Finite Group Databasebeta

← C127×#21474 #44585147

Subgroups — C127×#21474 #44585147

Order 32512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×H(5,𝔽2)×C127C211yesmaximal, characteristic
2C2×#455×C127C211yesmaximal, characteristic
3C4×#137×C127C212yesmaximal, complemented
4C127×#2326C211yesmaximal, characteristic, complemented
5C127×#1885C211yesmaximal, characteristic, complemented
6C127×#1882C212yesmaximal, complemented
7C127×#1859C212yesmaximal, complemented
8C127×#1856C211yesmaximal, characteristic, complemented
9C127×#1792C211yesmaximal, characteristic
10C127×#1759C212yesmaximal, complemented
11C4×#100×C127C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C127C2211yescharacteristic
13C2×#137×C127C2212yes—
14C2×#124×C127C2211yescharacteristic
15#520×C127—22no—
16#520×C127C2212yes—
17#518×C127—22no—
18#518×C127—22no—
19#518×C127C2211yescharacteristic
20C4×#41×C127—22no—
21C4×#41×C127C2212yes—
22C2×C4×Q8×C127C2211yescharacteristic, complemented
23C2×C4×D4×C127—22no—
24C2×C4×D4×C127—22no—
25C2×C4×D4×C127C2211yescharacteristic
26#455×C127C418yescomplemented
27#441×C127—22no—
28#440×C127—22no—
29C4×Q16×C127—22no—
30C4×SD16×C127—22no—
31#428×C127C2212yescomplemented
32C2×#104×C127C2212yes—
33C2×#104×C127C2212yes—
34#419×C127C2212yescomplemented
35#417×C127C2212yes—
36C2×#100×C127C2211yescharacteristic
37C2×#100×C127C2211yescharacteristic
38C4×M16×C127C2212yescomplemented
39#391×C127C2211yescharacteristic, complemented
40#390×C127C2211yescharacteristic
41#388×C127—22no—
42#385×C127—22no—
43#384×C127C2211yescharacteristic, complemented
44#376×C127C2211yescharacteristic, complemented
45#343×C127C2212yescomplemented
46#339×C127C2212yescomplemented
47#339×C127C2212yescomplemented
48H(5,𝔽2)×C127—24no—
49H(5,𝔽2)×C127—22no—
50H(5,𝔽2)×C127C2×C414yes—
51C2×#41×C127—22no—
52C2×#41×C127C2312yes—
53C22×Q8×C127C2311yescharacteristic
54C22×D4×C127—22no—
55C22×D4×C127—22no—
56C22×D4×C127C2311yescharacteristic
57C23×C4×C127—22no—
58C23×C4×C127D412yes—
59#137×C127—24no—
60#137×C127C2×C418yescomplemented
61C2×Q16×C127—22no—
62C2×SD16×C127—22no—
63C2×M16×C127C2312yes—
64C2×M16×C127C2312yes—
65#124×C127—22no—
66#124×C127C2×C414yescomplemented
67C4×Q8×C127—22no—
68C4×Q8×C127—22no—
69C4×Q8×C127—22no—
70C4×Q8×C127D412yes—
71C4×Q8×C127D412yes—
72C4×D4×C127—24no—
73C4×D4×C127—24no—
74C4×D4×C127—24no—
75C4×D4×C127—24no—
76C4×D4×C127—22no—
77C4×D4×C127—22no—
78C4×D4×C127—24no—
79C4×D4×C127—22no—
80C4×D4×C127—22no—
81C4×D4×C127—22no—
82C4×D4×C127—22no—
83C4×D4×C127—22no—
84C4×D4×C127D412yes—
85C4×D4×C127D412yes—
86#117×C127—24no—
87#117×C127—22no—
88#117×C127—22no—
89#117×C127C2312yes—
90C2×C4⋊C4×C127—22no—
91C2×C4⋊C4×C127—22no—
92C2×C4⋊C4×C127—21no—
93C2×C4⋊C4×C127—22no—
94C2×C4⋊C4×C127C2311yescharacteristic
95C2×C4⋊C4×C127C2311yescharacteristic
