Finite Group Databasebeta

← C127×#15463 #44579136

Subgroups — C127×#15463 #44579136

Order 32512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C127C211yesmaximal, characteristic
2C2×#397×C127C212yesmaximal, complemented
3C2×#386×C127C211yesmaximal, characteristic
4C127×#1442C212yesmaximal, complemented
5C127×#1441C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1438C211yesmaximal, characteristic
8C127×C22⋊#95C212yesmaximal, complemented
9C127×C22⋊#95C212yesmaximal, complemented
10C23×D4×C127C2211yescharacteristic
11C24×C4×C127C2211yescharacteristic
12C24×C4×C127C2211yescharacteristic
13C2×#123×C127—24no—
14C2×#123×C127—28no—
15C22×C4⋊C4×C127C2212yes—
16C22×C4⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C22⋊C4×C127C2211yescharacteristic
19C22×C22⋊C4×C127C2212yes—
20C22×C22⋊C4×C127C2212yes—
21C22×C22⋊C4×C127C2212yes—
22C22×C22⋊C4×C127C2212yes—
23#397×C127—28no—
24#386×C127—24no—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2212yes—
32C2×#95×C127C2212yes—
33C2×#95×C127C2211yescharacteristic
34C2×#95×C127C2211yescharacteristic
35C25×C127C2311yescharacteristic
36C25×C127C2311yescharacteristic
37C22×D4×C127—24no—
38C22×D4×C127—24no—
39C22×D4×C127—41no—
40C22×D4×C127—41no—
41C22×D4×C127D414yes—
42C23×C4×C127—22no—
43C23×C4×C127—24no—
44C23×C4×C127—22no—
45C23×C4×C127—24no—
46C23×C4×C127—24no—
47C23×C4×C127—41no—
48C23×C4×C127—41no—
49C23×C4×C127—41no—
50C23×C4×C127—41no—
51C23×C4×C127—42no—
52C23×C4×C127C2312yes—
53C23×C4×C127C2312yes—
54C23×C4×C127C2312yes—
55C23×C4×C127C2312yes—
56C23×C4×C127C2311yescharacteristic
57C23×C4×C127C2312yes—
58C23×C4×C127C2312yes—
59#123×C127—232no—
60C2×C4⋊C4×C127—24no—
61C2×C4⋊C4×C127—24no—
62C2×C4⋊C4×C127—24no—
63C2×C4⋊C4×C127—24no—
64C2×C4⋊C4×C127—24no—
65C2×C4⋊C4×C127—42no—
66C2×C22⋊C4×C127—22no—
67C2×C22⋊C4×C127—24no—
68C2×C22⋊C4×C127—24no—
69C2×C22⋊C4×C127—24no—
70C2×C22⋊C4×C127—24no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—24no—
73C2×C22⋊C4×C127—24no—
74C2×C22⋊C4×C127—24no—
75C2×C22⋊C4×C127—24no—
76C2×C22⋊C4×C127—24no—
77C2×C22⋊C4×C127—24no—
78C2×C22⋊C4×C127—42no—
79C2×C22⋊C4×C127—42no—
80C2×C22⋊C4×C127—42no—
81C2×C22⋊C4×C127D418yes—
82#95×C127—24no—
83#95×C127—24no—
84#95×C127—24no—
85#95×C127—24no—
86#95×C127—24no—
87#95×C127—41no—
88#95×C127—42no—
89#95×C127—42no—
90#95×C127—42no—
91#95×C127—41no—
92C24×C127—22no—
93C24×C127—22no—
94C24×C127—22no—
95C24×C127—22no—
96C24×C127—22no—
97C24×C127—22no—
98C24×C127—22no—
99C24×C127—22no—
100C24×C127—21no—
101C24×C127—21no—
102C24×C127—21no—
103C24×C127—21no—
104C24×C127—22no—
105C24×C127—22no—
106C24×C127C2×D414yes—
107C24×C127C2×D412yes—
108C24×C127C2×D412yes—
109C24×C127#4114yes—
110C24×C127C2411yescharacteristic
111C2×D4×C127—28no—
112C2×D4×C127—28no—
113C2×D4×C127—44no—
114C2×D4×C127—48no—
