360. #335C22×C127semidirect
361. #335C22×C127semidirect
362. #333C22×C127semidirect
363. #333C22×C127semidirect
364. #332C22×C127semidirect
366. Q8⋊C8C22×C127semidirect
369. D4⋊C8C22×C127semidirect
370. #322C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#33802
C127×#33312
C127×#32952
C127×#32942
C127×#17192
C127×#16972
C127×#16832
C127×#16822
C127×#16752
C127×#16442
C127×#16352
C127×#16292
C127×#16222
C127×#15812
C127×SD16⋊C82
#19774127