96C2×C22⋊C4×C127—22no—
97C2×C22⋊C4×C127—24no—
98C2×C22⋊C4×C127—22no—
99C2×C22⋊C4×C127—22no—
100C2×C22⋊C4×C127—22no—
101C2×C22⋊C4×C127D412yes—
102C2×C42×C127—22no—
103C2×C42×C127—22no—
104C2×C42×C127—21no—
105C2×C42×C127C2311yescharacteristic
106C2×C42×C127C2311yescharacteristic
107C2×C42×C127C2311yescharacteristic
108#107×C127—22no—
109#106×C127—22no—
110#104×C127—24no—
111#104×C127C2×C418yescomplemented
112#103×C127—22no—
113#103×C127—22no—
114#103×C127—22no—
115#102×C127—22no—
116#100×C127—22no—
117#100×C127C2×C414yescomplemented
118#97×C127—22no—
119C4×C8×C127—22no—
120#95×C127—21no—
121#95×C127—21no—
122#95×C127—22no—
123C24×C127—22no—
124C24×C127C2×D412yes—
125#41×C127—28no—
126#41×C127—28no—
127#41×C127—22no—
128#41×C127—22no—
129#41×C127C22×C414yes—
130#41×C127#4114yes—
131C2×Q8×C127—21no—
132C2×Q8×C127—22no—
133C2×Q8×C127—22no—
134C2×Q8×C127C22×C412yescomplemented
135C2×Q8×C127C2×D412yes—
136C2×D4×C127—24no—
137C2×D4×C127—28no—
138C2×D4×C127—28no—
139C2×D4×C127—28no—
140C2×D4×C127—22no—
141C2×D4×C127—28no—
142C2×D4×C127—22no—
143C2×D4×C127—22no—
144C2×D4×C127C22×C412yes—
145C2×D4×C127C2×D412yes—
146C2×D4×C127#4112yes—
147C22×C4×C127—22no—
148C22×C4×C127—22no—
149C22×C4×C127—21no—
150C22×C4×C127—21no—
151C22×C4×C127—22no—
152C22×C4×C127—24no—
153C22×C4×C127—22no—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127—21no—
159C22×C4×C127—21no—
160C22×C4×C127—82no—
161C22×C4×C127—84no—
162C22×C4×C127C2×D412yes—
163C22×C4×C127C2×D411yescharacteristic
164C22×C4×C127C2×D411yescharacteristic
165C22×C4×C127C2411yescharacteristic
166Q16×C127—28no—
167SD16×C127—28no—
168M16×C127—22no—
169M16×C127—22no—
170M16×C127—22no—
171M16×C127C22×C414yescomplemented
172C2×C8×C127—22no—
173C2×C8×C127—22no—
174C4⋊C4×C127—24no—
175C4⋊C4×C127—22no—
176C4⋊C4×C127—22no—
177C4⋊C4×C127—22no—
178C4⋊C4×C127—22no—
179C4⋊C4×C127—41no—
180C4⋊C4×C127—41no—
181C4⋊C4×C127—82no—
182C4⋊C4×C127C2×D412yes—
183C4⋊C4×C127C2×D412yes—
184C22⋊C4×C127—24no—
185C22⋊C4×C127—24no—
186C22⋊C4×C127—28no—
187C22⋊C4×C127—22no—
188C22⋊C4×C127—22no—
189C22⋊C4×C127—22no—
190C22⋊C4×C127—24no—
191C22⋊C4×C127—82no—
192C42×C127—24no—
193C42×C127—22no—
194C42×C127—22no—
195C42×C127—22no—
196C42×C127—21no—
197C42×C127—22no—
198C42×C127—42no—
199C42×C127C22×C412yescomplemented
200C42×C127C2×D412yes—
201C42×C127C2×D412yes—
202C23×C127—28no—
203C23×C127—22no—
204C23×C127—22no—
205C23×C127—22no—
206C23×C127—42no—
207C23×C127—42no—
208C23×C127—82no—
209C23×C127—82no—
210C23×C127C4×D414yes—
211C23×C127C22×D411yescharacteristic
212Q8×C127—22no—
213Q8×C127—24no—
214Q8×C127—42no—
215Q8×C127C4×D414yes—
216D4×C127—216no—
217D4×C127—24no—
218D4×C127—24no—
219D4×C127—24no—
220D4×C127—22no—
221D4×C127—24no—
222D4×C127C4×D414yes—
223C2×C4×C127—28no—
224C2×C4×C127—22no—
225C2×C4×C127—24no—
