Finite Group Databasebeta

← C127×#16682 #44580355

Subgroups — C127×#16682 #44580355

Order 32512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125×C127C211yesmaximal, characteristic, complemented
2C2×#399×C127C211yesmaximal, characteristic
3C2×#397×C127C211yesmaximal, characteristic
4C2×#388×C127C211yesmaximal, characteristic
5C127×#1445C211yesmaximal, characteristic, complemented
6C127×#1445C211yesmaximal, characteristic, complemented
7C127×#1444C212yesmaximal, complemented
8C127×#1441C212yesmaximal, complemented
9C127×#1439C212yesmaximal
10C127×#1439C212yesmaximal
11C127×#1434C211yesmaximal, characteristic, complemented
12C2×#125×C127—24no—
13C2×#125×C127—24no—
14C2×#125×C127—24no—
15C22×C4⋊C4×C127C2212yes—
16C22×C4⋊C4×C127C2212yes—
17C22×C4⋊C4×C127C2212yes—
18C22×C4⋊C4×C127C2211yescharacteristic
19C22×C4⋊C4×C127C2211yescharacteristic
20C22×C4⋊C4×C127C2211yescharacteristic
21C22×C22⋊C4×C127C2211yescharacteristic
22C22×C22⋊C4×C127C2211yescharacteristic
23C22×C22⋊C4×C127C2212yes—
24C22×C22⋊C4×C127C2212yes—
25C22×C22⋊C4×C127C2211yescharacteristic
26C22×C42×C127C2211yescharacteristic
27#399×C127—24no—
28#397×C127—24no—
29#388×C127—24no—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2212yes—
32C2×#95×C127C2211yescharacteristic
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C2×#95×C127C2211yescharacteristic
36C2×#95×C127C2211yescharacteristic
37C2×#95×C127C2212yes—
38C2×#95×C127C2212yes—
39C2×#95×C127C2212yes—
40C2×#95×C127C2212yes—
41C25×C127C2311yescharacteristic
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2312yes—
48C23×C4×C127C2312yes—
49C23×C4×C127C2312yes—
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2311yescharacteristic
52#125×C127—232no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—24no—
55C2×C4⋊C4×C127—24no—
56C2×C4⋊C4×C127—24no—
57C2×C4⋊C4×C127—24no—
58C2×C4⋊C4×C127—24no—
59C2×C4⋊C4×C127—22no—
60C2×C4⋊C4×C127—22no—
61C2×C4⋊C4×C127—22no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127—24no—
65C2×C4⋊C4×C127—24no—
66C2×C4⋊C4×C127—42no—
67C2×C4⋊C4×C127—42no—
68C2×C4⋊C4×C127D414yes—
69C2×C4⋊C4×C127D414yes—
70C2×C22⋊C4×C127—22no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—22no—
73C2×C22⋊C4×C127—24no—
74C2×C22⋊C4×C127—22no—
75C2×C22⋊C4×C127—22no—
76C2×C22⋊C4×C127—24no—
77C2×C22⋊C4×C127—22no—
78C2×C22⋊C4×C127—24no—
79C2×C22⋊C4×C127—22no—
80C2×C22⋊C4×C127—24no—
81C2×C22⋊C4×C127—42no—
82C2×C22⋊C4×C127—42no—
83C2×C22⋊C4×C127—42no—
84C2×C42×C127—22no—
85C2×C42×C127—22no—
86C2×C42×C127D414yes—
87#95×C127—22no—
88#95×C127—22no—
89#95×C127—24no—
90#95×C127—22no—
91#95×C127—22no—
92#95×C127—24no—
93#95×C127—24no—
94#95×C127—42no—
95#95×C127—42no—
96#95×C127—42no—
97#95×C127—42no—
98C24×C127—21no—
99C24×C127—21no—
100C24×C127—21no—
101C24×C127—21no—
102C24×C127—22no—
103C24×C127—22no—
104C24×C127—21no—
105C24×C127—22no—
106C24×C127—21no—
107C24×C127—22no—
108C24×C127#4112yes—
109C24×C127C2411yescharacteristic
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—21no—
113C22×C4×C127—22no—
114C22×C4×C127—21no—
115C22×C4×C127—21no—
