Finite Group Databasebeta

← C127×#16684 #44580357

Subgroups — C127×#16684 #44580357

Order 32512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394×C127C211yesmaximal, characteristic
2C2×#388×C127C212yesmaximal, complemented
3C2×#382×C127C211yesmaximal, characteristic, complemented
4C127×#1442C212yesmaximal, complemented
5C127×#1441C211yesmaximal, characteristic, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1439C212yesmaximal, complemented
8C127×#1439C212yesmaximal, complemented
9C127×#1438C212yesmaximal
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×C127—24no—
23#388×C127—28no—
24#382×C127—24no—
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2212yes—
32C2×#95×C127C2211yescharacteristic
33C2×#95×C127C2212yes—
34C25×C127C2312yes—
35C22×D4×C127—24no—
36C22×D4×C127—22no—
37C22×D4×C127—42no—
38C22×D4×C127—41no—
39C22×D4×C127—41no—
40C23×C4×C127C2312yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2312yes—
47C23×C4×C127C2311yescharacteristic
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—41no—
52C2×C4⋊C4×C127—41no—
53C2×C4⋊C4×C127—41no—
54C2×C4⋊C4×C127—42no—
55C2×C4⋊C4×C127—42no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127—41no—
58C2×C4⋊C4×C127D414yes—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127—24no—
64C2×C22⋊C4×C127—24no—
65C2×C22⋊C4×C127—24no—
66C2×C22⋊C4×C127—24no—
67C2×C22⋊C4×C127—24no—
68C2×C22⋊C4×C127—24no—
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×C127—42no—
75C2×C22⋊C4×C127—42no—
76C2×C22⋊C4×C127—42no—
77C2×C42×C127—22no—
78C2×C42×C127—22no—
79C2×C42×C127D414yescomplemented
80#95×C127—24no—
81#95×C127—24no—
82#95×C127—24no—
83#95×C127—24no—
84#95×C127—22no—
85#95×C127—24no—
86#95×C127—41no—
87#95×C127—42no—
88#95×C127—42no—
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×C127—22no—
103C24×C127#4114yes—
104C24×C127C2411yescharacteristic
105C2×D4×C127—48no—
106C2×D4×C127—44no—
107C2×D4×C127—44no—
108C2×D4×C127—48no—
109C2×D4×C127—44no—
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127—21no—
113C22×C4×C127—21no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
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—21no—
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—21no—
131C22×C4×C127—21no—
132C22×C4×C127—21no—
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×C127—22no—
155C22×C4×C127C2×D412yes—
156C22×C4×C127C2×D412yes—
157C22×C4×C127C2×D414yes—
158C22×C4×C127C2×D412yes—
159C22×C4×C127C2×D412yes—
160C22×C4×C127#4114yes—
161C22×C4×C127#4112yes—
162C4⋊C4×C127—44no—
163C4⋊C4×C127—44no—
164C4⋊C4×C127—48no—
165C4⋊C4×C127—44no—
166C22⋊C4×C127—216no—
167C22⋊C4×C127—48no—
168C22⋊C4×C127—48no—
169C22⋊C4×C127—48no—
170C22⋊C4×C127—48no—
171C22⋊C4×C127—48no—
172C42×C127—28no—
173C23×C127—24no—
174C23×C127—24no—
175C23×C127—24no—
176C23×C127—24no—
177C23×C127—24no—
178C23×C127—24no—
179C23×C127—24no—
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×C127—42no—
208C23×C127C22×D411yescharacteristic
209C23×C127C22×D411yescharacteristic
210C23×C127C2×#4111yescharacteristic
211C23×C127C2×#4112yes—
212C23×C127C2×#4112yes—
213C23×C127H(5,𝔽2)11yescharacteristic
214C23×C127H(5,𝔽2)11yescharacteristic
215C23×C127H(5,𝔽2)12yes—
216C23×C127H(5,𝔽2)12yes—
217C23×C1272−1+411yescharacteristic
218C23×C1272−1+411yescharacteristic
219D4×C127—416no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—22no—
223C2×C4×C127—22no—
224C2×C4×C127—22no—
225C2×C4×C127—22no—
226C2×C4×C127—24no—