226C2×C4×C127—24no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—22no—
232C2×C4×C127—41no—
233C2×C4×C127—41no—
234C2×C4×C127—44no—
235C2×C4×C127—44no—
236C2×C4×C127—41no—
237C2×C4×C127—41no—
238C2×C4×C127—41no—
239C2×C4×C127—41no—
240C2×C4×C127—42no—
241C2×C4×C127—41no—
242C2×C4×C127—42no—
243C2×C4×C127—41no—
244C2×C4×C127—41no—
245C2×C4×C127—41no—
246C2×C4×C127—82no—
247C2×C4×C127—82no—
248C2×C4×C127—82no—
249C2×C4×C127—84no—
250C2×C4×C127—82no—
251C2×C4×C127#12011yescharacteristic
252C2×C4×C127#12012yes—
253C2×C4×C127#12011yescharacteristic
254C2×C4×C127C23×C411yescharacteristic
255C2×C4×C127C22×D412yes—
256C2×C4×C127C2×#4112yes—
257C2×C4×C127C2×#4111yescharacteristic
258C8×C127—24no—
259C8×C127—42no—
260C22×C127—21no—
261C22×C127—21no—
262C22×C127—24no—
263C22×C127—24no—
264C22×C127—44no—
265C22×C127—42no—
266C22×C127—42no—
267C22×C127—42no—
268C22×C127—42no—
269C22×C127—82no—
270C22×C127—88no—
271C22×C127C2×C4×D411yescharacteristic
272C22×C127C2×#12011yescharacteristic
273C22×C127#52511yescharacteristic
274C4×C127—21no—
275C4×C127—21no—
276C4×C127—22no—
277C4×C127—24no—
278C4×C127—42no—
279C4×C127—42no—
280C4×C127—42no—
281C4×C127—42no—
282C4×C127—42no—
283C4×C127—81no—
284C4×C127—81no—
285C4×C127C2×C4×D412yes—
286C4×C127#52512yes—
287#21474C12711yesmaximal, characteristic, direct product factor, Sylow
288C2×C127—21no—
289C2×C127—21no—
290C2×C127—44no—
291C2×C127—44no—
292C2×C127C4×#12011yescharacteristic
293C2×C127#301911yescharacteristic
294C2×C127#302211yescharacteristic
295C4×H(5,𝔽2)C2×C12711yescharacteristic
296C2×#455C2×C12711yescharacteristic
297C4×#137C2×C12712yescomplemented
298#2326C2×C12711yescharacteristic, complemented
299#1885C2×C12711yescharacteristic, complemented
300#1882C2×C12712yescomplemented
301#1859C2×C12712yescomplemented
302#1856C2×C12711yescharacteristic, complemented
303#1792C2×C12711yescharacteristic
304#1759C2×C12712yescomplemented
305C4×#100C2×C12711yescharacteristic, complemented
306C127#2147411yescharacteristic, direct product factor, Sylow
307C2×H(5,𝔽2)C22×C12711yescharacteristic
308C2×#137C22×C12712yes—
309C2×#124C22×C12711yescharacteristic
310#520—22no—
311#520C22×C12712yes—
312#518—22no—
313#518—22no—
314#518C22×C12711yescharacteristic
315C4×#41—22no—
316C4×#41C22×C12712yes—
317C2×C4×Q8C22×C12711yescharacteristic, complemented
318C2×C4×D4—22no—
319C2×C4×D4—22no—
320C2×C4×D4C22×C12711yescharacteristic
321#455C4×C12718yescomplemented
322#441—22no—
323#440—22no—
324C4×Q16—22no—
325C4×SD16—22no—
326#428C22×C12712yescomplemented
327C2×#104C22×C12712yes—
328C2×#104C22×C12712yes—
329#419C22×C12712yescomplemented
330#417C22×C12712yes—
331C2×#100C22×C12711yescharacteristic
332C2×#100C22×C12711yescharacteristic
333C4×M16C22×C12712yescomplemented
334#391C22×C12711yescharacteristic, complemented
335#390C22×C12711yescharacteristic
