Finite Group Databasebeta

← C127×#18329 #44582002

Subgroups — C127×#18329 #44582002

Order 32512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#538×C127C211yesmaximal, characteristic
2C127×#3195C211yesmaximal, characteristic, complemented
3C127×#3193C211yesmaximal, characteristic, complemented
4C127×#2561C211yesmaximal, characteristic, complemented
5C127×#1612C214yesmaximal, complemented
6C127×#1606C214yesmaximal, complemented
7C127×#1508C211yesmaximal, characteristic, complemented
8C127×#1507C211yesmaximal, characteristic, complemented
9C127×#1462C211yesmaximal, characteristic, complemented
10#538×C127—22no—
11#538×C127—22no—
12#538×C127—22no—
13#538×C127C2214yes—
14C2×#126×C127—21no—
15C2×#124×C127—21no—
16C2×#123×C127—21no—
17C2×C22⋊Q8×C127C2211yescharacteristic
18C2×C22⋊Q8×C127C2211yescharacteristic
19C2×#121×C127—21no—
20C2×#121×C127C2211yescharacteristic
21C2×#120×C127—21no—
22C2×C4×D4×C127—21no—
23C22×C22⋊C4×C127C2211yescharacteristic
24#484×C127—21no—
25#483×C127—21no—
26#483×C127C2212yescomplemented
27#480×C127C2212yescomplemented
28#470×C127—22no—
29#468×C127—22no—
30#451×C127—21no—
31#451×C127C2212yescomplemented
32#450×C127—21no—
33#448×C127C2212yescomplemented
34#407×C127C2211yescharacteristic
35#407×C127C2211yescharacteristic
36#397×C127C2212yescomplemented
37#393×C127C2212yescomplemented
38#388×C127—22no—
39#381×C127—21no—
40#379×C127—21no—
41C4×C22⋊C4×C127C2211yescharacteristic
42#328×C127C2214yescomplemented
43#327×C127C2214yescomplemented
44#323×C127C2214yescomplemented
45C25×C127D411yescharacteristic
46C22×Q8×C127D411yescharacteristic
47C22×Q8×C127C2311yescharacteristic
48C22×D4×C127—21no—
49C22×D4×C127—21no—
50C22×D4×C127—21no—
51C22×D4×C127D411yescharacteristic
52C23×C4×C127—21no—
53C23×C4×C127D411yescharacteristic
54C23×C4×C127C2311yescharacteristic
55C2×Q16×C127—22no—
56C2×SD16×C127—22no—
57C2×M16×C127—21no—
58C2×M16×C127—21no—
59#126×C127—24no—
60#126×C127—24no—
61#124×C127—24no—
62#124×C127—24no—
63#123×C127—24no—
64#123×C127—24no—
65C22⋊Q8×C127—21no—
66C22⋊Q8×C127—21no—
67C22⋊Q8×C127—21no—
68C22⋊Q8×C127—21no—
69C22⋊Q8×C127—21no—
70C22⋊Q8×C127—21no—
71C22⋊Q8×C127D412yes—
72C22⋊Q8×C127C2312yes—
73#121×C127—24no—
74#121×C127—24no—
75#121×C127—21no—
76#121×C127—21no—
77#121×C127D412yes—
78#121×C127C2312yes—
79#120×C127—24no—
80#120×C127—24no—
81C4×D4×C127—24no—
82C4×D4×C127—24no—
83C2×C4⋊C4×C127—21no—
84C2×C4⋊C4×C127—21no—
85C2×C4⋊C4×C127—21no—
86C2×C4⋊C4×C127D411yescharacteristic
87C2×C4⋊C4×C127D411yescharacteristic
88C2×C4⋊C4×C127C2311yescharacteristic
89C2×C22⋊C4×C127—21no—
90C2×C22⋊C4×C127—21no—
91C2×C22⋊C4×C127—21no—
92C2×C22⋊C4×C127—21no—
93C2×C22⋊C4×C127—21no—
94C2×C22⋊C4×C127—21no—
95C2×C22⋊C4×C127—21no—
96C2×C22⋊C4×C127—21no—
97C2×C22⋊C4×C127—21no—
98C2×C22⋊C4×C127—21no—
99C2×C22⋊C4×C127—21no—
100C2×C22⋊C4×C127—22no—
101C2×C22⋊C4×C127—22no—
102C2×C22⋊C4×C127D412yes—
103C2×C22⋊C4×C127C2312yes—
104C2×C42×C127—21no—
105C2×C42×C127—21no—
106#107×C127—22no—
107#106×C127—22no—
108#103×C127—22no—
109#103×C127—22no—
110#103×C127—22no—
111#102×C127—22no—
112C22⋊C8×C127—21no—
113C22⋊C8×C127—21no—
114C22⋊C8×C127C2312yes—
115C22⋊C8×C127C2312yes—
116#95×C127—22no—
117#95×C127—22no—