116C22×C4×C127—21no—
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—21no—
126C22×C4×C127—21no—
127C22×C4×C127—21no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—21no—
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—22no—
141C22×C4×C127—22no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—21no—
145C22×C4×C127—21no—
146C22×C4×C127—22no—
147C22×C4×C127—21no—
148C22×C4×C127—21no—
149C22×C4×C127—22no—
150C22×C4×C127—22no—
151C22×C4×C127—22no—
152C22×C4×C127—21no—
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—22no—
160C22×C4×C127—21no—
161C22×C4×C127C2×D412yes—
162C22×C4×C127C2×D414yes—
163C22×C4×C127C2×D412yes—
164C22×C4×C127C2×D414yes—
165C22×C4×C127C2×D412yes—
166C22×C4×C127C2×D412yes—
167C22×C4×C127C2×D412yes—
168C22×C4×C127#4112yes—
169C22×C4×C127#4114yes—
170C4⋊C4×C127—216no—
171C4⋊C4×C127—216no—
172C4⋊C4×C127—28no—
173C4⋊C4×C127—28no—
174C4⋊C4×C127—48no—
175C4⋊C4×C127—44no—
176C22⋊C4×C127—28no—
177C22⋊C4×C127—48no—
178C22⋊C4×C127—48no—
179C22⋊C4×C127—44no—
180C22⋊C4×C127—44no—
181C42×C127—28no—
182C23×C127—22no—
183C23×C127—24no—
184C23×C127—22no—
185C23×C127—22no—
186C23×C127—24no—
187C23×C127—41no—
188C23×C127—41no—
189C23×C127—41no—
190C23×C127—41no—
191C23×C127—41no—
192C23×C127—42no—
193C23×C127—42no—
194C23×C127—41no—
195C23×C127—42no—
196C23×C127—42no—
197C23×C127—42no—
198C23×C127—42no—
199C23×C127—41no—
200C23×C127—41no—
201C23×C127—42no—
202C23×C127—42no—
203C23×C127—42no—
204C23×C127—42no—
205C23×C127C22×D411yescharacteristic
206C23×C127C22×D411yescharacteristic
207C23×C127C22×D411yescharacteristic
208C23×C127C2×#4111yescharacteristic
209C23×C127C2×#4111yescharacteristic
210C23×C127C2×#4112yes—
211C23×C127H(5,𝔽2)12yes—
212C23×C127H(5,𝔽2)11yescharacteristic
213C23×C1272−1+412yes—
214C23×C1272−1+412yes—
215C23×C1272−1+411yescharacteristic
216C2×C4×C127—22no—
217C2×C4×C127—22no—
218C2×C4×C127—24no—
219C2×C4×C127—24no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—24no—
223C2×C4×C127—24no—
224C2×C4×C127—22no—
225C2×C4×C127—22no—
226C2×C4×C127—22no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—24no—
230C2×C4×C127—24no—
231C2×C4×C127—24no—
232C2×C4×C127—24no—
233C2×C4×C127—24no—
234C2×C4×C127—24no—
235C2×C4×C127—24no—
236C2×C4×C127—24no—
237C2×C4×C127—42no—
238C2×C4×C127—42no—
239C2×C4×C127—41no—
240C2×C4×C127—42no—
241C2×C4×C127—41no—
242C2×C4×C127—42no—
243C2×C4×C127—42no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—41no—
247C2×C4×C127—42no—
248C2×C4×C127—41no—
249C2×C4×C127—41no—
250C2×C4×C127—42no—
251C2×C4×C127—41no—
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—41no—
259C2×C4×C127—41no—
260C2×C4×C127—41no—
261C2×C4×C127—41no—
262C2×C4×C127—41no—
263C2×C4×C127—42no—
264C2×C4×C127—41no—
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#12014yes—
276C22×C127—42no—
277C22×C127—42no—
278C22×C127—42no—
279C22×C127—44no—
280C22×C127—44no—
281C22×C127—81no—
282C22×C127—81no—
283C22×C127—81no—
284C22×C127—81no—
285C22×C127—82no—
286C22×C127—82no—
287C22×C127C2×#12011yescharacteristic