115C2×D4×C127—48no—
116C22×C4×C127—28no—
117C22×C4×C127—22no—
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—21no—
140C22×C4×C127—21no—
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—22no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127—22no—
159C22×C4×C127—44no—
160C22×C4×C127—48no—
161C22×C4×C127—48no—
162C22×C4×C127—48no—
163C22×C4×C127—48no—
164C22×C4×C127—44no—
165C22×C4×C127—44no—
166C22×C4×C127—44no—
167C22×C4×C127—44no—
168C22×C4×C127C2×D414yes—
169C22×C4×C127C2×D412yes—
170C22×C4×C127C2×D414yes—
171C22×C4×C127#4114yes—
172C22×C4×C127#4114yes—
173C4⋊C4×C127—216no—
174C4⋊C4×C127—48no—
175C22⋊C4×C127—28no—
176C22⋊C4×C127—216no—
177C22⋊C4×C127—48no—
178C22⋊C4×C127—48no—
179C22⋊C4×C127—48no—
180C22⋊C4×C127—44no—
181C23×C127—22no—
182C23×C127—22no—
183C23×C127—24no—
184C23×C127—24no—
185C23×C127—22no—
186C23×C127—22no—
187C23×C127—24no—
188C23×C127—24no—
189C23×C127—24no—
190C23×C127—24no—
191C23×C127—24no—
192C23×C127—24no—
193C23×C127—24no—
194C23×C127—24no—
195C23×C127—24no—
196C23×C127—24no—
197C23×C127—24no—
198C23×C127—24no—
199C23×C127—24no—
200C23×C127—24no—
201C23×C127—42no—
202C23×C127—42no—
203C23×C127—42no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—42no—
207C23×C127—42no—
208C23×C127—42no—
209C23×C127—42no—
210C23×C127—42no—
211C23×C127—41no—
212C23×C127—42no—
213C23×C127—41no—
214C23×C127—42no—
215C23×C127—41no—
216C23×C127—41no—
217C23×C127—41no—
218C23×C127—41no—
219C23×C127—41no—
220C23×C127—41no—
221C23×C127#12014yes—
222C23×C127#12314yes—
223C23×C127C22×D411yescharacteristic
224C23×C127C22×D412yes—
225C23×C127C2×#4112yes—
226C23×C127C2×#4112yes—
227C23×C127C2×#4112yes—
228C23×C127H(5,𝔽2)11yescharacteristic
229C23×C127H(5,𝔽2)11yescharacteristic
230C23×C127H(5,𝔽2)12yes—
231C23×C1272−1+412yes—
232D4×C127—416no—
233C2×C4×C127—24no—
234C2×C4×C127—24no—
235C2×C4×C127—22no—
236C2×C4×C127—22no—
237C2×C4×C127—22no—
238C2×C4×C127—24no—
239C2×C4×C127—24no—
240C2×C4×C127—24no—
241C2×C4×C127—24no—
242C2×C4×C127—24no—
243C2×C4×C127—24no—
244C2×C4×C127—24no—
245C2×C4×C127—24no—
246C2×C4×C127—24no—
247C2×C4×C127—24no—
248C2×C4×C127—416no—
249C2×C4×C127—416no—
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—
267C2×C4×C127—42no—
268C2×C4×C127—42no—
269C2×C4×C127—42no—
270C2×C4×C127—42no—
271C2×C4×C127—42no—
272C2×C4×C127—42no—
273C2×C4×C127—42no—
274C2×C4×C127—42no—
275C2×C4×C127—42no—
276C2×C4×C127—42no—
277C2×C4×C127—42no—
278C2×C4×C127—42no—
279C2×C4×C127—42no—
280C2×C4×C127—42no—
281C2×C4×C127—42no—
282C22×C127—24no—
283C22×C127—24no—
284C22×C127—28no—
285C22×C127—28no—
286C22×C127—44no—
287C22×C127—44no—
288C22×C127—44no—
289C22×C127—44no—
290C22×C127—44no—
291C22×C127—44no—
292C22×C127—42no—
293C22×C127—42no—
294C22×C127—42no—
295C22×C127—42no—
296C22×C127—44no—
297C22×C127—44no—
