Finite Group Databasebeta

← C127×#17159 #44580832

Subgroups — C127×#17159 #44580832

Order 32512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#397×C127C212yesmaximal, complemented
2C127×#1442C212yesmaximal, complemented
3C127×#1442C211yesmaximal, characteristic, complemented
4C127×#1442C211yesmaximal, characteristic, complemented
5C127×#1439C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1439C212yesmaximal, complemented
8C127×#1438C211yesmaximal, characteristic
9C127×#1438C212yesmaximal
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2212yes—
20#397×C127—28no—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2211yescharacteristic
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2212yes—
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2211yescharacteristic
32C25×C127C2312yes—
33C22×D4×C127—24no—
34C22×D4×C127—22no—
35C22×D4×C127—42no—
36C22×D4×C127—41no—
37C22×D4×C127—41no—
38C23×C4×C127C2312yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2311yescharacteristic
46C2×C4⋊C4×C127—24no—
47C2×C4⋊C4×C127—42no—
48C2×C4⋊C4×C127—42no—
49C2×C4⋊C4×C127—41no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—42no—
52C2×C4⋊C4×C127—42no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127—24no—
57C2×C22⋊C4×C127—24no—
58C2×C22⋊C4×C127—24no—
59C2×C22⋊C4×C127—42no—
60C2×C22⋊C4×C127—42no—
61C2×C22⋊C4×C127—42no—
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—
71#95×C127—24no—
72#95×C127—24no—
73#95×C127—24no—
74#95×C127—22no—
75#95×C127—42no—
76#95×C127—41no—
77#95×C127—41no—
78#95×C127—42no—
79#95×C127—42no—
80#95×C127—42no—
81#95×C127D414yescomplemented
82C24×C127—22no—
83C24×C127—22no—
84C24×C127—22no—
85C24×C127—22no—
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#4114yes—
97C24×C127C2411yescharacteristic
98C2×D4×C127—48no—
99C2×D4×C127—44no—
100C2×D4×C127—44no—
101C2×D4×C127—48no—
102C2×D4×C127—44no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
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—21no—
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—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×C127C2×D412yes—
150C22×C4×C127C2×D414yes—
151C22×C4×C127#4112yes—
152C22×C4×C127#4112yes—
153C4⋊C4×C127—48no—
154C4⋊C4×C127—44no—
155C4⋊C4×C127—44no—
156C22⋊C4×C127—48no—
157C22⋊C4×C127—48no—
158C22⋊C4×C127—48no—
159C22⋊C4×C127—48no—
160C22⋊C4×C127—48no—
161C22⋊C4×C127—48no—
162C23×C127—24no—
163C23×C127—24no—
164C23×C127—24no—
165C23×C127—24no—
166C23×C127—24no—
167C23×C127—24no—
168C23×C127—24no—
169C23×C127—42no—
170C23×C127—42no—
171C23×C127—42no—
172C23×C127—42no—
173C23×C127—42no—
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×C127C22×D411yescharacteristic
198C23×C127C2×#4111yescharacteristic
199C23×C127C2×#4112yes—
200C23×C127C2×#4111yescharacteristic
201C23×C127H(5,𝔽2)12yes—
202C23×C127H(5,𝔽2)12yes—
203C23×C127H(5,𝔽2)11yescharacteristic
204C23×C127H(5,𝔽2)11yescharacteristic
205C23×C127H(5,𝔽2)12yes—
206C23×C1272−1+411yescharacteristic
207C23×C1272−1+411yescharacteristic
208D4×C127—416no—
209C2×C4×C127—24no—
210C2×C4×C127—22no—
211C2×C4×C127—24no—
212C2×C4×C127—22no—
213C2×C4×C127—22no—
214C2×C4×C127—24no—
215C2×C4×C127—24no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—42no—
219C2×C4×C127—42no—
220C2×C4×C127—42no—
221C2×C4×C127—42no—
222C2×C4×C127—41no—
223C2×C4×C127—42no—
224C2×C4×C127—41no—
225C2×C4×C127—41no—
226C2×C4×C127—41no—
227C2×C4×C127—41no—
228C2×C4×C127—41no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127—42no—
235C2×C4×C127—42no—
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—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—
259C22×C127—44no—
260C22×C127—44no—
261C22×C127—44no—
262C22×C127—44no—
263C22×C127—44no—
264C22×C127—44no—
265C22×C127—44no—
266C22×C127—82no—
267C22×C127—82no—
268C22×C127—82no—
269C22×C127—82no—
270C22×C127—82no—
271C22×C127—82no—
272C22×C127—82no—
273C22×C127—82no—
274C22×C127#53512yes—
275C22×C127#53711yescharacteristic
276C22×C127#53812yes—
277C22×C127#53812yes—
278C22×C127#53811yescharacteristic
279C22×C127#53911yescharacteristic
280C22×C127#53912yes—
281C22×C127#53912yes—
282C22×C127#54012yes—
283C22×C127#54612yes—
284C22×C127#54711yescharacteristic
285C22×C127#55011yescharacteristic
286C22×C127#55111yescharacteristic
287C22×C127#55112yes—
288C22×C127#55312yes—
289C22×C127#55611yescharacteristic
290C22×C127#56011yescharacteristic
291C22×C127#56012yes—
292C22×C127H(3,𝔽4)11yescharacteristic
293C22×C127H(3,𝔽4)11yescharacteristic
294C22×C127#56212yes—
295C22×C127#56212yes—
296C22×C127#56311yescharacteristic
297C4×C127—42no—
298C4×C127—42no—
299C4×C127—44no—
300C4×C127—42no—
301C4×C127—82no—
302C4×C127—82no—
303C4×C127—82no—
304C4×C127—82no—
305#17159C12711yesmaximal, characteristic, direct product factor, Sylow
306C2×C127—84no—
307C2×C127#269311yescharacteristic
308C2×C127#270612yes—
309C2×C127#271412yes—