288C22×C127#52511yescharacteristic
289C22×C127#53211yescharacteristic
290C22×C127#53511yescharacteristic
291C22×C127#53611yescharacteristic
292C22×C127#53611yescharacteristic
293C22×C127#53612yes—
294C22×C127#53612yes—
295C22×C127#53711yescharacteristic
296C22×C127#53912yes—
297C22×C127#53912yes—
298C22×C127#53912yes—
299C22×C127#54112yes—
300C22×C127#54611yescharacteristic
301C22×C127#54612yes—
302C22×C127#54711yescharacteristic
303C22×C127#54811yescharacteristic
304C22×C127#55011yescharacteristic
305C22×C127#55012yes—
306C22×C127#55212yes—
307C22×C127#55212yes—
308C22×C127#56212yes—
309C22×C127#56312yes—
310C4×C127—24no—
311C4×C127—42no—
312C4×C127—42no—
313C4×C127—44no—
314C4×C127—42no—
315C4×C127—44no—
316C4×C127—44no—
317C4×C127—81no—
318C4×C127—81no—
319C4×C127—82no—
320#16682C12711yesmaximal, characteristic, direct product factor, Sylow
321C2×C127—82no—
322C2×C127#240711yescharacteristic
323C2×C127#243111yescharacteristic
324C2×C127#261111yescharacteristic
325C2×C127#261411yescharacteristic
326C2×C127#263311yescharacteristic
327C2×C127#266411yescharacteristic
328C2×C127#269012yes—
329C2×C127#269412yes—
330C2×C127#269412yes—
331C2×C127#272112yes—
332C2×C127#282011yescharacteristic
333C22×#125C2×C12711yescharacteristic, complemented
334C2×#399C2×C12711yescharacteristic
335C2×#397C2×C12711yescharacteristic
336C2×#388C2×C12711yescharacteristic
337#1445C2×C12711yescharacteristic, complemented
338#1445C2×C12711yescharacteristic, complemented
339#1444C2×C12712yescomplemented
340#1441C2×C12712yescomplemented
341#1439C2×C12712yes—
342#1439C2×C12712yes—
343#1434C2×C12711yescharacteristic, complemented
344C127#1668211yescharacteristic, direct product factor, Sylow
345C2×#125—24no—
346C2×#125—24no—
347C2×#125—24no—
348C22×C4⋊C4C22×C12712yes—
349C22×C4⋊C4C22×C12712yes—
350C22×C4⋊C4C22×C12712yes—
351C22×C4⋊C4C22×C12711yescharacteristic
352C22×C4⋊C4C22×C12711yescharacteristic
353C22×C4⋊C4C22×C12711yescharacteristic
354C22×C22⋊C4C22×C12711yescharacteristic
355C22×C22⋊C4C22×C12711yescharacteristic
356C22×C22⋊C4C22×C12712yes—
357C22×C22⋊C4C22×C12712yes—
358C22×C22⋊C4C22×C12711yescharacteristic
359C22×C42C22×C12711yescharacteristic
360#399—24no—
361#397—24no—
362#388—24no—
363C2×#95C22×C12711yescharacteristic
364C2×#95C22×C12712yes—
365C2×#95C22×C12711yescharacteristic
366C2×#95C22×C12712yes—
367C2×#95C22×C12712yes—
368C2×#95C22×C12711yescharacteristic
369C2×#95C22×C12711yescharacteristic
370C2×#95C22×C12712yes—
371C2×#95C22×C12712yes—
372C2×#95C22×C12712yes—
373C2×#95C22×C12712yes—
374C25C23×C12711yescharacteristic
375C23×C4C23×C12711yescharacteristic
376C23×C4C23×C12712yes—
377C23×C4C23×C12711yescharacteristic
378C23×C4C23×C12712yes—
379C23×C4C23×C12712yes—
380C23×C4C23×C12711yescharacteristic
381C23×C4C23×C12711yescharacteristic
382C23×C4C23×C12712yes—
383C23×C4C23×C12711yescharacteristic
384C23×C4C23×C12711yescharacteristic
385#125—232no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—42no—
401C2×C4⋊C4D4×C12714yes—
