Finite Group Databasebeta

← C127×#21012 #44584685

Subgroups — C127×#21012 #44584685

Order 32512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3449C211yesmaximal, characteristic
2C127×#3428C211yesmaximal, characteristic, complemented
3C2×#421×C127C212yesmaximal, complemented
4C2×#413×C127C211yesmaximal, characteristic, complemented
5C127×#1912C214yesmaximal, complemented
6C127×#1911C214yesmaximal, complemented
7C2×H(3,ℤ/4)×C127C212yesmaximal, complemented
8C22×#41×C127C2211yescharacteristic
9C22×#41×C127C2211yescharacteristic
10C22×M16×C127C2212yes—
11#537×C127—22no—
12#536×C127C412yescomplemented
13#536×C127C2212yes—
14#535×C127C412yescomplemented
15#535×C127C2212yes—
16#534×C127—22no—
17#532×C127C414yescomplemented
18#525×C127—22no—
19#525×C127—22no—
20#520×C127—24no—
21#519×C127C2212yescomplemented
22#518×C127—22no—
23#518×C127C2212yescomplemented
24C4×#41×C127—24no—
25C2×#117×C127—22no—
26C2×#117×C127C2211yescharacteristic
27C2×#117×C127C2212yes—
28#421×C127—24no—
29#421×C127C2218yes—
30C2×#104×C127—24no—
31#413×C127—22no—
32#413×C127—22no—
33C2×#101×C127C2212yescomplemented
34C2×#100×C127C2212yescomplemented
35H(3,ℤ/4)×C127C2218yescomplemented
36C2×#41×C127—22no—
37C2×#41×C127—22no—
38C2×#41×C127—21no—
39C2×#41×C127—21no—
40C2×#41×C127—22no—
41C2×#41×C127—22no—
42C2×#41×C127—22no—
43C2×#41×C127—22no—
44C2×#41×C127—22no—
45C2×#41×C127—22no—
46C2×#41×C127C2×C412yes—
47C2×#41×C127C2×C412yes—
48C2×#41×C127C2×C412yescomplemented
49C2×#41×C127D412yes—
50C2×#41×C127D412yes—
51C2×#41×C127C2312yes—
52C22×Q8×C127C2×C411yescharacteristic, complemented
53C22×Q8×C127C2311yescharacteristic
54C22×D4×C127—21no—
55C22×D4×C127C2×C411yescharacteristic, complemented
56C22×D4×C127D412yes—
57C22×D4×C127C2311yescharacteristic
58C23×C4×C127—21no—
59C23×C4×C127D412yes—
60C23×C4×C127C2311yescharacteristic
61C2×M16×C127—22no—
62C2×M16×C127—22no—
63C2×M16×C127—22no—
64C2×M16×C127—22no—
65C2×M16×C127—22no—
66C2×M16×C127C2314yes—
67C22×C8×C127—22no—
68C4⋊Q8×C127C2×C414yescomplemented
69#127×C127C2×C414yescomplemented
70#124×C127C2×C414yescomplemented
71#124×C127C2×C414yescomplemented
72#123×C127—24no—
73#123×C127—24no—
74C22⋊Q8×C127—24no—
75C22⋊Q8×C127—24no—
76#121×C127—28no—
77#121×C127—24no—
78#121×C127—24no—
79#120×C127—24no—
80#120×C127—24no—
81C4×Q8×C127—28no—
82C4×D4×C127—216no—
83C4×D4×C127—28no—
84C4×D4×C127—28no—
85#117×C127—24no—
86#117×C127—24no—
87#117×C127—22no—
88#117×C127—24no—
89#117×C127—22no—
90#117×C127—21no—
91#117×C127—21no—
92#117×C127C2×C412yescomplemented
93#117×C127D414yescomplemented
94#117×C127C2312yes—
95#117×C127C2314yes—
96C2×C4⋊C4×C127—24no—
97C2×C4⋊C4×C127—22no—
98C2×C4⋊C4×C127—21no—
99C2×C22⋊C4×C127—24no—
100C2×C22⋊C4×C127—22no—
101C2×C22⋊C4×C127—21no—
102C2×C42×C127—24no—
103C2×C42×C127C2×C412yescomplemented
104C2×C42×C127D414yescomplemented
105#104×C127—216no—
106#101×C127—24no—
107#100×C127—24no—
108C24×C127—21no—
109C24×C127C2×D412yes—