227C2×C4×C127—24no—
228C2×C4×C127—24no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—24no—
232C2×C4×C127—24no—
233C2×C4×C127—24no—
234C2×C4×C127—24no—
235C2×C4×C127—24no—
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—41no—
245C2×C4×C127—41no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—41no—
249C2×C4×C127—41no—
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#12114yes—
272C22×C127—44no—
273C22×C127—44no—
274C22×C127—44no—
275C22×C127—44no—
276C22×C127—44no—
277C22×C127—44no—
278C22×C127—44no—
279C22×C127—82no—
280C22×C127—82no—
281C22×C127—82no—
282C22×C127—82no—
283C22×C127—82no—
284C22×C127—82no—
285C22×C127—82no—
286C22×C127—82no—
287C22×C127C2×#12111yescharacteristic
288C22×C127#52512yes—
289C22×C127#53411yescharacteristic
290C22×C127#53512yes—
291C22×C127#53611yescharacteristic
292C22×C127#53711yescharacteristic
293C22×C127#53711yescharacteristic
294C22×C127#53812yes—
295C22×C127#53812yes—
296C22×C127#53911yescharacteristic
297C22×C127#53911yescharacteristic
298C22×C127#53912yes—
299C22×C127#53912yes—
300C22×C127#54612yes—
301C22×C127#55012yes—
302C22×C127#55111yescharacteristic
303C22×C127#55212yes—
304C22×C127#55312yes—
305C22×C127#55511yescharacteristic
306C22×C127#55512yes—
307C22×C127H(3,𝔽4)11yescharacteristic
308C22×C127H(3,𝔽4)11yescharacteristic
309C22×C127#56212yes—
310C4×C127—24no—
311C4×C127—44no—
312C4×C127—44no—
313C4×C127—42no—
314C4×C127—42no—
315C4×C127—82no—
316C4×C127—82no—
317C4×C127—82no—
318#16684C12711yesmaximal, characteristic, direct product factor, Sylow
319C2×C127—84no—
320C2×C127#241412yes—
321C2×C127#260911yescharacteristic
322C2×C127#261811yescharacteristic
323C2×C127#263612yes—
324C2×C127#268411yescharacteristic
325C2×C127#269412yes—
326C2×C127#273311yescharacteristic
327C2×C127#278112yes—
328C2×C127#278211yescharacteristic
329C2×C127#278511yescharacteristic
330C2×C127#281911yescharacteristic
331C2×#394C2×C12711yescharacteristic
332C2×#388C2×C12712yescomplemented
333C2×#382C2×C12711yescharacteristic, complemented
334#1442C2×C12712yescomplemented
335#1441C2×C12711yescharacteristic, complemented
336#1439C2×C12712yescomplemented
337#1439C2×C12712yescomplemented
338#1439C2×C12712yescomplemented
339#1438C2×C12712yes—
340C127#1668411yescharacteristic, direct product factor, Sylow
341C23×D4C22×C12711yescharacteristic
342C22×C4⋊C4C22×C12711yescharacteristic
343C22×C4⋊C4C22×C12712yes—
344C22×C4⋊C4C22×C12711yescharacteristic
345C22×C4⋊C4C22×C12711yescharacteristic
346C22×C22⋊C4C22×C12712yes—
347C22×C22⋊C4C22×C12712yes—
348C22×C22⋊C4C22×C12712yes—
349C22×C22⋊C4C22×C12712yes—
350C22×C22⋊C4C22×C12712yes—
351C22×C22⋊C4C22×C12712yes—
352C22×C42C22×C12711yescharacteristic
353#394—24no—
354#388—28no—
355#382—24no—
356C2×#95C22×C12711yescharacteristic
357C2×#95C22×C12712yes—
358C2×#95C22×C12712yes—
359C2×#95C22×C12712yes—
360C2×#95C22×C12712yes—
361C2×#95C22×C12712yes—
362C2×#95C22×C12712yes—
363C2×#95C22×C12711yescharacteristic
364C2×#95C22×C12712yes—
365C25C23×C12712yes—
366C22×D4—24no—
367C22×D4—22no—
368C22×D4—42no—
369C22×D4—41no—
370C22×D4—41no—
371C23×C4C23×C12712yes—
372C23×C4C23×C12712yes—
373C23×C4C23×C12712yes—
374C23×C4C23×C12711yescharacteristic
375C23×C4C23×C12712yes—
376C23×C4C23×C12711yescharacteristic
377C23×C4C23×C12712yes—
378C23×C4C23×C12711yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—41no—
384C2×C4⋊C4—41no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—42no—
