Finite Group Databasebeta

← C127×#20615 #44584288

Subgroups — C127×#20615 #44584288

Order 32512.

3718 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585×C127C211yesmaximal, characteristic
2C22×#104×C127C212yesmaximal, complemented
3C127×#2896C212yesmaximal, complemented
4C2×#413×C127C211yesmaximal, characteristic, complemented
5C127×#2297C211yesmaximal, characteristic, complemented
6C127×#1793C214yesmaximal, complemented
7C127×#1792C214yesmaximal, complemented
8#585×C127—28no—
9#585×C127—24no—
10#585×C127C418yescomplemented
11C2×2−1+4×C127—22no—
12C2×2−1+4×C127C2212yes—
13C2×H(5,𝔽2)×C127—22no—
14C2×H(5,𝔽2)×C127—22no—
15C2×H(5,𝔽2)×C127C2212yes—
16C22×#41×C127—24no—
17C22×#41×C127—22no—
18C22×#41×C127C2212yes—
19C22×#41×C127C2211yescharacteristic
20C22×M16×C127C2212yescomplemented
21C2×#117×C127C2211yescharacteristic, complemented
22C22×C42×C127C2211yescharacteristic, complemented
23C2×#104×C127—28no—
24C2×#104×C127C2218yescomplemented
25#419×C127—22no—
26#418×C127—22no—
27#417×C127C2218yescomplemented
28#416×C127—24no—
29#413×C127—24no—
30C2×#101×C127C2212yescomplemented
31C2×#100×C127C2212yescomplemented
32#391×C127C2212yescomplemented
33#390×C127C2212yescomplemented
34#386×C127—22no—
35C4×C22⋊C4×C127—22no—
362−1+4×C127—216no—
372−1+4×C127—28no—
382−1+4×C127—216no—
392−1+4×C127—24no—
402−1+4×C127C2×C418yescomplemented
41H(5,𝔽2)×C127—216no—
42H(5,𝔽2)×C127—216no—
43H(5,𝔽2)×C127—28no—
44H(5,𝔽2)×C127—216no—
45H(5,𝔽2)×C127—216no—
46H(5,𝔽2)×C127—24no—
47H(5,𝔽2)×C127C2×C418yescomplemented
48C2×#41×C127—24no—
49C2×#41×C127—28no—
50C2×#41×C127—24no—
51C2×#41×C127—232no—
52C2×#41×C127—216no—
53C2×#41×C127—232no—
54C2×#41×C127—24no—
55C2×#41×C127—24no—
56C2×#41×C127—216no—
57C2×#41×C127—22no—
58C2×#41×C127—22no—
59C2×#41×C127—24no—
60C2×#41×C127—22no—
61C2×#41×C127—22no—
62C2×#41×C127—24no—
63C2×#41×C127—24no—
64C2×#41×C127—24no—
65C2×#41×C127C2×C418yes—
66C2×#41×C127C2×C414yes—
67C2×#41×C127D418yes—
68C2×#41×C127D414yes—
69C2×#41×C127D414yes—
70C2×#41×C127C2314yes—
71C22×Q8×C127—22no—
72C22×Q8×C127—24no—
73C22×Q8×C127D412yes—
74C22×Q8×C127C2311yescharacteristic
75C22×D4×C127—28no—
76C22×D4×C127—22no—
77C22×D4×C127—24no—
78C22×D4×C127—22no—
79C22×D4×C127—24no—
80C22×D4×C127D412yes—
81C22×D4×C127D412yes—
82C22×D4×C127C2311yescharacteristic
83C23×C4×C127—24no—
84C23×C4×C127—22no—
85C23×C4×C127—21no—
86C23×C4×C127D412yes—
87C23×C4×C127C2311yescharacteristic
88C2×M16×C127—22no—
89C2×M16×C127—24no—
90C2×M16×C127—24no—
91C2×M16×C127C2314yes—
92C22×C8×C127—22no—
93#117×C127—22no—
94#117×C127—22no—
95C2×C4⋊C4×C127C2312yes—
96C2×C22⋊C4×C127—21no—
97C2×C22⋊C4×C127—24no—
98C2×C42×C127—24no—
99C2×C42×C127—21no—
100C2×C42×C127—21no—
101C2×C42×C127C2312yes—
102#104×C127—216no—
103#103×C127—28no—
104#102×C127—28no—
105#101×C127—24no—
106#100×C127—24no—
107#95×C127—24no—
108C24×C127—24no—
109C24×C127—22no—
110C24×C127C2×D412yes—
