Finite Group Databasebeta

← C127×#21009 #44584682

Subgroups — C127×#21009 #44584682

Order 32512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#568×C127C211yesmaximal, characteristic, complemented
2C127×#3449C211yesmaximal, characteristic
3C127×#2896C212yesmaximal, complemented
4C2×#410×C127C211yesmaximal, characteristic, complemented
5C127×#1910C214yesmaximal, complemented
6C127×#1909C214yesmaximal, complemented
7C2×H(3,ℤ/4)×C127C212yesmaximal, complemented
8C22×#41×C127C2211yescharacteristic
9C22×#41×C127C2211yescharacteristic
10#568×C127—22no—
11#568×C127—24no—
12#568×C127C2214yes—
13C2×#131×C127—24no—
14C22×M16×C127—22no—
15C22×M16×C127C2212yes—
16#537×C127—22no—
17#536×C127C412yescomplemented
18#536×C127C2212yes—
19#535×C127C412yescomplemented
20#535×C127C2212yes—
21#534×C127—22no—
22#532×C127C414yescomplemented
23#525×C127—22no—
24#525×C127—22no—
25C2×#117×C127C2212yes—
26C2×#117×C127C2211yescharacteristic
27#419×C127C2214yescomplemented
28#418×C127C2214yescomplemented
29#417×C127—24no—
30#416×C127—24no—
31#410×C127—22no—
32#410×C127C2214yes—
33C2×#99×C127—22no—
34H(3,ℤ/4)×C127C2218yescomplemented
35C2×#41×C127—22no—
36C2×#41×C127—22no—
37C2×#41×C127—21no—
38C2×#41×C127—21no—
39C2×#41×C127—22no—
40C2×#41×C127—22no—
41C2×#41×C127—22no—
42C2×#41×C127—22no—
43C2×#41×C127—22no—
44C2×#41×C127—22no—
45C2×#41×C127C2×C412yes—
46C2×#41×C127C2×C412yes—
47C2×#41×C127C2×C412yes—
48C2×#41×C127D412yes—
49C2×#41×C127D412yes—
50C2×#41×C127C2312yes—
51C22×Q8×C127C2×C411yescharacteristic
52C22×Q8×C127C2311yescharacteristic
53C22×D4×C127—21no—
54C22×D4×C127C2×C411yescharacteristic
55C22×D4×C127D412yes—
56C22×D4×C127C2311yescharacteristic
57C23×C4×C127—21no—
58C23×C4×C127D412yes—
59C23×C4×C127C2311yescharacteristic
60#131×C127—216no—
61#131×C127—216no—
62C2×M16×C127—24no—
63C2×M16×C127—28no—
64C2×M16×C127—22no—
65C2×M16×C127—28no—
66C2×M16×C127—22no—
67C2×M16×C127—22no—
68C2×M16×C127—22no—
69C2×M16×C127D414yescomplemented
70C2×M16×C127C2314yes—
71C22×C8×C127—24no—
72C22×C8×C127D414yescomplemented
73C4⋊Q8×C127C2×C414yescomplemented
74#127×C127C2×C414yescomplemented
75#124×C127C2×C414yescomplemented
76#124×C127C2×C414yescomplemented
77#123×C127—24no—
78#123×C127—24no—
79C22⋊Q8×C127—24no—
80C22⋊Q8×C127—24no—
81#121×C127—24no—
82#121×C127—28no—
83#121×C127—24no—
84#120×C127—24no—
85#120×C127—24no—
86C4×D4×C127—28no—
87#117×C127—21no—
88#117×C127—21no—
89#117×C127—22no—
90#117×C127—22no—
91#117×C127—22no—
92#117×C127C2×C412yes—
93#117×C127C2314yes—
94#117×C127C2312yes—
95C2×C4⋊C4×C127—21no—
96C2×C4⋊C4×C127—22no—
97C2×C22⋊C4×C127—21no—
98C2×C22⋊C4×C127—22no—
99C2×C42×C127—22no—
100C2×C42×C127C2×C412yes—
101#103×C127—28no—
102#102×C127—28no—
103#99×C127—28no—
104C24×C127—21no—
105C24×C127C22⋊C412yes—
106C24×C127C2×D411yescharacteristic
107#41×C127—24no—