118#95×C127C2312yes—
119C24×C127—21no—
120C24×C127—21no—
121C24×C127—21no—
122C24×C127—42no—
123C24×C127—42no—
124C24×C127—81no—
125C24×C127C2×D411yescharacteristic
126C24×C127C2×D412yes—
127C2×Q8×C127—21no—
128C2×Q8×C127—21no—
129C2×Q8×C127—21no—
130C2×Q8×C127—81no—
131C2×Q8×C127—81no—
132C2×Q8×C127C2×D412yes—
133C2×D4×C127—22no—
134C2×D4×C127—22no—
135C2×D4×C127—24no—
136C2×D4×C127—24no—
137C2×D4×C127—21no—
138C2×D4×C127—44no—
139C2×D4×C127—81no—
140C2×D4×C127—82no—
141C2×D4×C127—82no—
142C2×D4×C127C2×D412yes—
143C22×C4×C127—21no—
144C22×C4×C127—21no—
145C22×C4×C127—21no—
146C22×C4×C127—21no—
147C22×C4×C127—24no—
148C22×C4×C127—21no—
149C22×C4×C127—22no—
150C22×C4×C127—21no—
151C22×C4×C127—21no—
152C22×C4×C127—42no—
153C22×C4×C127—81no—
154C22×C4×C127—82no—
155C22×C4×C127C2×D412yes—
156C22×C4×C127C2×D411yescharacteristic
157C22×C4×C127C2×D411yescharacteristic
158C22×C4×C127C2×D411yescharacteristic
159C22×C4×C127C2×D411yescharacteristic
160C22×C4×C127C2×D411yescharacteristic
161C22×C4×C127C2411yescharacteristic
162Q16×C127—82no—
163SD16×C127—82no—
164M16×C127—81no—
165M16×C127—81no—
166C2×C8×C127—22no—
167C2×C8×C127—22no—
168C4⋊C4×C127—24no—
169C4⋊C4×C127—22no—
170C4⋊C4×C127—21no—
171C4⋊C4×C127—21no—
172C4⋊C4×C127—22no—
173C4⋊C4×C127—21no—
174C4⋊C4×C127—21no—
175C4⋊C4×C127—41no—
176C4⋊C4×C127—41no—
177C4⋊C4×C127C2×D412yes—
178C4⋊C4×C127C2×D412yes—
179C22⋊C4×C127—24no—
180C22⋊C4×C127—24no—
181C22⋊C4×C127—24no—
182C22⋊C4×C127—22no—
183C22⋊C4×C127—22no—
184C22⋊C4×C127—22no—
185C22⋊C4×C127—22no—
186C22⋊C4×C127—22no—
187C22⋊C4×C127—22no—
188C22⋊C4×C127—22no—
189C22⋊C4×C127—22no—
190C22⋊C4×C127—42no—
191C22⋊C4×C127—44no—
192C22⋊C4×C127—41no—
193C22⋊C4×C127—41no—
194C22⋊C4×C127—82no—
195C42×C127—24no—
196C42×C127—81no—
197C23×C127—22no—
198C23×C127—22no—
199C23×C127—22no—
200C23×C127—22no—
201C23×C127—22no—
202C23×C127—22no—
203C23×C127—41no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—42no—
207C23×C127—42no—
208C23×C127—42no—
209C23×C127—42no—
210C23×C127—42no—
211C23×C127—42no—
212C23×C127—42no—
213C23×C127—81no—
214C23×C127—81no—
215C23×C127—88no—
216C23×C127#12011yescharacteristic
217C23×C127#12011yescharacteristic
218C23×C127C22×D411yescharacteristic
219Q8×C127—42no—
220Q8×C127—42no—
221Q8×C127—42no—
222Q8×C127—81no—
223D4×C127—42no—
224D4×C127—44no—
225D4×C127—44no—
226D4×C127—81no—
227D4×C127—84no—
228D4×C127—84no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—22no—
232C2×C4×C127—22no—
233C2×C4×C127—41no—
234C2×C4×C127—41no—
235C2×C4×C127—41no—
236C2×C4×C127—41no—
237C2×C4×C127—41no—
238C2×C4×C127—41no—
239C2×C4×C127—42no—
240C2×C4×C127—42no—
241C2×C4×C127—42no—
242C2×C4×C127—42no—
243C2×C4×C127—42no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—42no—
247C2×C4×C127—41no—
248C2×C4×C127—41no—
249C2×C4×C127—42no—
250C2×C4×C127—42no—
251C2×C4×C127—81no—
252C2×C4×C127—84no—
253C2×C4×C127#12011yescharacteristic
254C2×C4×C127#12011yescharacteristic
255C2×C4×C127C22×D411yescharacteristic
256C2×C4×C127C22×D411yescharacteristic
257C8×C127—42no—
258C8×C127—42no—
259C22×C127—21no—
260C22×C127—21no—
261C22×C127—21no—
262C22×C127—21no—
263C22×C127—21no—
264C22×C127—21no—
265C22×C127—21no—
266C22×C127—44no—
267C22×C127—44no—
