Finite Group Databasebeta

← C127×#14666 #44578339

Subgroups — C127×#14666 #44578339

Order 32512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386×C127C211yesmaximal, characteristic
2C127×#1443C212yesmaximal, complemented
3C127×#1442C212yesmaximal, complemented
4C127×#1440C212yesmaximal, complemented
5C127×#1439C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1438C211yesmaximal, characteristic
8C127×#1438C211yesmaximal, characteristic
9C127×C22⋊#95C212yesmaximal, complemented
10C23×D4×C127C2211yescharacteristic
11C24×C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2212yes—
13C22×C22⋊C4×C127C2212yes—
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C22⋊C4×C127C2211yescharacteristic
19C22×C22⋊C4×C127C2212yes—
20#386×C127C418yescomplemented
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2212yes—
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2212yes—
32C25×C127C2311yescharacteristic
33C25×C127C2311yescharacteristic
34C22×D4×C127—24no—
35C22×D4×C127—42no—
36C22×D4×C127—41no—
37C22×D4×C127—41no—
38C22×D4×C127C2×C414yes—
39C23×C4×C127—41no—
40C23×C4×C127—41no—
41C23×C4×C127—42no—
42C23×C4×C127—42no—
43C23×C4×C127C2×C414yes—
44C23×C4×C127C2312yes—
45C23×C4×C127C2312yes—
46C23×C4×C127C2312yes—
47C23×C4×C127C2312yes—
48C23×C4×C127C2312yes—
49C23×C4×C127C2312yes—
50C23×C4×C127C2311yescharacteristic
51C2×C4⋊C4×C127—42no—
52C2×C4⋊C4×C127—42no—
53C2×C4⋊C4×C127—42no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127—42no—
57C2×C22⋊C4×C127—42no—
58C2×C22⋊C4×C127—42no—
59C2×C22⋊C4×C127—42no—
60C2×C22⋊C4×C127—42no—
61C2×C22⋊C4×C127—42no—
62C2×C22⋊C4×C127—42no—
63C2×C22⋊C4×C127—41no—
64C2×C22⋊C4×C127—41no—
65C2×C22⋊C4×C127—42no—
66C2×C22⋊C4×C127—42no—
67C2×C22⋊C4×C127—42no—
68C2×C22⋊C4×C127—42no—
69C2×C22⋊C4×C127—42no—
70C2×C22⋊C4×C127C2×C414yes—
71C2×C22⋊C4×C127C2×C418yes—
72#95×C127—42no—
73#95×C127—42no—
74#95×C127—42no—
75#95×C127—42no—
76#95×C127—42no—
77#95×C127—42no—
78#95×C127—42no—
79#95×C127—41no—
80#95×C127—41no—
81#95×C127—42no—
82#95×C127C2×C418yes—
83C24×C127—22no—
84C24×C127—22no—
85C24×C127—22no—
86C24×C127—22no—
87C24×C127—21no—
88C24×C127—21no—
89C24×C127—22no—
90C24×C127—22no—
91C24×C127—22no—
92C24×C127—22no—
93C24×C127—22no—
94C24×C127—21no—
95C24×C127—22no—
96C24×C127—21no—
97C24×C127—22no—
98C24×C127C22×C412yes—
99C24×C127C22×C412yes—
100C24×C127#4114yes—
101C24×C127C2411yescharacteristic
102C2×D4×C127—28no—
103C2×D4×C127—44no—
104C2×D4×C127—44no—
105C2×D4×C127—48no—
106C2×D4×C127—48no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—21no—
116C22×C4×C127—22no—
117C22×C4×C127—21no—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—22no—
133C22×C4×C127—22no—
134C22×C4×C127—22no—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—22no—
140C22×C4×C127—22no—
141C22×C4×C127—22no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—22no—
145C22×C4×C127—22no—
146C22×C4×C127—22no—
147C22×C4×C127—22no—
148C22×C4×C127—22no—
149C22×C4×C127—22no—
150C22×C4×C127—22no—
151C22×C4×C127—48no—
152C22×C4×C127—44no—
153C22×C4×C127—48no—
154C22×C4×C127—44no—
155C22×C4×C127—44no—
156C22×C4×C127C22×C412yes—
157C22×C4×C127C22×C414yes—
158C22×C4×C127C22×C414yes—
159C4⋊C4×C127—48no—
160C22⋊C4×C127—48no—
161C22⋊C4×C127—44no—
162C22⋊C4×C127—48no—
163C22⋊C4×C127—48no—
164C22⋊C4×C127—44no—
165C22⋊C4×C127—48no—
166C22⋊C4×C127—44no—
167C23×C127—22no—
168C23×C127—22no—
169C23×C127—24no—
170C23×C127—24no—
171C23×C127—24no—
172C23×C127—24no—
173C23×C127—24no—
174C23×C127—24no—
175C23×C127—24no—
176C23×C127—24no—
177C23×C127—22no—
178C23×C127—24no—
179C23×C127—22no—
180C23×C127—22no—
181C23×C127—24no—
182C23×C127—42no—
183C23×C127—41no—
