Finite Group Databasebeta

← C127×#17121 #44580794

Subgroups — C127×#17121 #44580794

Order 32512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#397×C127C212yesmaximal, complemented
2C2×#396×C127C212yesmaximal, complemented
3C127×#1445C211yesmaximal, characteristic, complemented
4C127×#1444C211yesmaximal, characteristic, complemented
5C127×#1442C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1439C212yesmaximal, complemented
8C127×#1438C212yesmaximal
9C127×#1438C211yesmaximal, characteristic
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
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—
21#397×C127—28no—
22#396×C127—28no—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2212yes—
29C2×#95×C127C2211yescharacteristic
30C2×#95×C127C2212yes—
31C2×#95×C127C2211yescharacteristic
32C2×#95×C127C2211yescharacteristic
33C25×C127C2312yes—
34C22×D4×C127—24no—
35C22×D4×C127—22no—
36C22×D4×C127—42no—
37C22×D4×C127—41no—
38C22×D4×C127—41no—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2312yes—
44C23×C4×C127C2312yes—
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2311yescharacteristic
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—24no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—41no—
52C2×C4⋊C4×C127—42no—
53C2×C4⋊C4×C127—42no—
54C2×C4⋊C4×C127—42no—
55C2×C4⋊C4×C127—41no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127D414yescomplemented
58C2×C22⋊C4×C127—24no—
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—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×C127—42no—
75C2×C22⋊C4×C127—42no—
76#95×C127—24no—
77#95×C127—24no—
78#95×C127—24no—
79#95×C127—24no—
80#95×C127—41no—
81#95×C127—42no—
82#95×C127—41no—
83#95×C127—41no—
84#95×C127—42no—
85#95×C127D414yescomplemented
86C24×C127—22no—
87C24×C127—22no—
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#4114yes—
101C24×C127C2411yescharacteristic
102C2×D4×C127—48no—
103C2×D4×C127—44no—
104C2×D4×C127—44no—
105C2×D4×C127—48no—
106C2×D4×C127—44no—
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—21no—
114C22×C4×C127—21no—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
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—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—22no—
152C22×C4×C127C2×D412yes—
153C22×C4×C127C2×D414yes—
154C22×C4×C127C2×D414yes—
155C22×C4×C127C2×D412yes—
156C22×C4×C127#4112yes—
157C4⋊C4×C127—44no—
158C4⋊C4×C127—48no—
159C4⋊C4×C127—48no—
160C4⋊C4×C127—44no—
161C22⋊C4×C127—48no—
162C22⋊C4×C127—48no—
163C22⋊C4×C127—48no—
164C22⋊C4×C127—48no—
165C22⋊C4×C127—48no—
166C22⋊C4×C127—48no—
167C23×C127—24no—
168C23×C127—24no—
169C23×C127—24no—
170C23×C127—24no—
171C23×C127—24no—
172C23×C127—24no—
173C23×C127—24no—
174C23×C127—42no—
175C23×C127—42no—
176C23×C127—42no—
177C23×C127—42no—
178C23×C127—42no—
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×C127C22×D411yescharacteristic
203C23×C127C22×D411yescharacteristic
204C23×C127C2×#4112yes—
205C23×C127C2×#4111yescharacteristic
206C23×C127H(5,𝔽2)11yescharacteristic
207C23×C127H(5,𝔽2)11yescharacteristic
208C23×C127H(5,𝔽2)12yes—
209C23×C127H(5,𝔽2)12yes—
210C23×C127H(5,𝔽2)11yescharacteristic
211C23×C1272−1+411yescharacteristic
212C23×C1272−1+412yes—
213D4×C127—416no—
214C2×C4×C127—24no—
215C2×C4×C127—22no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—22no—
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—24no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127—41no—
235C2×C4×C127—41no—
236C2×C4×C127—41no—
237C2×C4×C127—41no—
238C2×C4×C127—42no—
239C2×C4×C127—41no—
240C2×C4×C127—41no—
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—
264C22×C127—44no—
265C22×C127—44no—
266C22×C127—44no—
267C22×C127—44no—
268C22×C127—44no—
269C22×C127—44no—
270C22×C127—44no—
271C22×C127—82no—
272C22×C127—82no—
273C22×C127—82no—
274C22×C127—82no—
275C22×C127—82no—
276C22×C127—82no—
277C22×C127—82no—
278C22×C127—82no—
279C22×C127#53411yescharacteristic
280C22×C127#53512yes—
281C22×C127#53712yes—
282C22×C127#53711yescharacteristic
283C22×C127#53812yes—
284C22×C127#53811yescharacteristic
285C22×C127#53912yes—
286C22×C127#53912yes—
287C22×C127#54212yes—
288C22×C127D4211yescharacteristic
289C22×C127#54612yes—
290C22×C127#54711yescharacteristic
291C22×C127#54812yes—
292C22×C127#55011yescharacteristic
293C22×C127#55111yescharacteristic
294C22×C127#55112yes—
295C22×C127#56011yescharacteristic
296C22×C127#56012yes—
297C22×C127#56011yescharacteristic
298C22×C127#56211yescharacteristic
299C22×C127#56212yes—
300C22×C127#56312yes—
301C22×C127#56311yescharacteristic
302C4×C127—44no—
303C4×C127—44no—
304C4×C127—42no—
305C4×C127—42no—
306C4×C127—42no—
307C4×C127—82no—
308C4×C127—82no—
309C4×C127—82no—
310#17121C12711yesmaximal, characteristic, direct product factor, Sylow
311C2×C127—84no—