111C24×C127C2×D411yescharacteristic
112#41×C127—232no—
113#41×C127—216no—
114#41×C127—264no—
115#41×C127—216no—
116#41×C127—216no—
117#41×C127—232no—
118#41×C127—216no—
119#41×C127—216no—
120#41×C127—28no—
121#41×C127—28no—
122#41×C127—216no—
123#41×C127—28no—
124#41×C127—28no—
125#41×C127—28no—
126#41×C127—28no—
127#41×C127—28no—
128#41×C127—28no—
129#41×C127—28no—
130#41×C127—24no—
131#41×C127—24no—
132#41×C127C22⋊C4116yescomplemented
133#41×C127C22×C418yes—
134#41×C127C2×D418yes—
135C2×Q8×C127—24no—
136C2×Q8×C127—216no—
137C2×Q8×C127—216no—
138C2×Q8×C127—216no—
139C2×Q8×C127—22no—
140C2×Q8×C127—24no—
141C2×Q8×C127—22no—
142C2×Q8×C127—22no—
143C2×Q8×C127C22⋊C418yescomplemented
144C2×Q8×C127C22×C412yes—
145C2×Q8×C127C2×D414yes—
146C2×Q8×C127C2×D412yes—
147C2×D4×C127—28no—
148C2×D4×C127—24no—
149C2×D4×C127—216no—
150C2×D4×C127—264no—
151C2×D4×C127—216no—
152C2×D4×C127—216no—
153C2×D4×C127—216no—
154C2×D4×C127—28no—
155C2×D4×C127—24no—
156C2×D4×C127—24no—
157C2×D4×C127—28no—
158C2×D4×C127—24no—
159C2×D4×C127—24no—
160C2×D4×C127—216no—
161C2×D4×C127—22no—
162C2×D4×C127—24no—
163C2×D4×C127—22no—
164C2×D4×C127—22no—
165C2×D4×C127C22⋊C418yes—
166C2×D4×C127C22⋊C418yescomplemented
167C2×D4×C127C22×C412yes—
168C2×D4×C127C2×D412yes—
169C2×D4×C127C2×D414yes—
170C22×C4×C127—24no—
171C22×C4×C127—28no—
172C22×C4×C127—232no—
173C22×C4×C127—22no—
174C22×C4×C127—28no—
175C22×C4×C127—28no—
176C22×C4×C127—22no—
177C22×C4×C127—24no—
178C22×C4×C127—24no—
179C22×C4×C127—22no—
180C22×C4×C127—24no—
181C22×C4×C127—22no—
182C22×C4×C127—22no—
183C22×C4×C127—44no—
184C22×C4×C127—42no—
185C22×C4×C127C22⋊C418yes—
186C22×C4×C127C22×C412yes—
187C22×C4×C127C2×D412yes—
188C22×C4×C127C2×D411yescharacteristic
189C22×C4×C127C2×D414yes—
190C22×C4×C127C2×D412yes—
191C22×C4×C127C2×D412yes—
192C22×C4×C127C2×D412yes—
193C22×C4×C127C2411yescharacteristic
194M16×C127—24no—
195M16×C127—24no—
196M16×C127—44no—
197C2×C8×C127—24no—
198C2×C8×C127—44no—
199C4⋊C4×C127—24no—
200C22⋊C4×C127—48no—
201C22⋊C4×C127—42no—
202C42×C127—24no—
203C42×C127—42no—
204C42×C127—41no—
205C42×C127—41no—
206C23×C127—216no—
207C23×C127—22no—
208C23×C127—24no—
209C23×C127—22no—
210C23×C127—28no—
211C23×C127—42no—
212C23×C127—44no—
213C23×C127—84no—
214C23×C127—88no—
215C23×C127C2×C22⋊C414yes—
216C23×C127C2×C22⋊C412yes—
217C23×C127#12012yes—
218C23×C127#12012yes—
219C23×C127C22×D411yescharacteristic
220Q8×C127—216no—
221Q8×C127—24no—
222Q8×C127—24no—
223Q8×C127—24no—
224Q8×C127—28no—
225Q8×C127C2×C22⋊C418yes—
226D4×C127—232no—
227D4×C127—216no—
228D4×C127—28no—
229D4×C127—28no—
230D4×C127—216no—
231D4×C127—216no—
232D4×C127—24no—
233D4×C127—24no—
234D4×C127—24no—
235D4×C127—28no—
236D4×C127C2×C22⋊C418yes—
237C2×C4×C127—216no—
238C2×C4×C127—232no—
239C2×C4×C127—24no—
240C2×C4×C127—22no—
241C2×C4×C127—22no—
242C2×C4×C127—28no—
243C2×C4×C127—28no—
244C2×C4×C127—28no—
245C2×C4×C127—216no—