268C22×C127—42no—
269C22×C127—42no—
270C22×C127—42no—
271C22×C127—42no—
272C22×C127—42no—
273C22×C127—42no—
274C22×C127—82no—
275C22×C127—84no—
276C22×C127—84no—
277C22×C127—84no—
278C22×C127#45512yes—
279C22×C127C2×#12011yescharacteristic
280C22×C127#57512yes—
281C4×C127—21no—
282C4×C127—21no—
283C4×C127—21no—
284C4×C127—21no—
285C4×C127—42no—
286C4×C127—42no—
287C4×C127—42no—
288C4×C127—42no—
289C4×C127—82no—
290C4×C127—81no—
291C4×C127—81no—
292#18329C12711yesmaximal, characteristic, direct product factor, Sylow
293C2×C127—22no—
294C2×C127—22no—
295C2×C127—41no—
296C2×C127—82no—
297C2×C127—82no—
298C2×C127#300511yescharacteristic
299C2×C127C2×#45511yescharacteristic
300C2×C127#302611yescharacteristic
301C2×#538C2×C12711yescharacteristic
302#3195C2×C12711yescharacteristic, complemented
303#3193C2×C12711yescharacteristic, complemented
304#2561C2×C12711yescharacteristic, complemented
305#1612C2×C12714yescomplemented
306#1606C2×C12714yescomplemented
307#1508C2×C12711yescharacteristic, complemented
308#1507C2×C12711yescharacteristic, complemented
309#1462C2×C12711yescharacteristic, complemented
310C127#1832911yescharacteristic, direct product factor, Sylow
311#538—22no—
312#538—22no—
313#538—22no—
314#538C22×C12714yes—
315C2×#126—21no—
316C2×#124—21no—
317C2×#123—21no—
318C2×C22⋊Q8C22×C12711yescharacteristic
319C2×C22⋊Q8C22×C12711yescharacteristic
320C2×#121—21no—
321C2×#121C22×C12711yescharacteristic
322C2×#120—21no—
323C2×C4×D4—21no—
324C22×C22⋊C4C22×C12711yescharacteristic
325#484—21no—
326#483—21no—
327#483C22×C12712yescomplemented
328#480C22×C12712yescomplemented
329#470—22no—
330#468—22no—
331#451—21no—
332#451C22×C12712yescomplemented
333#450—21no—
334#448C22×C12712yescomplemented
335#407C22×C12711yescharacteristic
336#407C22×C12711yescharacteristic
337#397C22×C12712yescomplemented
338#393C22×C12712yescomplemented
339#388—22no—
340#381—21no—
341#379—21no—
342C4×C22⋊C4C22×C12711yescharacteristic
343#328C22×C12714yescomplemented
344#327C22×C12714yescomplemented
345#323C22×C12714yescomplemented
346C25D4×C12711yescharacteristic
347C22×Q8D4×C12711yescharacteristic
348C22×Q8C23×C12711yescharacteristic
349C22×D4—21no—
350C22×D4—21no—
351C22×D4—21no—
352C22×D4D4×C12711yescharacteristic
353C23×C4—21no—
354C23×C4D4×C12711yescharacteristic
355C23×C4C23×C12711yescharacteristic
356C2×Q16—22no—
357C2×SD16—22no—
358C2×M16—21no—
359C2×M16—21no—
360#126—24no—
361#126—24no—
362#124—24no—
363#124—24no—
364#123—24no—
365#123—24no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8D4×C12712yes—
373C22⋊Q8C23×C12712yes—
374#121—24no—
375#121—24no—
376#121—21no—
377#121—21no—
378#121D4×C12712yes—
379#121C23×C12712yes—
380#120—24no—
381#120—24no—
382C4×D4—24no—
383C4×D4—24no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4D4×C12711yescharacteristic
388C2×C4⋊C4D4×C12711yescharacteristic
389C2×C4⋊C4C23×C12711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4D4×C12712yes—
404C2×C22⋊C4C23×C12712yes—
405C2×C42—21no—
406C2×C42—21no—
407#107—22no—
408#106—22no—
409#103—22no—
410#103—22no—
411#103—22no—
412#102—22no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8C23×C12712yes—
416C22⋊C8C23×C12712yes—
417#95—22no—