312C2×C127#267612yes—
313C2×C127#268211yescharacteristic
314C2×C127#269311yescharacteristic
315C2×C127#270512yes—
316C2×C127#272312yes—
317C2×C127#272811yescharacteristic
318C2×C127#273511yescharacteristic
319C2×C127#281511yescharacteristic
320C2×C127#282311yescharacteristic
321C2×C127#282712yes—
322C2×C127#283011yescharacteristic
323C2×#397C2×C12712yescomplemented
324C2×#396C2×C12712yescomplemented
325#1445C2×C12711yescharacteristic, complemented
326#1444C2×C12711yescharacteristic, complemented
327#1442C2×C12712yescomplemented
328#1439C2×C12712yescomplemented
329#1439C2×C12712yescomplemented
330#1438C2×C12712yes—
331#1438C2×C12711yescharacteristic
332C127#1712111yescharacteristic, direct product factor, Sylow
333C23×D4C22×C12711yescharacteristic
334C22×C4⋊C4C22×C12712yes—
335C22×C4⋊C4C22×C12712yes—
336C22×C4⋊C4C22×C12711yescharacteristic
337C22×C4⋊C4C22×C12711yescharacteristic
338C22×C22⋊C4C22×C12712yes—
339C22×C22⋊C4C22×C12712yes—
340C22×C22⋊C4C22×C12712yes—
341C22×C22⋊C4C22×C12712yes—
342C22×C22⋊C4C22×C12712yes—
343C22×C22⋊C4C22×C12712yes—
344#397—28no—
345#396—28no—
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×C12711yescharacteristic
355C2×#95C22×C12711yescharacteristic
356C25C23×C12712yes—
357C22×D4—24no—
358C22×D4—22no—
359C22×D4—42no—
360C22×D4—41no—
361C22×D4—41no—
362C23×C4C23×C12712yes—
363C23×C4C23×C12712yes—
364C23×C4C23×C12712yes—
365C23×C4C23×C12711yescharacteristic
366C23×C4C23×C12712yes—
367C23×C4C23×C12712yes—
368C23×C4C23×C12711yescharacteristic
369C23×C4C23×C12711yescharacteristic
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4D4×C12714yescomplemented
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
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—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—41no—
404#95—42no—
405#95—41no—
406#95—41no—
407#95—42no—
408#95D4×C12714yescomplemented
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24#41×C12714yes—
424C24C24×C12711yescharacteristic
425C2×D4—48no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—44no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
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—21no—
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×C4C2×D4×C12712yes—
476C22×C4C2×D4×C12714yes—
477C22×C4C2×D4×C12714yes—
478C22×C4C2×D4×C12712yes—
479C22×C4#41×C12712yes—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
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—
525C23C22×D4×C12711yescharacteristic
526C23C22×D4×C12711yescharacteristic
527C23C2×#41×C12712yes—
528C23C2×#41×C12711yescharacteristic
529C23H(5,𝔽2)×C12711yescharacteristic
530C23H(5,𝔽2)×C12711yescharacteristic
531C23H(5,𝔽2)×C12712yes—
532C23H(5,𝔽2)×C12712yes—
533C23H(5,𝔽2)×C12711yescharacteristic
534C232−1+4×C12711yescharacteristic
535C232−1+4×C12712yes—
536D4—416no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
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—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22#534×C12711yescharacteristic
603C22#535×C12712yes—
604C22#537×C12712yes—
605C22#537×C12711yescharacteristic
606C22#538×C12712yes—
607C22#538×C12711yescharacteristic
608C22#539×C12712yes—
609C22#539×C12712yes—
610C22#542×C12712yes—
611C22D42×C12711yescharacteristic
612C22#546×C12712yes—
613C22#547×C12711yescharacteristic
614C22#548×C12712yes—
615C22#550×C12711yescharacteristic
616C22#551×C12711yescharacteristic
617C22#551×C12712yes—
618C22#560×C12712yes—
619C22#560×C12711yescharacteristic
620C22#560×C12711yescharacteristic
621C22#562×C12711yescharacteristic
622C22#562×C12712yes—
623C22#563×C12712yes—
624C22#563×C12711yescharacteristic
625C4—44no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—82no—
633C2—84no—
634C2C127×#267612yes—
635C2C127×#268211yescharacteristic
636C2C127×#269311yescharacteristic
637C2C127×#270512yes—
638C2C127×#272312yes—
639C2C127×#272811yescharacteristic
640C2C127×#273511yescharacteristic
641C2C127×#281511yescharacteristic
642C2C127×#282311yescharacteristic
643C2C127×#282712yes—
644C2C127×#283011yescharacteristic
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
1. C2×#397×C127C2semidirect
2. C2×#396×C127C2semidirect
3. C127×#1445C2semidirect
4. C127×#1444C2semidirect
5. C127×#1442C2semidirect
6. C127×#1439C2semidirect
7. C127×#1439C2semidirect
57. C2×C4⋊C4×C127D4semidirect
85. #95×C127D4semidirect
310. #17121C127direct (N × H)
323. C2×#397C2×C127semidirect
324. C2×#396C2×C127semidirect
325. #1445C2×C127semidirect
326. #1444C2×C127semidirect
327. #1442C2×C127semidirect
328. #1439C2×C127semidirect
329. #1439C2×C127semidirect
332. C127#17121direct (N × H)
380. C2×C4⋊C4D4×C127semidirect
408. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#397×C1272
C2×#396×C1272
C127×#14452
C127×#14442
C127×#14422
C127×#14392
C127×#14382
#17121127