298C22×C127—44no—
299C22×C127—44no—
300C22×C127C2×#12011yescharacteristic
301C22×C127C2×#12311yescharacteristic
302C22×C127#52912yes—
303C22×C127#53212yes—
304C22×C127#53411yescharacteristic
305C22×C127#53712yes—
306C22×C127#53812yes—
307C22×C127#53912yes—
308C22×C127#53912yes—
309C22×C127#54612yes—
310C22×C127#54612yes—
311C22×C127#54612yes—
312C22×C127#54612yes—
313C22×C127#54712yes—
314C22×C127#54712yes—
315C22×C127#55112yes—
316C22×C127#55111yescharacteristic
317C22×C127#55212yes—
318C22×C127#55911yescharacteristic
319C22×C127#56011yescharacteristic
320C22×C127#56311yescharacteristic
321C4×C127—44no—
322C4×C127—42no—
323C4×C127—44no—
324C4×C127—44no—
325C4×C127—44no—
326C4×C127—82no—
327C4×C127—82no—
328#15463C12711yesmaximal, characteristic, direct product factor, Sylow
329C2×C127—44no—
330C2×C127—44no—
331C2×C127#240511yescharacteristic
332C2×C127#241912yes—
333C2×C127#243012yes—
334C2×C127#255312yes—
335C2×C127#255712yes—
336C2×C127#269312yes—
337C2×C127#269512yes—
338C2×C127#269711yescharacteristic
339C2×C127#272411yescharacteristic
340C22×#123C2×C12711yescharacteristic
341C2×#397C2×C12712yescomplemented
342C2×#386C2×C12711yescharacteristic
343#1442C2×C12712yescomplemented
344#1441C2×C12712yescomplemented
345#1439C2×C12712yescomplemented
346#1438C2×C12711yescharacteristic
347C22⋊#95C2×C12712yescomplemented
348C22⋊#95C2×C12712yescomplemented
349C127#1546311yescharacteristic, direct product factor, Sylow
350C23×D4C22×C12711yescharacteristic
351C24×C4C22×C12711yescharacteristic
352C24×C4C22×C12711yescharacteristic
353C2×#123—24no—
354C2×#123—28no—
355C22×C4⋊C4C22×C12712yes—
356C22×C4⋊C4C22×C12712yes—
357C22×C22⋊C4C22×C12711yescharacteristic
358C22×C22⋊C4C22×C12711yescharacteristic
359C22×C22⋊C4C22×C12712yes—
360C22×C22⋊C4C22×C12712yes—
361C22×C22⋊C4C22×C12712yes—
362C22×C22⋊C4C22×C12712yes—
363#397—28no—
364#386—24no—
365C2×#95C22×C12712yes—
366C2×#95C22×C12712yes—
367C2×#95C22×C12712yes—
368C2×#95C22×C12712yes—
369C2×#95C22×C12712yes—
370C2×#95C22×C12712yes—
371C2×#95C22×C12712yes—
372C2×#95C22×C12712yes—
373C2×#95C22×C12711yescharacteristic
374C2×#95C22×C12711yescharacteristic
375C25C23×C12711yescharacteristic
376C25C23×C12711yescharacteristic
377C22×D4—24no—
378C22×D4—24no—
379C22×D4—41no—
380C22×D4—41no—
381C22×D4D4×C12714yes—
382C23×C4—22no—
383C23×C4—24no—
384C23×C4—22no—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—41no—
388C23×C4—41no—
389C23×C4—41no—
390C23×C4—41no—
391C23×C4—42no—
392C23×C4C23×C12712yes—
393C23×C4C23×C12712yes—
394C23×C4C23×C12712yes—
395C23×C4C23×C12712yes—
396C23×C4C23×C12711yescharacteristic
397C23×C4C23×C12712yes—
398C23×C4C23×C12712yes—
399#123—232no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—42no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—42no—