246C2×C4×C127—28no—
247C2×C4×C127—28no—
248C2×C4×C127—24no—
249C2×C4×C127—42no—
250C2×C4×C127—48no—
251C2×C4×C127—42no—
252C2×C4×C127—44no—
253C2×C4×C127—42no—
254C2×C4×C127C2×C22⋊C414yes—
255C2×C4×C127C2×C22⋊C414yes—
256C2×C4×C127C2×C22⋊C414yes—
257C2×C4×C127C2×C22⋊C412yes—
258C2×C4×C127#12014yes—
259C2×C4×C127#12014yes—
260C2×C4×C127#12011yescharacteristic
261C2×C4×C127#12012yes—
262C2×C4×C127#12011yescharacteristic
263C2×C4×C127C23×C411yescharacteristic
264C2×C4×C127C22×D412yes—
265C2×C4×C127C22×D412yes—
266C2×C4×C127C22×D411yescharacteristic
267C8×C127—44no—
268C22×C127—22no—
269C22×C127—24no—
270C22×C127—24no—
271C22×C127—28no—
272C22×C127—42no—
273C22×C127—42no—
274C22×C127—44no—
275C22×C127—44no—
276C22×C127—48no—
277C22×C127—82no—
278C22×C127—88no—
279C22×C127—816no—
280C22×C127#37914yes—
281C22×C127C22×C22⋊C411yescharacteristic
282C22×C127C2×#12011yescharacteristic
283C22×C127C2×#12011yescharacteristic
284C4×C127—28no—
285C4×C127—24no—
286C4×C127—44no—
287C4×C127—82no—
288C4×C127#37912yes—
289C4×C127#37912yes—
290C4×C127C22×C22⋊C412yes—
291C4×C127C2×#12012yes—
292#20615C12711yesmaximal, characteristic, direct product factor, Sylow
293C2×C127—21no—
294C2×C127—21no—
295C2×C127—24no—
296C2×C127—44no—
297C2×C127—48no—
298C2×C127C2×#37911yescharacteristic
299C2×C127#301812yes—
300C2×#585C2×C12711yescharacteristic
301C22×#104C2×C12712yescomplemented
302#2896C2×C12712yescomplemented
303C2×#413C2×C12711yescharacteristic, complemented
304#2297C2×C12711yescharacteristic, complemented
305#1793C2×C12714yescomplemented
306#1792C2×C12714yescomplemented
307C127#2061511yescharacteristic, direct product factor, Sylow
308#585—28no—
309#585—24no—
310#585C4×C12718yescomplemented
311C2×2−1+4—22no—
312C2×2−1+4C22×C12712yes—
313C2×H(5,𝔽2)—22no—
314C2×H(5,𝔽2)—22no—
315C2×H(5,𝔽2)C22×C12712yes—
316C22×#41—24no—
317C22×#41—22no—
318C22×#41C22×C12712yes—
319C22×#41C22×C12711yescharacteristic
320C22×M16C22×C12712yescomplemented
321C2×#117C22×C12711yescharacteristic, complemented
322C22×C42C22×C12711yescharacteristic, complemented
323C2×#104—28no—
324C2×#104C22×C12718yescomplemented
325#419—22no—
326#418—22no—
327#417C22×C12718yescomplemented
328#416—24no—
329#413—24no—
330C2×#101C22×C12712yescomplemented
331C2×#100C22×C12712yescomplemented
332#391C22×C12712yescomplemented
333#390C22×C12712yescomplemented
334#386—22no—
335C4×C22⋊C4—22no—
3362−1+4—216no—
3372−1+4—28no—
3382−1+4—216no—
3392−1+4—24no—
3402−1+4C2×C4×C12718yescomplemented
341H(5,𝔽2)—216no—
342H(5,𝔽2)—216no—
343H(5,𝔽2)—28no—
344H(5,𝔽2)—216no—
345H(5,𝔽2)—216no—
346H(5,𝔽2)—24no—
347H(5,𝔽2)C2×C4×C12718yescomplemented
348C2×#41—24no—
349C2×#41—28no—
350C2×#41—24no—
351C2×#41—232no—
352C2×#41—216no—
353C2×#41—232no—
354C2×#41—24no—
355C2×#41—24no—
356C2×#41—216no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—24no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—24no—
363C2×#41—24no—
