Finite Group Databasebeta

← C127×#14632 #44578305

Subgroups — C127×#14632 #44578305

Order 32512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394×C127C211yesmaximal, characteristic
2C2×#388×C127C212yesmaximal, complemented
3C127×#1442C212yesmaximal, complemented
4C127×#1441C211yesmaximal, characteristic, complemented
5C127×#1439C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1439C212yesmaximal, complemented
8C127×#1438C212yesmaximal
9C127×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2212yes—
20C22×C22⋊C4×C127C2212yes—
21C22×C42×C127C2211yescharacteristic
22#394×C127C418yescomplemented
23#388×C127—28no—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2211yescharacteristic
30C2×#95×C127C2212yes—
31C2×#95×C127C2212yes—
32C2×#95×C127C2211yescharacteristic
33C25×C127C2312yes—
34C22×D4×C127—42no—
35C22×D4×C127—42no—
36C22×D4×C127—41no—
37C22×D4×C127—41no—
38C22×D4×C127C2×C414yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2311yescharacteristic
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—41no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—41no—
52C2×C4⋊C4×C127—41no—
53C2×C4⋊C4×C127—42no—
54C2×C4⋊C4×C127—42no—
55C2×C4⋊C4×C127—42no—
56C2×C4⋊C4×C127—41no—
57C2×C4⋊C4×C127C2×C414yes—
58C2×C22⋊C4×C127—24no—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—42no—
63C2×C22⋊C4×C127—42no—
64C2×C22⋊C4×C127—42no—
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×C127—42no—
71C2×C22⋊C4×C127—42no—
72C2×C22⋊C4×C127—42no—
73C2×C22⋊C4×C127—42no—
74C2×C22⋊C4×C127C2×C418yes—
75C2×C22⋊C4×C127C2×C418yes—
76C2×C42×C127—41no—
77C2×C42×C127—41no—
78C2×C42×C127D414yescomplemented
79#95×C127—24no—
80#95×C127—24no—
81#95×C127—42no—
82#95×C127—42no—
83#95×C127—42no—
84#95×C127—41no—
85#95×C127—42no—
86#95×C127—42no—
87#95×C127C2×C414yes—
88C24×C127—22no—
89C24×C127—22no—
90C24×C127—22no—
91C24×C127—22no—
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—22no—
101C24×C127—22no—
102C24×C127C22×C414yes—
103C24×C127C2411yescharacteristic
104C2×D4×C127—44no—
105C2×D4×C127—48no—
106C2×D4×C127—44no—
107C2×D4×C127—48no—
108C2×D4×C127—44no—
109C22×C4×C127—21no—
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127—21no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—21no—
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—21no—
129C22×C4×C127—21no—
130C22×C4×C127—21no—
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—22no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127C22×C414yes—
155C22×C4×C127C22×C412yes—
156C22×C4×C127C22×C412yes—
157C22×C4×C127C22×C412yes—
158C22×C4×C127C2×D414yes—
159C22×C4×C127C2×D412yes—
160C22×C4×C127#4112yes—
161C4⋊C4×C127—44no—
162C4⋊C4×C127—44no—
163C4⋊C4×C127—48no—
164C4⋊C4×C127—44no—
165C22⋊C4×C127—216no—
166C22⋊C4×C127—48no—
167C22⋊C4×C127—48no—
168C22⋊C4×C127—48no—
169C22⋊C4×C127—48no—
170C22⋊C4×C127—48no—
171C42×C127—44no—
172C23×C127—24no—
173C23×C127—24no—
174C23×C127—24no—
175C23×C127—24no—
176C23×C127—24no—
177C23×C127—24no—
178C23×C127—24no—
179C23×C127—42no—
180C23×C127—42no—
181C23×C127—42no—
182C23×C127—42no—
183C23×C127—42no—
184C23×C127—42no—
185C23×C127—42no—
186C23×C127—42no—
187C23×C127—42no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127—42no—
192C23×C127—42no—
193C23×C127—42no—
194C23×C127—42no—
195C23×C127—42no—
196C23×C127—42no—
197C23×C127—42no—
198C23×C127—42no—
199C23×C127—42no—
200C23×C127—42no—
201C23×C127—42no—
202C23×C127—42no—
203C23×C127—42no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—42no—
207C23×C127C23×C411yescharacteristic
208C23×C127C22×D411yescharacteristic
209C23×C127C2×#4111yescharacteristic
210C23×C127H(5,𝔽2)11yescharacteristic
211C23×C127H(5,𝔽2)12yes—
212C23×C127H(5,𝔽2)11yescharacteristic