110C24×C127C2×D411yescharacteristic
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—24no—
120#41×C127—24no—
121#41×C127—24no—
122#41×C127—24no—
123#41×C127—41no—
124#41×C127—41no—
125#41×C127—42no—
126#41×C127—42no—
127#41×C127—41no—
128#41×C127—41no—
129C2×Q8×C127—21no—
130C2×Q8×C127—22no—
131C2×Q8×C127—22no—
132C2×Q8×C127—22no—
133C2×Q8×C127—22no—
134C2×Q8×C127C22×C412yes—
135C2×Q8×C127C22×C412yes—
136C2×Q8×C127C2×D412yes—
137C2×D4×C127—24no—
138C2×D4×C127—24no—
139C2×D4×C127—22no—
140C2×D4×C127—22no—
141C2×D4×C127—28no—
142C2×D4×C127—21no—
143C2×D4×C127—22no—
144C2×D4×C127—22no—
145C2×D4×C127—22no—
146C2×D4×C127—22no—
147C2×D4×C127—42no—
148C2×D4×C127—42no—
149C2×D4×C127C22⋊C414yes—
150C2×D4×C127C22⋊C414yes—
151C2×D4×C127C22×C412yes—
152C2×D4×C127C22×C412yes—
153C2×D4×C127C2×D412yes—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127—21no—
159C22×C4×C127—28no—
160C22×C4×C127—22no—
161C22×C4×C127—24no—
162C22×C4×C127—22no—
163C22×C4×C127—22no—
164C22×C4×C127—42no—
165C22×C4×C127—42no—
166C22×C4×C127C22⋊C414yes—
167C22×C4×C127C22⋊C414yes—
168C22×C4×C127C22×C411yescharacteristic
169C22×C4×C127C22×C411yescharacteristic
170C22×C4×C127C2×D412yes—
171C22×C4×C127C2×D411yescharacteristic
172C22×C4×C127C2×D411yescharacteristic
173C22×C4×C127C2×D411yescharacteristic
174C22×C4×C127C2×D412yes—
175C22×C4×C127C2×D412yes—
176C22×C4×C127#4112yes—
177C22×C4×C127#4112yes—
178C22×C4×C127C2411yescharacteristic
179M16×C127—24no—
180M16×C127—24no—
181M16×C127—42no—
182M16×C127—42no—
183C2×C8×C127—24no—
184C2×C8×C127—42no—
185C2×C8×C127—42no—
186C4⋊C4×C127—28no—
187C4⋊C4×C127—22no—
188C4⋊C4×C127—28no—
189C4⋊C4×C127—24no—
190C4⋊C4×C127—24no—
191C4⋊C4×C127—22no—
192C22⋊C4×C127—28no—
193C22⋊C4×C127—22no—
194C22⋊C4×C127—28no—
195C22⋊C4×C127—24no—
196C22⋊C4×C127—24no—
197C22⋊C4×C127—22no—
198C22⋊C4×C127—28no—
199C42×C127—28no—
200C42×C127—22no—
201C42×C127—24no—
202C42×C127C22×C414yes—
203C42×C127C2×D418yes—
204C23×C127—22no—
205C23×C127—22no—
206C23×C127—22no—
207C23×C127—42no—
208C23×C127—81no—
209C23×C127—82no—
210C23×C127—82no—
211C23×C127C2×C22⋊C414yes—
212C23×C127C2×C22⋊C411yescharacteristic
213C23×C127C2×C22⋊C411yescharacteristic
214C23×C127#12012yes—
215C23×C127#12312yes—
216C23×C127C22×D411yescharacteristic
217Q8×C127—24no—
218Q8×C127—24no—
219Q8×C127—24no—
220Q8×C127—41no—
221Q8×C127—41no—
222D4×C127—28no—
223D4×C127—28no—
224D4×C127—24no—
225D4×C127—24no—
226D4×C127—24no—
227D4×C127—42no—
228D4×C127—42no—
229D4×C127—44no—
230D4×C127—41no—
231D4×C127—41no—
232D4×C127—88no—
233C2×C4×C127—22no—
234C2×C4×C127—22no—
235C2×C4×C127—24no—
236C2×C4×C127—24no—
237C2×C4×C127—24no—
238C2×C4×C127—24no—
239C2×C4×C127—24no—
240C2×C4×C127—22no—
241C2×C4×C127—28no—
242C2×C4×C127—42no—
243C2×C4×C127—44no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—84no—
247C2×C4×C127—88no—
248C2×C4×C127—82no—
249C2×C4×C127C2×C22⋊C414yes—
250C2×C4×C127C2×C22⋊C411yescharacteristic