364C2×#41—24no—
365C2×#41C2×C4×C12718yes—
366C2×#41C2×C4×C12714yes—
367C2×#41D4×C12718yes—
368C2×#41D4×C12714yes—
369C2×#41D4×C12714yes—
370C2×#41C23×C12714yes—
371C22×Q8—22no—
372C22×Q8—24no—
373C22×Q8D4×C12712yes—
374C22×Q8C23×C12711yescharacteristic
375C22×D4—28no—
376C22×D4—22no—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4D4×C12712yes—
381C22×D4D4×C12712yes—
382C22×D4C23×C12711yescharacteristic
383C23×C4—24no—
384C23×C4—22no—
385C23×C4—21no—
386C23×C4D4×C12712yes—
387C23×C4C23×C12711yescharacteristic
388C2×M16—22no—
389C2×M16—24no—
390C2×M16—24no—
391C2×M16C23×C12714yes—
392C22×C8—22no—
393#117—22no—
394#117—22no—
395C2×C4⋊C4C23×C12712yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—24no—
398C2×C42—24no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42C23×C12712yes—
402#104—216no—
403#103—28no—
404#102—28no—
405#101—24no—
406#100—24no—
407#95—24no—
408C24—24no—
409C24—22no—
410C24C2×D4×C12712yes—
411C24C2×D4×C12711yescharacteristic
412#41—232no—
413#41—216no—
414#41—264no—
415#41—216no—
416#41—216no—
417#41—232no—
418#41—216no—
419#41—216no—
420#41—28no—
421#41—28no—
422#41—216no—
423#41—28no—
424#41—28no—
425#41—28no—
426#41—28no—
427#41—28no—
428#41—28no—
429#41—28no—
430#41—24no—
431#41—24no—
432#41C22⋊C4×C127116yescomplemented
433#41C22×C4×C12718yes—
434#41C2×D4×C12718yes—
435C2×Q8—24no—
436C2×Q8—216no—
437C2×Q8—216no—
438C2×Q8—216no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8C22⋊C4×C12718yescomplemented
444C2×Q8C22×C4×C12712yes—
445C2×Q8C2×D4×C12714yes—
446C2×Q8C2×D4×C12712yes—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—216no—
450C2×D4—264no—
451C2×D4—216no—
452C2×D4—216no—
453C2×D4—216no—
454C2×D4—28no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—28no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—216no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4C22⋊C4×C12718yes—
466C2×D4C22⋊C4×C12718yescomplemented
467C2×D4C22×C4×C12712yes—
468C2×D4C2×D4×C12712yes—
469C2×D4C2×D4×C12714yes—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—232no—
473C22×C4—22no—
474C22×C4—28no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4C22⋊C4×C12718yes—
486C22×C4C22×C4×C12712yes—
487C22×C4C2×D4×C12712yes—
488C22×C4C2×D4×C12711yescharacteristic
489C22×C4C2×D4×C12714yes—
490C22×C4C2×D4×C12712yes—
491C22×C4C2×D4×C12712yes—
492C22×C4C2×D4×C12712yes—
493C22×C4C24×C12711yescharacteristic
494M16—24no—
495M16—24no—
496M16—44no—
497C2×C8—24no—
498C2×C8—44no—
499C4⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—42no—
502C42—24no—
503C42—42no—
504C42—41no—
505C42—41no—
506C23—216no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—28no—
511C23—42no—
512C23—44no—
513C23—84no—
514C23—88no—
515C23C2×C22⋊C4×C12714yes—
516C23C2×C22⋊C4×C12712yes—
517C23#120×C12712yes—
518C23#120×C12712yes—
519C23C22×D4×C12711yescharacteristic
520Q8—216no—
521Q8—24no—
522Q8—24no—