213C23×C127H(5,𝔽2)12yes—
214C23×C127H(5,𝔽2)11yescharacteristic
215C23×C127H(5,𝔽2)12yes—
216C23×C1272−1+411yescharacteristic
217C23×C1272−1+412yes—
218D4×C127—416no—
219C2×C4×C127—22no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—22no—
223C2×C4×C127—22no—
224C2×C4×C127—22no—
225C2×C4×C127—24no—
226C2×C4×C127—24no—
227C2×C4×C127—24no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—24no—
231C2×C4×C127—24no—
232C2×C4×C127—24no—
233C2×C4×C127—24no—
234C2×C4×C127—24no—
235C2×C4×C127—42no—
236C2×C4×C127—42no—
237C2×C4×C127—42no—
238C2×C4×C127—42no—
239C2×C4×C127—41no—
240C2×C4×C127—42no—
241C2×C4×C127—42no—
242C2×C4×C127—42no—
243C2×C4×C127—41no—
244C2×C4×C127—42no—
245C2×C4×C127—41no—
246C2×C4×C127—41no—
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—
267C2×C4×C127—42no—
268C2×C4×C127—42no—
269C2×C4×C127—42no—
270C2×C4×C127C4×D414yes—
271C22×C127—44no—
272C22×C127—44no—
273C22×C127—44no—
274C22×C127—44no—
275C22×C127—44no—
276C22×C127—44no—
277C22×C127—44no—
278C22×C127—82no—
279C22×C127—82no—
280C22×C127—82no—
281C22×C127—82no—
282C22×C127—82no—
283C22×C127—82no—
284C22×C127—82no—
285C22×C127—82no—
286C22×C127C2×C4×D411yescharacteristic
287C22×C127#51812yes—
288C22×C127#51811yescharacteristic
289C22×C127#52012yes—
290C22×C127#52011yescharacteristic
291C22×C127#53411yescharacteristic
292C22×C127#53512yes—
293C22×C127#53711yescharacteristic
294C22×C127#53712yes—
295C22×C127#53811yescharacteristic
296C22×C127#53911yescharacteristic
297C22×C127#54012yes—
298C22×C127#54212yes—
299C22×C127#56011yescharacteristic
300C22×C127#56012yes—
301C22×C127#56012yes—
302C22×C127H(3,𝔽4)11yescharacteristic
303C22×C127H(3,𝔽4)11yescharacteristic
304C22×C127#56212yes—
305C22×C127#56212yes—
306C22×C127#56212yes—
307C22×C127#56311yescharacteristic
308C22×C127#56312yes—
309C4×C127—24no—
310C4×C127—44no—
311C4×C127—42no—
312C4×C127—42no—
313C4×C127—44no—
314C4×C127—82no—
315C4×C127—82no—
316C4×C127—82no—
317#14632C12711yesmaximal, characteristic, direct product factor, Sylow
318C2×C127—84no—
319C2×C127#232412yes—
320C2×C127#232511yescharacteristic
321C2×C127#237511yescharacteristic
322C2×C127#237612yes—
323C2×C127#237911yescharacteristic
324C2×C127#281312yes—
325C2×C127#281911yescharacteristic
326C2×C127#282311yescharacteristic
327C2×C127#282811yescharacteristic
328C2×C127#283211yescharacteristic
329C2×C127#283712yes—
330C2×#394C2×C12711yescharacteristic
331C2×#388C2×C12712yescomplemented
332#1442C2×C12712yescomplemented
333#1441C2×C12711yescharacteristic, complemented
334#1439C2×C12712yescomplemented
335#1439C2×C12712yescomplemented
336#1439C2×C12712yescomplemented
337#1438C2×C12712yes—
338C4⋊#95C2×C12711yescharacteristic, complemented
339C127#1463211yescharacteristic, direct product factor, Sylow
340C23×D4C22×C12711yescharacteristic
341C22×C4⋊C4C22×C12711yescharacteristic
342C22×C4⋊C4C22×C12712yes—
343C22×C4⋊C4C22×C12711yescharacteristic
344C22×C4⋊C4C22×C12711yescharacteristic
345C22×C22⋊C4C22×C12712yes—
346C22×C22⋊C4C22×C12712yes—
347C22×C22⋊C4C22×C12712yes—
348C22×C22⋊C4C22×C12712yes—
349C22×C22⋊C4C22×C12712yes—
350C22×C22⋊C4C22×C12712yes—
351C22×C42C22×C12711yescharacteristic
352#394C4×C12718yescomplemented
353#388—28no—
354C2×#95C22×C12712yes—
355C2×#95C22×C12712yes—
356C2×#95C22×C12712yes—
357C2×#95C22×C12712yes—
358C2×#95C22×C12712yes—
359C2×#95C22×C12711yescharacteristic
360C2×#95C22×C12712yes—
361C2×#95C22×C12712yes—
362C2×#95C22×C12711yescharacteristic
363C25C23×C12712yes—
364C22×D4—42no—
365C22×D4—42no—
366C22×D4—41no—
367C22×D4—41no—
368C22×D4C2×C4×C12714yes—
369C23×C4C23×C12712yes—
370C23×C4C23×C12712yes—
371C23×C4C23×C12712yes—
372C23×C4C23×C12711yescharacteristic
373C23×C4C23×C12712yes—