402C2×C4⋊C4D4×C12714yes—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42D4×C12714yes—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—22no—
424#95—22no—
425#95—24no—
426#95—24no—
427#95—42no—
428#95—42no—
429#95—42no—
430#95—42no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24—22no—
437C24—21no—
438C24—22no—
439C24—21no—
440C24—22no—
441C24#41×C12712yes—
442C24C24×C12711yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
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—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4C2×D4×C12712yes—
495C22×C4C2×D4×C12714yes—
496C22×C4C2×D4×C12712yes—
497C22×C4C2×D4×C12714yes—
498C22×C4C2×D4×C12712yes—
499C22×C4C2×D4×C12712yes—
500C22×C4C2×D4×C12712yes—
501C22×C4#41×C12712yes—
502C22×C4#41×C12714yes—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C42—28no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C22×D4×C12711yescharacteristic
539C23C22×D4×C12711yescharacteristic
540C23C22×D4×C12711yescharacteristic
541C23C2×#41×C12711yescharacteristic
542C23C2×#41×C12711yescharacteristic
543C23C2×#41×C12712yes—
544C23H(5,𝔽2)×C12712yes—
545C23H(5,𝔽2)×C12711yescharacteristic
546C232−1+4×C12712yes—
547C232−1+4×C12712yes—
548C232−1+4×C12711yescharacteristic
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
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#120×C12714yes—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—82no—
619C22—81no—
620C22C2×#120×C12711yescharacteristic
621C22#525×C12711yescharacteristic
622C22#532×C12711yescharacteristic
623C22#535×C12711yescharacteristic
624C22#536×C12711yescharacteristic
625C22#536×C12711yescharacteristic
626C22#536×C12712yes—
627C22#536×C12712yes—
628C22#537×C12711yescharacteristic
629C22#539×C12712yes—
630C22#539×C12712yes—
631C22#539×C12712yes—
632C22#541×C12712yes—
633C22#546×C12711yescharacteristic
634C22#546×C12712yes—
635C22#547×C12711yescharacteristic
636C22#548×C12711yescharacteristic
637C22#550×C12711yescharacteristic
638C22#550×C12712yes—
639C22#552×C12712yes—
640C22#552×C12712yes—
641C22#562×C12712yes—
642C22#563×C12712yes—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—81no—
651C4—82no—
652C4—81no—
653C2—82no—
654C2C127×#240711yescharacteristic
655C2C127×#243111yescharacteristic
656C2C127×#261111yescharacteristic
657C2C127×#261411yescharacteristic
658C2C127×#263311yescharacteristic
659C2C127×#266411yescharacteristic
660C2C127×#269012yes—
661C2C127×#269412yes—
662C2C127×#269412yes—
663C2C127×#272112yes—
664C2C127×#282011yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#125×C127C2semidirect
5. C127×#1445C2semidirect
6. C127×#1445C2semidirect
7. C127×#1444C2semidirect
8. C127×#1441C2semidirect
11. C127×#1434C2semidirect
320. #16682C127direct (N × H)
333. C22×#125C2×C127semidirect
337. #1445C2×C127semidirect
338. #1445C2×C127semidirect
339. #1444C2×C127semidirect
340. #1441C2×C127semidirect
343. #1434C2×C127semidirect
344. C127#16682direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#125×C1272
C2×#399×C1272
C2×#397×C1272
C2×#388×C1272
C127×#14452
C127×#14442
C127×#14412
C127×#14392
C127×#14342
#16682127