108#41×C127—24no—
109#41×C127—24no—
110#41×C127—24no—
111#41×C127—24no—
112#41×C127—24no—
113#41×C127—24no—
114#41×C127—24no—
115#41×C127—24no—
116#41×C127—24no—
117#41×C127—24no—
118#41×C127—24no—
119#41×C127—42no—
120#41×C127—41no—
121#41×C127—41no—
122#41×C127—41no—
123#41×C127—42no—
124#41×C127—41no—
125C2×Q8×C127—22no—
126C2×Q8×C127—22no—
127C2×Q8×C127—21no—
128C2×Q8×C127—22no—
129C2×Q8×C127—22no—
130C2×Q8×C127C22⋊C412yes—
131C2×Q8×C127C22×C412yes—
132C2×Q8×C127C22×C412yes—
133C2×D4×C127—24no—
134C2×D4×C127—24no—
135C2×D4×C127—22no—
136C2×D4×C127—22no—
137C2×D4×C127—28no—
138C2×D4×C127—22no—
139C2×D4×C127—22no—
140C2×D4×C127—22no—
141C2×D4×C127—21no—
142C2×D4×C127—22no—
143C2×D4×C127—42no—
144C2×D4×C127—42no—
145C2×D4×C127C22⋊C412yes—
146C2×D4×C127C22⋊C414yes—
147C2×D4×C127C22×C412yes—
148C2×D4×C127C22×C412yes—
149C2×D4×C127C2×D414yes—
150C22×C4×C127—22no—
151C22×C4×C127—22no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127—21no—
155C22×C4×C127—22no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127—42no—
159C22×C4×C127—42no—
160C22×C4×C127—42no—
161C22×C4×C127C22⋊C414yes—
162C22×C4×C127C22⋊C412yes—
163C22×C4×C127C22×C411yescharacteristic
164C22×C4×C127C22×C411yescharacteristic
165C22×C4×C127C2×D414yes—
166C22×C4×C127C2×D411yescharacteristic
167C22×C4×C127C2×D411yescharacteristic
168C22×C4×C127C2×D411yescharacteristic
169C22×C4×C127C2×D412yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127#4112yes—
172C22×C4×C127#4112yes—
173C22×C4×C127C2411yescharacteristic
174M16×C127—216no—
175M16×C127—24no—
176M16×C127—24no—
177M16×C127—24no—
178M16×C127—24no—
179C2×C8×C127—216no—
180C2×C8×C127—24no—
181C2×C8×C127—24no—
182C2×C8×C127C2×D418yes—
183C4⋊C4×C127—22no—
184C4⋊C4×C127—28no—
185C4⋊C4×C127—22no—
186C4⋊C4×C127—24no—
187C4⋊C4×C127—42no—
188C22⋊C4×C127—22no—
189C22⋊C4×C127—28no—
190C22⋊C4×C127—22no—
191C22⋊C4×C127—28no—
192C22⋊C4×C127—24no—
193C22⋊C4×C127—42no—
194C42×C127—22no—
195C42×C127—24no—
196C42×C127—42no—
197C42×C127C22×C414yes—
198C23×C127—22no—
199C23×C127—22no—
200C23×C127—22no—
201C23×C127—42no—
202C23×C127—81no—
203C23×C127—82no—
204C23×C127—82no—
205C23×C127C2×C22⋊C414yes—
206C23×C127C2×C22⋊C411yescharacteristic
207C23×C127C2×C22⋊C411yescharacteristic
208C23×C127#12112yes—
209C23×C127#12112yes—
210C23×C127C22×D411yescharacteristic
211Q8×C127—24no—
212Q8×C127—24no—
213Q8×C127—24no—
214Q8×C127—41no—
215Q8×C127—41no—
216D4×C127—28no—
217D4×C127—28no—
218D4×C127—24no—
219D4×C127—24no—
220D4×C127—24no—
221D4×C127—42no—
222D4×C127—42no—
223D4×C127—44no—
224D4×C127—41no—
225D4×C127—41no—
226D4×C127—88no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—24no—
230C2×C4×C127—24no—
231C2×C4×C127—24no—
232C2×C4×C127—24no—
233C2×C4×C127—24no—
234C2×C4×C127—22no—
235C2×C4×C127—42no—
236C2×C4×C127—42no—