374C23×C4C23×C12711yescharacteristic
375C23×C4C23×C12712yes—
376C23×C4C23×C12711yescharacteristic
377C2×C4⋊C4—24no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—42no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—41no—
387C2×C4⋊C4C2×C4×C12714yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—42no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4C2×C4×C12718yes—
405C2×C22⋊C4C2×C4×C12718yes—
406C2×C42—41no—
407C2×C42—41no—
408C2×C42D4×C12714yescomplemented
409#95—24no—
410#95—24no—
411#95—42no—
412#95—42no—
413#95—42no—
414#95—41no—
415#95—42no—
416#95—42no—
417#95C2×C4×C12714yes—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C22×C4×C12714yes—
433C24C24×C12711yescharacteristic
434C2×D4—44no—
435C2×D4—48no—
436C2×D4—44no—
437C2×D4—48no—
438C2×D4—44no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
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—21no—
459C22×C4—21no—
460C22×C4—21no—
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—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4C22×C4×C12714yes—
485C22×C4C22×C4×C12712yes—
486C22×C4C22×C4×C12712yes—
487C22×C4C22×C4×C12712yes—
488C22×C4C2×D4×C12714yes—
489C22×C4C2×D4×C12712yes—
490C22×C4#41×C12712yes—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C42—44no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×C4×C12711yescharacteristic
538C23C22×D4×C12711yescharacteristic
539C23C2×#41×C12711yescharacteristic
540C23H(5,𝔽2)×C12711yescharacteristic
541C23H(5,𝔽2)×C12712yes—
542C23H(5,𝔽2)×C12711yescharacteristic
543C23H(5,𝔽2)×C12712yes—
544C23H(5,𝔽2)×C12711yescharacteristic
545C23H(5,𝔽2)×C12712yes—
546C232−1+4×C12711yescharacteristic
547C232−1+4×C12712yes—
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
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—
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×C4C4×D4×C12714yes—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×C4×D4×C12711yescharacteristic
617C22#518×C12712yes—
618C22#518×C12711yescharacteristic
619C22#520×C12712yes—
620C22#520×C12711yescharacteristic
621C22#534×C12711yescharacteristic
622C22#535×C12712yes—
623C22#537×C12712yes—
624C22#537×C12711yescharacteristic
625C22#538×C12711yescharacteristic
626C22#539×C12711yescharacteristic
627C22#540×C12712yes—
628C22#542×C12712yes—
629C22#560×C12711yescharacteristic
630C22#560×C12712yes—
631C22#560×C12712yes—
632C22H(3,𝔽4)×C12711yescharacteristic
633C22H(3,𝔽4)×C12711yescharacteristic
634C22#562×C12712yes—
635C22#562×C12712yes—
636C22#562×C12712yes—
637C22#563×C12711yescharacteristic
638C22#563×C12712yes—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C2—84no—
648C2C127×#232412yes—
649C2C127×#232511yescharacteristic
650C2C127×#237511yescharacteristic
651C2C127×#237612yes—
652C2C127×#237911yescharacteristic
653C2C127×#281312yes—
654C2C127×#281911yescharacteristic
655C2C127×#282311yescharacteristic
656C2C127×#282811yescharacteristic
657C2C127×#283211yescharacteristic
658C2C127×#283712yes—
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. C2×#388×C127C2semidirect
3. C127×#1442C2semidirect
4. C127×#1441C2semidirect
5. C127×#1439C2semidirect
6. C127×#1439C2semidirect
7. C127×#1439C2semidirect
9. C127×C4⋊#95C2semidirect
22. #394×C127C4semidirect
78. C2×C42×C127D4semidirect
317. #14632C127direct (N × H)
331. C2×#388C2×C127semidirect
332. #1442C2×C127semidirect
333. #1441C2×C127semidirect
334. #1439C2×C127semidirect
335. #1439C2×C127semidirect
336. #1439C2×C127semidirect
338. C4⋊#95C2×C127semidirect
339. C127#14632direct (N × H)
352. #394C4×C127semidirect
408. C2×C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#394×C1272
C2×#388×C1272
C127×#14422
C127×#14412
C127×#14392
C127×#14382
C127×C4⋊#952
#14632127