418#95—22no—
419#95C23×C12712yes—
420C24—21no—
421C24—21no—
422C24—21no—
423C24—42no—
424C24—42no—
425C24—81no—
426C24C2×D4×C12711yescharacteristic
427C24C2×D4×C12712yes—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×Q8C2×D4×C12712yes—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—21no—
439C2×D4—44no—
440C2×D4—81no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4C2×D4×C12712yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—82no—
456C22×C4C2×D4×C12712yes—
457C22×C4C2×D4×C12711yescharacteristic
458C22×C4C2×D4×C12711yescharacteristic
459C22×C4C2×D4×C12711yescharacteristic
460C22×C4C2×D4×C12711yescharacteristic
461C22×C4C2×D4×C12711yescharacteristic
462C22×C4C24×C12711yescharacteristic
463Q16—82no—
464SD16—82no—
465M16—81no—
466M16—81no—
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4C2×D4×C12712yes—
479C4⋊C4C2×D4×C12712yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—82no—
496C42—24no—
497C42—81no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—81no—
515C23—81no—
516C23—88no—
517C23#120×C12711yescharacteristic
518C23#120×C12711yescharacteristic
519C23C22×D4×C12711yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—81no—
524D4—42no—
525D4—44no—
526D4—44no—
527D4—81no—
528D4—84no—
529D4—84no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—81no—
553C2×C4—84no—
554C2×C4#120×C12711yescharacteristic
555C2×C4#120×C12711yescharacteristic
556C2×C4C22×D4×C12711yescharacteristic
557C2×C4C22×D4×C12711yescharacteristic
558C8—42no—
559C8—42no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—82no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22#455×C12712yes—
580C22C2×#120×C12711yescharacteristic
581C22#575×C12712yes—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—82no—
591C4—81no—
592C4—81no—
593C2—22no—
594C2—22no—
595C2—41no—
596C2—82no—
597C2—82no—
598C2C127×#300511yescharacteristic
599C2C2×#455×C12711yescharacteristic
600C2C127×#302611yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radicalC2×#538×C127C2
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×#3195C2semidirect
3. C127×#3193C2semidirect
4. C127×#2561C2semidirect
5. C127×#1612C2semidirect
6. C127×#1606C2semidirect
7. C127×#1508C2semidirect
8. C127×#1507C2semidirect
9. C127×#1462C2semidirect
26. #483×C127C22semidirect
27. #480×C127C22semidirect
31. #451×C127C22semidirect
33. #448×C127C22semidirect
36. #397×C127C22semidirect
37. #393×C127C22semidirect
42. #328×C127C22semidirect
43. #327×C127C22semidirect
44. #323×C127C22semidirect
292. #18329C127direct (N × H)
302. #3195C2×C127semidirect
303. #3193C2×C127semidirect
304. #2561C2×C127semidirect
305. #1612C2×C127semidirect
306. #1606C2×C127semidirect
307. #1508C2×C127semidirect
308. #1507C2×C127semidirect
309. #1462C2×C127semidirect
310. C127#18329direct (N × H)
327. #483C22×C127semidirect
328. #480C22×C127semidirect
332. #451C22×C127semidirect
334. #448C22×C127semidirect
337. #397C22×C127semidirect
338. #393C22×C127semidirect
343. #328C22×C127semidirect
344. #327C22×C127semidirect
345. #323C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#538×C1272
C127×#31952
C127×#31932
C127×#25612
C127×#16122
C127×#16062
C127×#15082
C127×#15072
C127×#14622
#18329127