237C2×C4×C127—44no—
238C2×C4×C127—42no—
239C2×C4×C127—44no—
240C2×C4×C127—82no—
241C2×C4×C127—84no—
242C2×C4×C127C2×C22⋊C414yes—
243C2×C4×C127C2×C22⋊C411yescharacteristic
244C2×C4×C127C2×C22⋊C411yescharacteristic
245C2×C4×C127C4×D412yes—
246C2×C4×C127C4×D412yes—
247C2×C4×C127C4×D412yes—
248C2×C4×C127C4×D412yes—
249C2×C4×C127#12012yes—
250C2×C4×C127#12011yescharacteristic
251C2×C4×C127#12011yescharacteristic
252C2×C4×C127#12112yes—
253C2×C4×C127#12112yes—
254C2×C4×C127#12311yescharacteristic
255C2×C4×C127#12312yes—
256C2×C4×C127#12311yescharacteristic
257C2×C4×C127C23×C411yescharacteristic
258C2×C4×C127C22×D412yes—
259C2×C4×C127C22×D411yescharacteristic
260C2×C4×C127C2×#4112yes—
261C8×C127—28no—
262C22×C127—24no—
263C22×C127—41no—
264C22×C127—42no—
265C22×C127—41no—
266C22×C127—41no—
267C22×C127—42no—
268C22×C127—44no—
269C22×C127—44no—
270C22×C127—44no—
271C22×C127—42no—
272C22×C127—82no—
273C22×C127—84no—
274C22×C127#38611yescharacteristic
275C22×C127#38611yescharacteristic
276C22×C127#38612yes—
277C22×C127C22×C22⋊C411yescharacteristic
278C22×C127C2×#12112yes—
279C4×C127—24no—
280C4×C127—44no—
281C4×C127—42no—
282C4×C127—84no—
283C4×C127#38611yescharacteristic
284C4×C127#38612yes—
285C4×C127#38611yescharacteristic
286C4×C127C2×C4×D412yes—
287C4×C127C2×#12011yescharacteristic
288C4×C127C2×#12311yescharacteristic
289#21009C12711yesmaximal, characteristic, direct product factor, Sylow
290C2×C127—21no—
291C2×C127—22no—
292C2×C127—21no—
293C2×C127—21no—
294C2×C127—22no—
295C2×C127—44no—
296C2×C127—82no—
297C2×C127C2×#38611yescharacteristic
298C2×#568C2×C12711yescharacteristic, complemented
299#3449C2×C12711yescharacteristic
300#2896C2×C12712yescomplemented
301C2×#410C2×C12711yescharacteristic, complemented
302#1910C2×C12714yescomplemented
303#1909C2×C12714yescomplemented
304C2×H(3,ℤ/4)C2×C12712yescomplemented
305C127#2100911yescharacteristic, direct product factor, Sylow
306C22×#41C22×C12711yescharacteristic
307C22×#41C22×C12711yescharacteristic
308#568—22no—
309#568—24no—
310#568C22×C12714yes—
311C2×#131—24no—
312C22×M16—22no—
313C22×M16C22×C12712yes—
314#537—22no—
315#536C4×C12712yescomplemented
316#536C22×C12712yes—
317#535C4×C12712yescomplemented
318#535C22×C12712yes—
319#534—22no—
320#532C4×C12714yescomplemented
321#525—22no—
322#525—22no—
323C2×#117C22×C12712yes—
324C2×#117C22×C12711yescharacteristic
325#419C22×C12714yescomplemented
326#418C22×C12714yescomplemented
327#417—24no—
328#416—24no—
329#410—22no—
330#410C22×C12714yes—
331C2×#99—22no—
332H(3,ℤ/4)C22×C12718yescomplemented
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41C2×C4×C12712yes—
344C2×#41C2×C4×C12712yes—
345C2×#41C2×C4×C12712yes—
346C2×#41D4×C12712yes—
347C2×#41D4×C12712yes—
348C2×#41C23×C12712yes—
349C22×Q8C2×C4×C12711yescharacteristic
350C22×Q8C23×C12711yescharacteristic