336#388—22no—
337#385—22no—
338#384C22×C12711yescharacteristic, complemented
339#376C22×C12711yescharacteristic, complemented
340#343C22×C12712yescomplemented
341#339C22×C12712yescomplemented
342#339C22×C12712yescomplemented
343H(5,𝔽2)—24no—
344H(5,𝔽2)—22no—
345H(5,𝔽2)C2×C4×C12714yes—
346C2×#41—22no—
347C2×#41C23×C12712yes—
348C22×Q8C23×C12711yescharacteristic
349C22×D4—22no—
350C22×D4—22no—
351C22×D4C23×C12711yescharacteristic
352C23×C4—22no—
353C23×C4D4×C12712yes—
354#137—24no—
355#137C2×C4×C12718yescomplemented
356C2×Q16—22no—
357C2×SD16—22no—
358C2×M16C23×C12712yes—
359C2×M16C23×C12712yes—
360#124—22no—
361#124C2×C4×C12714yescomplemented
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8D4×C12712yes—
366C4×Q8D4×C12712yes—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—24no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4D4×C12712yes—
380C4×D4D4×C12712yes—
381#117—24no—
382#117—22no—
383#117—22no—
384#117C23×C12712yes—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4C23×C12711yescharacteristic
390C2×C4⋊C4C23×C12711yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4D4×C12712yes—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—21no—
400C2×C42C23×C12711yescharacteristic
401C2×C42C23×C12711yescharacteristic
402C2×C42C23×C12711yescharacteristic
403#107—22no—
404#106—22no—
405#104—24no—
406#104C2×C4×C12718yescomplemented
407#103—22no—
408#103—22no—
409#103—22no—
410#102—22no—
411#100—22no—
412#100C2×C4×C12714yescomplemented
413#97—22no—
414C4×C8—22no—
415#95—21no—
416#95—21no—
417#95—22no—
418C24—22no—
419C24C2×D4×C12712yes—
420#41—28no—
421#41—28no—
422#41—22no—
423#41—22no—
424#41C22×C4×C12714yes—
425#41#41×C12714yes—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8C22×C4×C12712yescomplemented
430C2×Q8C2×D4×C12712yes—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—28no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4C22×C4×C12712yes—
440C2×D4C2×D4×C12712yes—
441C2×D4#41×C12712yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—82no—
456C22×C4—84no—
457C22×C4C2×D4×C12712yes—
458C22×C4C2×D4×C12711yescharacteristic
459C22×C4C2×D4×C12711yescharacteristic
460C22×C4C24×C12711yescharacteristic
461Q16—28no—
462SD16—28no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16C22×C4×C12714yescomplemented
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—82no—
477C4⋊C4C2×D4×C12712yes—
478C4⋊C4C2×D4×C12712yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—28no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—24no—
486C22⋊C4—82no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—22no—
493C42—42no—
494C42C22×C4×C12712yescomplemented
495C42C2×D4×C12712yes—
496C42C2×D4×C12712yes—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—82no—
504C23—82no—
505C23C4×D4×C12714yes—