523Q8—24no—
524Q8—28no—
525Q8C2×C22⋊C4×C12718yes—
526D4—232no—
527D4—216no—
528D4—28no—
529D4—28no—
530D4—216no—
531D4—216no—
532D4—24no—
533D4—24no—
534D4—24no—
535D4—28no—
536D4C2×C22⋊C4×C12718yes—
537C2×C4—216no—
538C2×C4—232no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—216no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4C2×C22⋊C4×C12714yes—
555C2×C4C2×C22⋊C4×C12714yes—
556C2×C4C2×C22⋊C4×C12714yes—
557C2×C4C2×C22⋊C4×C12712yes—
558C2×C4#120×C12714yes—
559C2×C4#120×C12714yes—
560C2×C4#120×C12711yescharacteristic
561C2×C4#120×C12711yescharacteristic
562C2×C4#120×C12712yes—
563C2×C4C23×C4×C12711yescharacteristic
564C2×C4C22×D4×C12712yes—
565C2×C4C22×D4×C12712yes—
566C2×C4C22×D4×C12711yescharacteristic
567C8—44no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—28no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—48no—
577C22—82no—
578C22—88no—
579C22—816no—
580C22#379×C12714yes—
581C22C22×C22⋊C4×C12711yescharacteristic
582C22C2×#120×C12711yescharacteristic
583C22C2×#120×C12711yescharacteristic
584C4—28no—
585C4—24no—
586C4—44no—
587C4—82no—
588C4#379×C12712yes—
589C4#379×C12712yes—
590C4C22×C22⋊C4×C12712yes—
591C4C2×#120×C12712yes—
592C2—21no—
593C2—21no—
594C2—24no—
595C2—44no—
596C2—48no—
597C2C2×#379×C12711yescharacteristic
598C2C127×#301812yes—
subgroupG/H
centerC2×C4×C127#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C2×#120
prime radicalC2×#585×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#104×C127C2semidirect
3. C127×#2896C2semidirect
4. C2×#413×C127C2semidirect
5. C127×#2297C2semidirect
6. C127×#1793C2semidirect
7. C127×#1792C2semidirect
10. #585×C127C4semidirect
20. C22×M16×C127C22semidirect
21. C2×#117×C127C22semidirect
22. C22×C42×C127C22semidirect
24. C2×#104×C127C22semidirect
27. #417×C127C22semidirect
30. C2×#101×C127C22semidirect
31. C2×#100×C127C22semidirect
32. #391×C127C22semidirect
33. #390×C127C22semidirect
40. 2−1+4×C127C2×C4semidirect
47. H(5,𝔽2)×C127C2×C4semidirect
132. #41×C127C22⋊C4semidirect
143. C2×Q8×C127C22⋊C4semidirect
166. C2×D4×C127C22⋊C4semidirect
292. #20615C127direct (N × H)
301. C22×#104C2×C127semidirect
302. #2896C2×C127semidirect
303. C2×#413C2×C127semidirect
304. #2297C2×C127semidirect
305. #1793C2×C127semidirect
306. #1792C2×C127semidirect
307. C127#20615direct (N × H)
310. #585C4×C127semidirect
320. C22×M16C22×C127semidirect
321. C2×#117C22×C127semidirect
322. C22×C42C22×C127semidirect
324. C2×#104C22×C127semidirect
327. #417C22×C127semidirect
330. C2×#101C22×C127semidirect
331. C2×#100C22×C127semidirect
332. #391C22×C127semidirect
333. #390C22×C127semidirect
340. 2−1+4C2×C4×C127semidirect
347. H(5,𝔽2)C2×C4×C127semidirect
432. #41C22⋊C4×C127semidirect
443. C2×Q8C22⋊C4×C127semidirect
466. C2×D4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#585×C1272
C22×#104×C1272
C127×#28962
C2×#413×C1272
C127×#22972
C127×#17932
C127×#17922
#20615127