351C22×D4—21no—
352C22×D4C2×C4×C12711yescharacteristic
353C22×D4D4×C12712yes—
354C22×D4C23×C12711yescharacteristic
355C23×C4—21no—
356C23×C4D4×C12712yes—
357C23×C4C23×C12711yescharacteristic
358#131—216no—
359#131—216no—
360C2×M16—24no—
361C2×M16—28no—
362C2×M16—22no—
363C2×M16—28no—
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—22no—
367C2×M16D4×C12714yescomplemented
368C2×M16C23×C12714yes—
369C22×C8—24no—
370C22×C8D4×C12714yescomplemented
371C4⋊Q8C2×C4×C12714yescomplemented
372#127C2×C4×C12714yescomplemented
373#124C2×C4×C12714yescomplemented
374#124C2×C4×C12714yescomplemented
375#123—24no—
376#123—24no—
377C22⋊Q8—24no—
378C22⋊Q8—24no—
379#121—24no—
380#121—28no—
381#121—24no—
382#120—24no—
383#120—24no—
384C4×D4—28no—
385#117—21no—
386#117—21no—
387#117—22no—
388#117—22no—
389#117—22no—
390#117C2×C4×C12712yes—
391#117C23×C12714yes—
392#117C23×C12712yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—22no—
397C2×C42—22no—
398C2×C42C2×C4×C12712yes—
399#103—28no—
400#102—28no—
401#99—28no—
402C24—21no—
403C24C22⋊C4×C12712yes—
404C24C2×D4×C12711yescharacteristic
405#41—24no—
406#41—24no—
407#41—24no—
408#41—24no—
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—41no—
418#41—42no—
419#41—41no—
420#41—41no—
421#41—41no—
422#41—42no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8C22⋊C4×C12712yes—
429C2×Q8C22×C4×C12712yes—
430C2×Q8C22×C4×C12712yes—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—28no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C22⋊C4×C12712yes—
444C2×D4C22⋊C4×C12714yes—
445C2×D4C22×C4×C12712yes—
446C2×D4C22×C4×C12712yes—
447C2×D4C2×D4×C12714yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4C22⋊C4×C12714yes—
460C22×C4C22⋊C4×C12712yes—
461C22×C4C22×C4×C12711yescharacteristic
462C22×C4C22×C4×C12711yescharacteristic
463C22×C4C2×D4×C12714yes—
464C22×C4C2×D4×C12711yescharacteristic
465C22×C4C2×D4×C12711yescharacteristic
466C22×C4C2×D4×C12711yescharacteristic
467C22×C4C2×D4×C12712yes—
468C22×C4C2×D4×C12712yes—
469C22×C4#41×C12712yes—
470C22×C4#41×C12712yes—
471C22×C4C24×C12711yescharacteristic
472M16—216no—
473M16—24no—
474M16—24no—
475M16—24no—
476M16—24no—
477C2×C8—216no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8C2×D4×C12718yes—
481C4⋊C4—22no—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—42no—
486C22⋊C4—22no—
487C22⋊C4—28no—
488C22⋊C4—22no—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C42—22no—
493C42—24no—
494C42—42no—
495C42C22×C4×C12714yes—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—81no—
501C23—82no—
502C23—82no—
503C23C2×C22⋊C4×C12714yes—
504C23C2×C22⋊C4×C12711yescharacteristic
505C23C2×C22⋊C4×C12711yescharacteristic
506C23#121×C12712yes—
507C23#121×C12712yes—
508C23C22×D4×C12711yescharacteristic
509Q8—24no—
510Q8—24no—
511Q8—24no—
512Q8—41no—