387C2×C4⋊C4—42no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4D4×C12714yes—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42D4×C12714yescomplemented
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24#41×C12714yes—
435C24C24×C12711yescharacteristic
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—48no—
440C2×D4—44no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
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—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
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×C4—22no—
485C22×C4—22no—
486C22×C4C2×D4×C12712yes—
487C22×C4C2×D4×C12712yes—
488C22×C4C2×D4×C12714yes—
489C22×C4C2×D4×C12712yes—
490C22×C4C2×D4×C12712yes—
491C22×C4#41×C12714yes—
492C22×C4#41×C12712yes—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C22⋊C4—216no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C42—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
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—
537C23—42no—
538C23—42no—
539C23C22×D4×C12711yescharacteristic
540C23C22×D4×C12711yescharacteristic
541C23C2×#41×C12711yescharacteristic
542C23C2×#41×C12712yes—
543C23C2×#41×C12712yes—
544C23H(5,𝔽2)×C12711yescharacteristic
545C23H(5,𝔽2)×C12711yescharacteristic
546C23H(5,𝔽2)×C12712yes—
547C23H(5,𝔽2)×C12712yes—
548C232−1+4×C12711yescharacteristic
549C232−1+4×C12711yescharacteristic
550D4—416no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
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—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
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×C4—42no—
601C2×C4—42no—
602C2×C4#121×C12714yes—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#121×C12711yescharacteristic
619C22#525×C12712yes—
620C22#534×C12711yescharacteristic
621C22#535×C12712yes—
622C22#536×C12711yescharacteristic
623C22#537×C12711yescharacteristic
624C22#537×C12711yescharacteristic
625C22#538×C12712yes—
626C22#538×C12712yes—
627C22#539×C12711yescharacteristic
628C22#539×C12711yescharacteristic
629C22#539×C12712yes—
630C22#539×C12712yes—
631C22#546×C12712yes—
632C22#550×C12712yes—
633C22#551×C12711yescharacteristic
634C22#552×C12712yes—
635C22#553×C12712yes—
636C22#555×C12711yescharacteristic
637C22#555×C12712yes—
638C22H(3,𝔽4)×C12711yescharacteristic
639C22H(3,𝔽4)×C12711yescharacteristic
640C22#562×C12712yes—
641C4—24no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4—82no—
649C2—84no—
650C2C127×#241412yes—
651C2C127×#260911yescharacteristic
652C2C127×#261811yescharacteristic
653C2C127×#263612yes—
654C2C127×#268411yescharacteristic
655C2C127×#269412yes—
656C2C127×#273311yescharacteristic
657C2C127×#278112yes—
658C2C127×#278211yescharacteristic
659C2C127×#278511yescharacteristic
660C2C127×#281911yescharacteristic
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×#388×C127C2semidirect
3. C2×#382×C127C2semidirect
4. C127×#1442C2semidirect
5. C127×#1441C2semidirect
6. C127×#1439C2semidirect
7. C127×#1439C2semidirect
8. C127×#1439C2semidirect
79. C2×C42×C127D4semidirect
318. #16684C127direct (N × H)
332. C2×#388C2×C127semidirect
333. C2×#382C2×C127semidirect
334. #1442C2×C127semidirect
335. #1441C2×C127semidirect
336. #1439C2×C127semidirect
337. #1439C2×C127semidirect
338. #1439C2×C127semidirect
340. C127#16684direct (N × H)
410. 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
C2×#382×C1272
C127×#14422
C127×#14412
C127×#14392
C127×#14382
#16684127