251C2×C4×C127C2×C22⋊C411yescharacteristic
252C2×C4×C127C4×D412yes—
253C2×C4×C127C4×D412yes—
254C2×C4×C127C4×D412yes—
255C2×C4×C127C4×D412yes—
256C2×C4×C127#12011yescharacteristic
257C2×C4×C127#12011yescharacteristic
258C2×C4×C127#12112yes—
259C2×C4×C127#12112yes—
260C2×C4×C127#12112yes—
261C2×C4×C127#12112yes—
262C2×C4×C127#12311yescharacteristic
263C2×C4×C127#12311yescharacteristic
264C2×C4×C127C23×C411yescharacteristic
265C2×C4×C127C22×D412yes—
266C2×C4×C127C22×D411yescharacteristic
267C2×C4×C127C2×#4112yes—
268C8×C127—44no—
269C22×C127—24no—
270C22×C127—41no—
271C22×C127—42no—
272C22×C127—41no—
273C22×C127—41no—
274C22×C127—42no—
275C22×C127—44no—
276C22×C127—44no—
277C22×C127—44no—
278C22×C127—42no—
279C22×C127—82no—
280C22×C127—84no—
281C22×C127#38612yes—
282C22×C127#38612yes—
283C22×C127C22×C22⋊C411yescharacteristic
284C22×C127C2×#12011yescharacteristic
285C22×C127C2×#12311yescharacteristic
286C4×C127—24no—
287C4×C127—48no—
288C4×C127—44no—
289C4×C127—42no—
290C4×C127#38611yescharacteristic
291C4×C127#38611yescharacteristic
292C4×C127#38611yescharacteristic
293C4×C127#38611yescharacteristic
294C4×C127C2×C4×D412yes—
295C4×C127C2×#12112yes—
296#21012C12711yesmaximal, characteristic, direct product factor, Sylow
297C2×C127—21no—
298C2×C127—21no—
299C2×C127—21no—
300C2×C127—22no—
301C2×C127—22no—
302C2×C127—44no—
303C2×C127—82no—
304C2×C127C2×#38611yescharacteristic
305#3449C2×C12711yescharacteristic
306#3428C2×C12711yescharacteristic, complemented
307C2×#421C2×C12712yescomplemented
308C2×#413C2×C12711yescharacteristic, complemented
309#1912C2×C12714yescomplemented
310#1911C2×C12714yescomplemented
311C2×H(3,ℤ/4)C2×C12712yescomplemented
312C127#2101211yescharacteristic, direct product factor, Sylow
313C22×#41C22×C12711yescharacteristic
314C22×#41C22×C12711yescharacteristic
315C22×M16C22×C12712yes—
316#537—22no—
317#536C4×C12712yescomplemented
318#536C22×C12712yes—
319#535C4×C12712yescomplemented
320#535C22×C12712yes—
321#534—22no—
322#532C4×C12714yescomplemented
323#525—22no—
324#525—22no—
325#520—24no—
326#519C22×C12712yescomplemented
327#518—22no—
328#518C22×C12712yescomplemented
329C4×#41—24no—
330C2×#117—22no—
331C2×#117C22×C12711yescharacteristic
332C2×#117C22×C12712yes—
333#421—24no—
334#421C22×C12718yes—
335C2×#104—24no—
336#413—22no—
337#413—22no—
338C2×#101C22×C12712yescomplemented
339C2×#100C22×C12712yescomplemented
340H(3,ℤ/4)C22×C12718yescomplemented
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41C2×C4×C12712yes—
352C2×#41C2×C4×C12712yes—
353C2×#41C2×C4×C12712yescomplemented
354C2×#41D4×C12712yes—
355C2×#41D4×C12712yes—
356C2×#41C23×C12712yes—
357C22×Q8C2×C4×C12711yescharacteristic, complemented
358C22×Q8C23×C12711yescharacteristic
359C22×D4—21no—
360C22×D4C2×C4×C12711yescharacteristic, complemented
361C22×D4D4×C12712yes—
362C22×D4C23×C12711yescharacteristic
363C23×C4—21no—
364C23×C4D4×C12712yes—
365C23×C4C23×C12711yescharacteristic