513Q8—41no—
514D4—28no—
515D4—28no—
516D4—24no—
517D4—24no—
518D4—24no—
519D4—42no—
520D4—42no—
521D4—44no—
522D4—41no—
523D4—41no—
524D4—88no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4C2×C22⋊C4×C12714yes—
541C2×C4C2×C22⋊C4×C12711yescharacteristic
542C2×C4C2×C22⋊C4×C12711yescharacteristic
543C2×C4C4×D4×C12712yes—
544C2×C4C4×D4×C12712yes—
545C2×C4C4×D4×C12712yes—
546C2×C4C4×D4×C12712yes—
547C2×C4#120×C12711yescharacteristic
548C2×C4#120×C12712yes—
549C2×C4#120×C12711yescharacteristic
550C2×C4#121×C12712yes—
551C2×C4#121×C12712yes—
552C2×C4#123×C12711yescharacteristic
553C2×C4#123×C12711yescharacteristic
554C2×C4#123×C12712yes—
555C2×C4C23×C4×C12711yescharacteristic
556C2×C4C22×D4×C12712yes—
557C2×C4C22×D4×C12711yescharacteristic
558C2×C4C2×#41×C12712yes—
559C8—28no—
560C22—24no—
561C22—41no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—82no—
571C22—84no—
572C22#386×C12711yescharacteristic
573C22#386×C12711yescharacteristic
574C22#386×C12712yes—
575C22C22×C22⋊C4×C12711yescharacteristic
576C22C2×#121×C12712yes—
577C4—24no—
578C4—44no—
579C4—42no—
580C4—84no—
581C4#386×C12711yescharacteristic
582C4#386×C12711yescharacteristic
583C4#386×C12712yes—
584C4C2×C4×D4×C12712yes—
585C4C2×#120×C12711yescharacteristic
586C4C2×#123×C12711yescharacteristic
587C2—21no—
588C2—22no—
589C2—21no—
590C2—21no—
591C2—22no—
592C2—44no—
593C2—82no—
594C2C2×#386×C12711yescharacteristic
subgroupG/H
centerC4×C127#386
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC2×C127C2×#386
prime radicalC127×#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#568×C127C2semidirect
3. C127×#2896C2semidirect
4. C2×#410×C127C2semidirect
5. C127×#1910C2semidirect
6. C127×#1909C2semidirect
7. C2×H(3,ℤ/4)×C127C2semidirect
17. #536×C127C4semidirect
19. #535×C127C4semidirect
22. #532×C127C4semidirect
27. #419×C127C22semidirect
28. #418×C127C22semidirect
34. H(3,ℤ/4)×C127C22semidirect
69. C2×M16×C127D4semidirect
72. C22×C8×C127D4semidirect
73. C4⋊Q8×C127C2×C4semidirect
74. #127×C127C2×C4semidirect
75. #124×C127C2×C4semidirect
76. #124×C127C2×C4semidirect
289. #21009C127direct (N × H)
298. C2×#568C2×C127semidirect
300. #2896C2×C127semidirect
301. C2×#410C2×C127semidirect
302. #1910C2×C127semidirect
303. #1909C2×C127semidirect
304. C2×H(3,ℤ/4)C2×C127semidirect
305. C127#21009direct (N × H)
315. #536C4×C127semidirect
317. #535C4×C127semidirect
320. #532C4×C127semidirect
325. #419C22×C127semidirect
326. #418C22×C127semidirect
332. H(3,ℤ/4)C22×C127semidirect
367. C2×M16D4×C127semidirect
370. C22×C8D4×C127semidirect
371. C4⋊Q8C2×C4×C127semidirect
372. #127C2×C4×C127semidirect
373. #124C2×C4×C127semidirect
374. #124C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#568×C1272
C127×#34492
C127×#28962
C2×#410×C1272
C127×#19102
C127×#19092
C2×H(3,ℤ/4)×C1272
#21009127