310C2×C127#272711yescharacteristic
311C2×C127#273611yescharacteristic
312C2×C127#278211yescharacteristic
313C2×C127#278612yes—
314C2×C127#281911yescharacteristic
315C2×C127#282411yescharacteristic
316C2×C127#282712yes—
317C2×C127#282811yescharacteristic
318C2×#397C2×C12712yescomplemented
319#1442C2×C12712yescomplemented
320#1442C2×C12711yescharacteristic, complemented
321#1442C2×C12711yescharacteristic, complemented
322#1439C2×C12712yescomplemented
323#1439C2×C12712yescomplemented
324#1439C2×C12712yescomplemented
325#1438C2×C12711yescharacteristic
326#1438C2×C12712yes—
327C127#1715911yescharacteristic, direct product factor, Sylow
328C23×D4C22×C12711yescharacteristic
329C22×C4⋊C4C22×C12712yes—
330C22×C4⋊C4C22×C12711yescharacteristic
331C22×C4⋊C4C22×C12711yescharacteristic
332C22×C22⋊C4C22×C12712yes—
333C22×C22⋊C4C22×C12712yes—
334C22×C22⋊C4C22×C12712yes—
335C22×C22⋊C4C22×C12712yes—
336C22×C22⋊C4C22×C12712yes—
337C22×C22⋊C4C22×C12712yes—
338#397—28no—
339C2×#95C22×C12712yes—
340C2×#95C22×C12712yes—
341C2×#95C22×C12712yes—
342C2×#95C22×C12711yescharacteristic
343C2×#95C22×C12711yescharacteristic
344C2×#95C22×C12712yes—
345C2×#95C22×C12711yescharacteristic
346C2×#95C22×C12712yes—
347C2×#95C22×C12712yes—
348C2×#95C22×C12712yes—
349C2×#95C22×C12711yescharacteristic
350C25C23×C12712yes—
351C22×D4—24no—
352C22×D4—22no—
353C22×D4—42no—
354C22×D4—41no—
355C22×D4—41no—
356C23×C4C23×C12712yes—
357C23×C4C23×C12712yes—
358C23×C4C23×C12712yes—
359C23×C4C23×C12712yes—
360C23×C4C23×C12712yes—
361C23×C4C23×C12711yescharacteristic
362C23×C4C23×C12711yescharacteristic
363C23×C4C23×C12711yescharacteristic
364C2×C4⋊C4—24no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—42no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—42no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—42no—
394#95—41no—
395#95—41no—
396#95—42no—
397#95—42no—
398#95—42no—
399#95D4×C12714yescomplemented
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24#41×C12714yes—
415C24C24×C12711yescharacteristic
416C2×D4—48no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—48no—
420C2×D4—44no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
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—22no—
440C22×C4—22no—
441C22×C4—22no—
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—22no—
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×C4C2×D4×C12712yes—
468C22×C4C2×D4×C12714yes—
469C22×C4#41×C12712yes—
470C22×C4#41×C12712yes—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
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—
515C23C22×D4×C12711yescharacteristic
516C23C2×#41×C12711yescharacteristic
517C23C2×#41×C12712yes—
518C23C2×#41×C12711yescharacteristic
519C23H(5,𝔽2)×C12712yes—
520C23H(5,𝔽2)×C12712yes—
521C23H(5,𝔽2)×C12711yescharacteristic
522C23H(5,𝔽2)×C12711yescharacteristic
523C23H(5,𝔽2)×C12712yes—
524C232−1+4×C12711yescharacteristic
525C232−1+4×C12711yescharacteristic
526D4—416no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
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—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
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—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22#535×C12712yes—
593C22#537×C12711yescharacteristic
594C22#538×C12712yes—
595C22#538×C12712yes—
596C22#538×C12711yescharacteristic
597C22#539×C12711yescharacteristic
598C22#539×C12712yes—
599C22#539×C12712yes—
600C22#540×C12712yes—
601C22#546×C12712yes—
602C22#547×C12711yescharacteristic
603C22#550×C12711yescharacteristic
604C22#551×C12711yescharacteristic
605C22#551×C12712yes—
606C22#553×C12712yes—
607C22#556×C12711yescharacteristic
608C22#560×C12711yescharacteristic
609C22#560×C12712yes—
610C22H(3,𝔽4)×C12711yescharacteristic
611C22H(3,𝔽4)×C12711yescharacteristic
612C22#562×C12712yes—
613C22#562×C12712yes—
614C22#563×C12711yescharacteristic
615C4—42no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C2—84no—
624C2C127×#269311yescharacteristic
625C2C127×#270612yes—
626C2C127×#271412yes—
627C2C127×#272711yescharacteristic
628C2C127×#273611yescharacteristic
629C2C127×#278211yescharacteristic
630C2C127×#278612yes—
631C2C127×#281911yescharacteristic
632C2C127×#282411yescharacteristic
633C2C127×#282712yes—
634C2C127×#282811yescharacteristic
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. C127×#1442C2semidirect
3. C127×#1442C2semidirect
4. C127×#1442C2semidirect
5. C127×#1439C2semidirect
6. C127×#1439C2semidirect
7. C127×#1439C2semidirect
81. #95×C127D4semidirect
305. #17159C127direct (N × H)
318. C2×#397C2×C127semidirect
319. #1442C2×C127semidirect
320. #1442C2×C127semidirect
321. #1442C2×C127semidirect
322. #1439C2×C127semidirect
323. #1439C2×C127semidirect
324. #1439C2×C127semidirect
327. C127#17159direct (N × H)
399. #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
C127×#14422
C127×#14392
C127×#14382
#17159127