184C23×C127—42no—
185C23×C127—41no—
186C23×C127—42no—
187C23×C127—41no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127—41no—
192C23×C127—42no—
193C23×C127—42no—
194C23×C127—42no—
195C23×C127—42no—
196C23×C127—42no—
197C23×C127—42no—
198C23×C127—41no—
199C23×C127—42no—
200C23×C127—42no—
201C23×C127—41no—
202C23×C127—42no—
203C23×C127—41no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—42no—
207C23×C127—41no—
208C23×C127#11714yes—
209C23×C127C23×C411yescharacteristic
210C23×C127C2×#4112yes—
211C23×C127H(5,𝔽2)11yescharacteristic
212C23×C127H(5,𝔽2)12yes—
213C23×C127H(5,𝔽2)11yescharacteristic
214C23×C127H(5,𝔽2)12yes—
215C23×C127H(5,𝔽2)12yes—
216C23×C127H(5,𝔽2)12yes—
217C23×C1272−1+412yes—
218D4×C127—416no—
219C2×C4×C127—22no—
220C2×C4×C127—24no—
221C2×C4×C127—24no—
222C2×C4×C127—24no—
223C2×C4×C127—24no—
224C2×C4×C127—24no—
225C2×C4×C127—22no—
226C2×C4×C127—22no—
227C2×C4×C127—24no—
228C2×C4×C127—416no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127—42no—
235C2×C4×C127—42no—
236C2×C4×C127—42no—
237C2×C4×C127—42no—
238C2×C4×C127—42no—
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—42no—
248C2×C4×C127—42no—
249C2×C4×C127—42no—
250C2×C4×C127—42no—
251C2×C4×C127—42no—
252C2×C4×C127—42no—
253C2×C4×C127—42no—
254C2×C4×C127—42no—
255C2×C4×C127—42no—
256C2×C4×C127—42no—
257C2×C4×C127—42no—
258C2×C4×C127—42no—
259C2×C4×C127—42no—
260C2×C4×C127—42no—
261C2×C4×C127—42no—
262C2×C4×C127—42no—
263C2×C4×C127—42no—
264C2×C4×C127—42no—
265C2×C4×C127—42no—
266C2×C4×C127—42no—
267C22×C127—24no—
268C22×C127—28no—
269C22×C127—44no—
270C22×C127—44no—
271C22×C127—44no—
272C22×C127—44no—
273C22×C127—42no—
274C22×C127—42no—
275C22×C127—44no—
276C22×C127—42no—
277C22×C127—44no—
278C22×C127—42no—
279C22×C127—42no—
280C22×C127—44no—
281C22×C127—82no—
282C22×C127—82no—
283C22×C127—82no—
284C22×C127—82no—
285C22×C127C2×#11711yescharacteristic
286C22×C127#51811yescharacteristic
287C22×C127#51811yescharacteristic
288C22×C127#51812yes—
289C22×C127#52012yes—
290C22×C127#53812yes—
291C22×C127#53812yes—
292C22×C127#53812yes—
293C22×C127#53812yes—
294C22×C127#53912yes—
295C22×C127#53912yes—
296C22×C127#56012yes—
297C22×C127#56011yescharacteristic
298C22×C127#56012yes—
299C22×C127#56012yes—
300C22×C127#56011yescharacteristic
301C22×C127H(3,𝔽4)11yescharacteristic
302C22×C127#56212yes—
303C22×C127#56212yes—
304C22×C127#56212yes—
305C22×C127#56311yescharacteristic
306C4×C127—42no—
307C4×C127—44no—
308C4×C127—44no—
309C4×C127—82no—
310C4×C127—82no—
311C4×C127—82no—
312C4×C127—82no—
313#14666C12711yesmaximal, characteristic, direct product factor, Sylow
314C2×C127—44no—
315C2×C127—82no—
316C2×C127#231211yescharacteristic
317C2×C127#231812yes—
318C2×C127#237511yescharacteristic
319C2×C127#237611yescharacteristic
320C2×C127#237712yes—
321C2×C127#282412yes—
322C2×C127#282512yes—
323C2×C127#282612yes—
324C2×C127#282712yes—
325C2×#386C2×C12711yescharacteristic
326#1443C2×C12712yescomplemented
327#1442C2×C12712yescomplemented
328#1440C2×C12712yescomplemented
329#1439C2×C12712yescomplemented
330#1439C2×C12712yescomplemented
331#1438C2×C12711yescharacteristic
332#1438C2×C12711yescharacteristic
333C22⋊#95C2×C12712yescomplemented
334C127#1466611yescharacteristic, direct product factor, Sylow
335C23×D4C22×C12711yescharacteristic
336C24×C4C22×C12711yescharacteristic
337C22×C4⋊C4C22×C12712yes—
338C22×C22⋊C4C22×C12712yes—
339C22×C22⋊C4C22×C12712yes—
340C22×C22⋊C4C22×C12711yescharacteristic
341C22×C22⋊C4C22×C12712yes—
342C22×C22⋊C4C22×C12711yescharacteristic
343C22×C22⋊C4C22×C12711yescharacteristic
344C22×C22⋊C4C22×C12712yes—
345#386C4×C12718yescomplemented
346C2×#95C22×C12712yes—