366C2×M16—22no—
367C2×M16—22no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—22no—
371C2×M16C23×C12714yes—
372C22×C8—22no—
373C4⋊Q8C2×C4×C12714yescomplemented
374#127C2×C4×C12714yescomplemented
375#124C2×C4×C12714yescomplemented
376#124C2×C4×C12714yescomplemented
377#123—24no—
378#123—24no—
379C22⋊Q8—24no—
380C22⋊Q8—24no—
381#121—28no—
382#121—24no—
383#121—24no—
384#120—24no—
385#120—24no—
386C4×Q8—28no—
387C4×D4—216no—
388C4×D4—28no—
389C4×D4—28no—
390#117—24no—
391#117—24no—
392#117—22no—
393#117—24no—
394#117—22no—
395#117—21no—
396#117—21no—
397#117C2×C4×C12712yescomplemented
398#117D4×C12714yescomplemented
399#117C23×C12712yes—
400#117C23×C12714yes—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—21no—
407C2×C42—24no—
408C2×C42C2×C4×C12712yescomplemented
409C2×C42D4×C12714yescomplemented
410#104—216no—
411#101—24no—
412#100—24no—
413C24—21no—
414C24C2×D4×C12712yes—
415C24C2×D4×C12711yescharacteristic
416#41—24no—
417#41—24no—
418#41—24no—
419#41—24no—
420#41—24no—
421#41—24no—
422#41—24no—
423#41—24no—
424#41—24no—
425#41—24no—
426#41—24no—
427#41—24no—
428#41—41no—
429#41—41no—
430#41—42no—
431#41—42no—
432#41—41no—
433#41—41no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C22×C4×C12712yes—
440C2×Q8C22×C4×C12712yes—
441C2×Q8C2×D4×C12712yes—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—28no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C22⋊C4×C12714yes—
455C2×D4C22⋊C4×C12714yes—
456C2×D4C22×C4×C12712yes—
457C2×D4C22×C4×C12712yes—
458C2×D4C2×D4×C12712yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—28no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4C22⋊C4×C12714yes—
472C22×C4C22⋊C4×C12714yes—
473C22×C4C22×C4×C12711yescharacteristic
474C22×C4C22×C4×C12711yescharacteristic
475C22×C4C2×D4×C12712yes—
476C22×C4C2×D4×C12711yescharacteristic
477C22×C4C2×D4×C12711yescharacteristic
478C22×C4C2×D4×C12711yescharacteristic
479C22×C4C2×D4×C12712yes—
480C22×C4C2×D4×C12712yes—
481C22×C4#41×C12712yes—
482C22×C4#41×C12712yes—
483C22×C4C24×C12711yescharacteristic
484M16—24no—
485M16—24no—
486M16—42no—
487M16—42no—
488C2×C8—24no—
489C2×C8—42no—
490C2×C8—42no—
491C4⋊C4—28no—
492C4⋊C4—22no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C22⋊C4—28no—
498C22⋊C4—22no—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—28no—
504C42—28no—
505C42—22no—
506C42—24no—
507C42C22×C4×C12714yes—
508C42C2×D4×C12718yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—42no—
513C23—81no—
514C23—82no—
515C23—82no—
516C23C2×C22⋊C4×C12714yes—
517C23C2×C22⋊C4×C12711yescharacteristic
518C23C2×C22⋊C4×C12711yescharacteristic
519C23#120×C12712yes—
520C23#123×C12712yes—
521C23C22×D4×C12711yescharacteristic
522Q8—24no—
523Q8—24no—
524Q8—24no—
525Q8—41no—
526Q8—41no—
527D4—28no—
528D4—28no—
529D4—24no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—44no—
535D4—41no—
536D4—41no—
537D4—88no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—88no—