506C23C22×D4×C12711yescharacteristic
507Q8—22no—
508Q8—24no—
509Q8—42no—
510Q8C4×D4×C12714yes—
511D4—216no—
512D4—24no—
513D4—24no—
514D4—24no—
515D4—22no—
516D4—24no—
517D4C4×D4×C12714yes—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#120×C12711yescharacteristic
547C2×C4#120×C12712yes—
548C2×C4#120×C12711yescharacteristic
549C2×C4C23×C4×C12711yescharacteristic
550C2×C4C22×D4×C12712yes—
551C2×C4C2×#41×C12712yes—
552C2×C4C2×#41×C12711yescharacteristic
553C8—24no—
554C8—42no—
555C22—21no—
556C22—21no—
557C22—24no—
558C22—24no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—88no—
566C22C2×C4×D4×C12711yescharacteristic
567C22C2×#120×C12711yescharacteristic
568C22#525×C12711yescharacteristic
569C4—21no—
570C4—21no—
571C4—22no—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4C2×C4×D4×C12712yes—
581C4#525×C12712yes—
582C2—21no—
583C2—21no—
584C2—44no—
585C2—44no—
586C2C4×#120×C12711yescharacteristic
587C2C127×#301911yescharacteristic
588C2C127×#302211yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×H(5,𝔽2)×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#137×C127C2semidirect
4. C127×#2326C2semidirect
5. C127×#1885C2semidirect
6. C127×#1882C2semidirect
7. C127×#1859C2semidirect
8. C127×#1856C2semidirect
10. C127×#1759C2semidirect
11. C4×#100×C127C2semidirect
22. C2×C4×Q8×C127C22semidirect
26. #455×C127C4semidirect
31. #428×C127C22semidirect
34. #419×C127C22semidirect
38. C4×M16×C127C22semidirect
39. #391×C127C22semidirect
43. #384×C127C22semidirect
44. #376×C127C22semidirect
45. #343×C127C22semidirect
46. #339×C127C22semidirect
47. #339×C127C22semidirect
60. #137×C127C2×C4semidirect
66. #124×C127C2×C4semidirect
111. #104×C127C2×C4semidirect
117. #100×C127C2×C4semidirect
134. C2×Q8×C127C22×C4semidirect
171. M16×C127C22×C4semidirect
199. C42×C127C22×C4semidirect
287. #21474C127direct (N × H)
297. C4×#137C2×C127semidirect
298. #2326C2×C127semidirect
299. #1885C2×C127semidirect
300. #1882C2×C127semidirect
301. #1859C2×C127semidirect
302. #1856C2×C127semidirect
304. #1759C2×C127semidirect
305. C4×#100C2×C127semidirect
306. C127#21474direct (N × H)
317. C2×C4×Q8C22×C127semidirect
321. #455C4×C127semidirect
326. #428C22×C127semidirect
329. #419C22×C127semidirect
333. C4×M16C22×C127semidirect
334. #391C22×C127semidirect
338. #384C22×C127semidirect
339. #376C22×C127semidirect
340. #343C22×C127semidirect
341. #339C22×C127semidirect
342. #339C22×C127semidirect
355. #137C2×C4×C127semidirect
361. #124C2×C4×C127semidirect
406. #104C2×C4×C127semidirect
412. #100C2×C4×C127semidirect
429. C2×Q8C22×C4×C127semidirect
466. M16C22×C4×C127semidirect
494. C42C22×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)×C1272
C2×#455×C1272
C4×#137×C1272
C127×#23262
C127×#18852
C127×#18822
C127×#18592
C127×#18562
C127×#17922
C127×#17592
C4×#100×C1272
#21474127