419C2×C22⋊C4—42no—
420C2×C22⋊C4—42no—
421C2×C22⋊C4D4×C12718yes—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—41no—
428#95—42no—
429#95—42no—
430#95—42no—
431#95—41no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24—21no—
443C24—21no—
444C24—22no—
445C24—22no—
446C24C2×D4×C12714yes—
447C24C2×D4×C12712yes—
448C24C2×D4×C12712yes—
449C24#41×C12714yes—
450C24C24×C12711yescharacteristic
451C2×D4—28no—
452C2×D4—28no—
453C2×D4—44no—
454C2×D4—48no—
455C2×D4—48no—
456C22×C4—28no—
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—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4C2×D4×C12714yes—
509C22×C4C2×D4×C12712yes—
510C22×C4C2×D4×C12714yes—
511C22×C4#41×C12714yes—
512C22×C4#41×C12714yes—
513C4⋊C4—216no—
514C4⋊C4—48no—
515C22⋊C4—28no—
516C22⋊C4—216no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23#120×C12714yes—
562C23#123×C12714yes—
563C23C22×D4×C12711yescharacteristic
564C23C22×D4×C12712yes—
565C23C2×#41×C12712yes—
566C23C2×#41×C12712yes—
567C23C2×#41×C12712yes—
568C23H(5,𝔽2)×C12711yescharacteristic
569C23H(5,𝔽2)×C12711yescharacteristic
570C23H(5,𝔽2)×C12712yes—
571C232−1+4×C12712yes—
572D4—416no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C22—24no—
623C22—24no—
624C22—28no—
625C22—28no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22C2×#120×C12711yescharacteristic
641C22C2×#123×C12711yescharacteristic
642C22#529×C12712yes—
643C22#532×C12712yes—
644C22#534×C12711yescharacteristic
645C22#537×C12712yes—
646C22#538×C12712yes—
647C22#539×C12712yes—
648C22#539×C12712yes—
649C22#546×C12712yes—
650C22#546×C12712yes—
651C22#546×C12712yes—
652C22#546×C12712yes—
653C22#547×C12712yes—
654C22#547×C12712yes—
655C22#551×C12712yes—
656C22#551×C12711yescharacteristic
657C22#552×C12712yes—
658C22#559×C12711yescharacteristic
659C22#560×C12711yescharacteristic
660C22#563×C12711yescharacteristic
661C4—44no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—82no—
667C4—82no—
668C2—44no—
669C2—44no—
670C2C127×#240511yescharacteristic
671C2C127×#241912yes—
672C2C127×#243012yes—
673C2C127×#255312yes—
674C2C127×#255712yes—
675C2C127×#269312yes—
676C2C127×#269512yes—
677C2C127×#269711yescharacteristic
678C2C127×#272411yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#397×C127C2semidirect
4. C127×#1442C2semidirect
5. C127×#1441C2semidirect
6. C127×#1439C2semidirect
8. C127×C22⋊#95C2semidirect
9. C127×C22⋊#95C2semidirect
328. #15463C127direct (N × H)
341. C2×#397C2×C127semidirect
343. #1442C2×C127semidirect
344. #1441C2×C127semidirect
345. #1439C2×C127semidirect
347. C22⋊#95C2×C127semidirect
348. C22⋊#95C2×C127semidirect
349. C127#15463direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#123×C1272
C2×#397×C1272
C2×#386×C1272
C127×#14422
C127×#14412
C127×#14392
C127×#14382
C127×C22⋊#952
#15463127