553C2×C4—82no—
554C2×C4C2×C22⋊C4×C12714yes—
555C2×C4C2×C22⋊C4×C12711yescharacteristic
556C2×C4C2×C22⋊C4×C12711yescharacteristic
557C2×C4C4×D4×C12712yes—
558C2×C4C4×D4×C12712yes—
559C2×C4C4×D4×C12712yes—
560C2×C4C4×D4×C12712yes—
561C2×C4#120×C12711yescharacteristic
562C2×C4#120×C12711yescharacteristic
563C2×C4#121×C12712yes—
564C2×C4#121×C12712yes—
565C2×C4#121×C12712yes—
566C2×C4#121×C12712yes—
567C2×C4#123×C12711yescharacteristic
568C2×C4#123×C12711yescharacteristic
569C2×C4C23×C4×C12711yescharacteristic
570C2×C4C22×D4×C12712yes—
571C2×C4C22×D4×C12711yescharacteristic
572C2×C4C2×#41×C12712yes—
573C8—44no—
574C22—24no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—82no—
585C22—84no—
586C22#386×C12712yes—
587C22#386×C12712yes—
588C22C22×C22⋊C4×C12711yescharacteristic
589C22C2×#120×C12711yescharacteristic
590C22C2×#123×C12711yescharacteristic
591C4—24no—
592C4—48no—
593C4—44no—
594C4—42no—
595C4#386×C12711yescharacteristic
596C4#386×C12711yescharacteristic
597C4#386×C12711yescharacteristic
598C4#386×C12711yescharacteristic
599C4C2×C4×D4×C12712yes—
600C4C2×#121×C12712yes—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—22no—
605C2—22no—
606C2—44no—
607C2—82no—
608C2C2×#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
2. C127×#3428C2semidirect
3. C2×#421×C127C2semidirect
4. C2×#413×C127C2semidirect
5. C127×#1912C2semidirect
6. C127×#1911C2semidirect
7. C2×H(3,ℤ/4)×C127C2semidirect
12. #536×C127C4semidirect
14. #535×C127C4semidirect
17. #532×C127C4semidirect
21. #519×C127C22semidirect
23. #518×C127C22semidirect
33. C2×#101×C127C22semidirect
34. C2×#100×C127C22semidirect
35. H(3,ℤ/4)×C127C22semidirect
48. C2×#41×C127C2×C4semidirect
52. C22×Q8×C127C2×C4semidirect
55. C22×D4×C127C2×C4semidirect
68. C4⋊Q8×C127C2×C4semidirect
69. #127×C127C2×C4semidirect
70. #124×C127C2×C4semidirect
71. #124×C127C2×C4semidirect
92. #117×C127C2×C4semidirect
93. #117×C127D4semidirect
103. C2×C42×C127C2×C4semidirect
104. C2×C42×C127D4semidirect
296. #21012C127direct (N × H)
306. #3428C2×C127semidirect
307. C2×#421C2×C127semidirect
308. C2×#413C2×C127semidirect
309. #1912C2×C127semidirect
310. #1911C2×C127semidirect
311. C2×H(3,ℤ/4)C2×C127semidirect
312. C127#21012direct (N × H)
317. #536C4×C127semidirect
319. #535C4×C127semidirect
322. #532C4×C127semidirect
326. #519C22×C127semidirect
328. #518C22×C127semidirect
338. C2×#101C22×C127semidirect
339. C2×#100C22×C127semidirect
340. H(3,ℤ/4)C22×C127semidirect
353. C2×#41C2×C4×C127semidirect
357. C22×Q8C2×C4×C127semidirect
360. C22×D4C2×C4×C127semidirect
373. C4⋊Q8C2×C4×C127semidirect
374. #127C2×C4×C127semidirect
375. #124C2×C4×C127semidirect
376. #124C2×C4×C127semidirect
397. #117C2×C4×C127semidirect
398. #117D4×C127semidirect
408. C2×C42C2×C4×C127semidirect
409. C2×C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34492
C127×#34282
C2×#421×C1272
C2×#413×C1272
C127×#19122
C127×#19112
C2×H(3,ℤ/4)×C1272
#21012127