347C2×#95C22×C12712yes—
348C2×#95C22×C12712yes—
349C2×#95C22×C12712yes—
350C2×#95C22×C12711yescharacteristic
351C2×#95C22×C12712yes—
352C2×#95C22×C12711yescharacteristic
353C2×#95C22×C12712yes—
354C2×#95C22×C12712yes—
355C2×#95C22×C12712yes—
356C2×#95C22×C12712yes—
357C25C23×C12711yescharacteristic
358C25C23×C12711yescharacteristic
359C22×D4—24no—
360C22×D4—42no—
361C22×D4—41no—
362C22×D4—41no—
363C22×D4C2×C4×C12714yes—
364C23×C4—41no—
365C23×C4—41no—
366C23×C4—42no—
367C23×C4—42no—
368C23×C4C2×C4×C12714yes—
369C23×C4C23×C12712yes—
370C23×C4C23×C12712yes—
371C23×C4C23×C12712yes—
372C23×C4C23×C12712yes—
373C23×C4C23×C12712yes—
374C23×C4C23×C12712yes—
375C23×C4C23×C12711yescharacteristic
376C2×C4⋊C4—42no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4—42no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—42no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—41no—
389C2×C22⋊C4—41no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
392C2×C22⋊C4—42no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4C2×C4×C12714yes—
396C2×C22⋊C4C2×C4×C12718yes—
397#95—42no—
398#95—42no—
399#95—42no—
400#95—42no—
401#95—42no—
402#95—42no—
403#95—42no—
404#95—41no—
405#95—41no—
406#95—42no—
407#95C2×C4×C12718yes—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—21no—
420C24—22no—
421C24—21no—
422C24—22no—
423C24C22×C4×C12712yes—
424C24C22×C4×C12712yes—
425C24#41×C12714yes—
426C24C24×C12711yescharacteristic
427C2×D4—28no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—48no—
431C2×D4—48no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4C22×C4×C12712yes—
482C22×C4C22×C4×C12714yes—
483C22×C4C22×C4×C12714yes—
484C4⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23#117×C12714yes—
534C23C23×C4×C12711yescharacteristic
535C23C2×#41×C12712yes—
536C23H(5,𝔽2)×C12711yescharacteristic
537C23H(5,𝔽2)×C12712yes—
538C23H(5,𝔽2)×C12711yescharacteristic
539C23H(5,𝔽2)×C12712yes—
540C23H(5,𝔽2)×C12712yes—
541C23H(5,𝔽2)×C12712yes—
542C232−1+4×C12712yes—
543D4—416no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C22—24no—
593C22—28no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C2×#117×C12711yescharacteristic
611C22#518×C12711yescharacteristic
612C22#518×C12711yescharacteristic
613C22#518×C12712yes—
614C22#520×C12712yes—
615C22#538×C12712yes—
616C22#538×C12712yes—
617C22#538×C12712yes—
618C22#538×C12712yes—
619C22#539×C12712yes—
620C22#539×C12712yes—
621C22#560×C12712yes—
622C22#560×C12711yescharacteristic
623C22#560×C12712yes—
624C22#560×C12712yes—
625C22#560×C12711yescharacteristic
626C22H(3,𝔽4)×C12711yescharacteristic
627C22#562×C12712yes—
628C22#562×C12712yes—
629C22#562×C12712yes—
630C22#563×C12711yescharacteristic
631C4—42no—
632C4—44no—
633C4—44no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C2—44no—
639C2—82no—
640C2C127×#231211yescharacteristic
641C2C127×#231812yes—
642C2C127×#237511yescharacteristic
643C2C127×#237611yescharacteristic
644C2C127×#237712yes—
645C2C127×#282412yes—
646C2C127×#282512yes—
647C2C127×#282612yes—
648C2C127×#282712yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1443C2semidirect
3. C127×#1442C2semidirect
4. C127×#1440C2semidirect
5. C127×#1439C2semidirect
6. C127×#1439C2semidirect
9. C127×C22⋊#95C2semidirect
20. #386×C127C4semidirect
313. #14666C127direct (N × H)
326. #1443C2×C127semidirect
327. #1442C2×C127semidirect
328. #1440C2×C127semidirect
329. #1439C2×C127semidirect
330. #1439C2×C127semidirect
333. C22⋊#95C2×C127semidirect
334. C127#14666direct (N × H)
345. #386C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#386×C1272
C127×#14432
C127×#14422
C127×#14402
C127×#14392
C127×#14382
C127×C22⋊#952
#14666127