Finite Group Databasebeta

← C127×#14641 #44578314

Subgroups — C127×#14641 #44578314

Order 32512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394×C127C211yesmaximal, characteristic
2C127×#1445C211yesmaximal, characteristic, complemented
3C127×#1442C212yesmaximal, complemented
4C127×#1440C212yesmaximal, complemented
5C127×#1439C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1438C212yesmaximal
8C127×#1437C212yesmaximal, complemented
9C127×#1436C211yesmaximal, characteristic, complemented
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C22⋊C4×C127C2212yes—
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C42×C127C2211yescharacteristic
20#394×C127C418yescomplemented
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2211yescharacteristic
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2211yescharacteristic
32C25×C127C2312yes—
33C22×D4×C127—42no—
34C22×D4×C127—42no—
35C22×D4×C127—41no—
36C22×D4×C127—41no—
37C22×D4×C127C2×C414yes—
38C23×C4×C127C2312yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2311yescharacteristic
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C2×C4⋊C4×C127—41no—
47C2×C4⋊C4×C127—41no—
48C2×C4⋊C4×C127—42no—
49C2×C4⋊C4×C127—42no—
50C2×C4⋊C4×C127—42no—
51C2×C4⋊C4×C127C2×C414yes—
52C2×C22⋊C4×C127—24no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—42no—
55C2×C22⋊C4×C127—42no—
56C2×C22⋊C4×C127—42no—
57C2×C22⋊C4×C127—42no—
58C2×C22⋊C4×C127—42no—
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×C127C2×C418yes—
69C2×C22⋊C4×C127C2×C418yes—
70C2×C42×C127—42no—
71C2×C42×C127—41no—
72#95×C127—24no—
73#95×C127—42no—
74#95×C127—42no—
75#95×C127—42no—
76#95×C127—42no—
77#95×C127—42no—
78#95×C127—41no—
79#95×C127—41no—
80#95×C127—41no—
81#95×C127—42no—
82#95×C127C2×C414yes—
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—22no—
97C24×C127C22×C414yes—
98C24×C127C2411yescharacteristic
99C2×D4×C127—44no—
100C2×D4×C127—48no—
101C2×D4×C127—48no—
102C2×D4×C127—44no—
103C2×D4×C127—44no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
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—21no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—21no—
125C22×C4×C127—21no—
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×C127C22×C414yes—
150C22×C4×C127C22×C412yes—
151C22×C4×C127C22×C412yes—
152C22×C4×C127C22×C412yes—
153C22×C4×C127#4114yes—
154C4⋊C4×C127—44no—
155C4⋊C4×C127—48no—
156C22⋊C4×C127—48no—
157C22⋊C4×C127—48no—
158C22⋊C4×C127—48no—
159C22⋊C4×C127—48no—
160C22⋊C4×C127—48no—
161C22⋊C4×C127—48no—
162C42×C127—44no—
163C23×C127—24no—
164C23×C127—24no—
165C23×C127—24no—
166C23×C127—24no—
167C23×C127—24no—
168C23×C127—24no—
169C23×C127—24no—
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×C127—42no—
198C23×C127C23×C411yescharacteristic
199C23×C127C2×#4112yes—
200C23×C127H(5,𝔽2)11yescharacteristic
201C23×C127H(5,𝔽2)12yes—
202C23×C127H(5,𝔽2)12yes—
203C23×C127H(5,𝔽2)11yescharacteristic
204C23×C127H(5,𝔽2)12yes—
205C23×C127H(5,𝔽2)12yes—
206C23×C1272−1+412yes—
207D4×C127—416no—
208C2×C4×C127—22no—
209C2×C4×C127—22no—
210C2×C4×C127—24no—
211C2×C4×C127—24no—
212C2×C4×C127—24no—
213C2×C4×C127—22no—
214C2×C4×C127—22no—
215C2×C4×C127—22no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—24no—
219C2×C4×C127—24no—
220C2×C4×C127—24no—
221C2×C4×C127—42no—
222C2×C4×C127—42no—
223C2×C4×C127—42no—
224C2×C4×C127—42no—
225C2×C4×C127—42no—
226C2×C4×C127—41no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—41no—
230C2×C4×C127—42no—
231C2×C4×C127—41no—
232C2×C4×C127—41no—
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—
258C22×C127—44no—
259C22×C127—44no—
260C22×C127—44no—
261C22×C127—44no—
262C22×C127—44no—
263C22×C127—44no—
264C22×C127—44no—
265C22×C127—82no—
266C22×C127—82no—
267C22×C127—82no—
268C22×C127—82no—
269C22×C127—82no—
270C22×C127—82no—
271C22×C127—82no—
272C22×C127—82no—
273C22×C127C4×#4111yescharacteristic
274C22×C127#51811yescharacteristic
275C22×C127#51811yescharacteristic
276C22×C127#51812yes—
277C22×C127#52012yes—
278C22×C127#53812yes—
279C22×C127#53812yes—
280C22×C127#53912yes—
281C22×C127#54012yes—
282C22×C127#54012yes—
283C22×C127#54212yes—
284C22×C127#56012yes—
285C22×C127#56012yes—
286C22×C127#56012yes—
287C22×C127#56011yescharacteristic
288C22×C127H(3,𝔽4)11yescharacteristic
289C22×C127H(3,𝔽4)11yescharacteristic
290C22×C127#56212yes—
291C22×C127#56212yes—
292C22×C127#56212yes—
293C22×C127#56311yescharacteristic
294C4×C127—44no—
295C4×C127—42no—
296C4×C127—42no—
297C4×C127—42no—
298C4×C127—44no—
299C4×C127—82no—
300C4×C127—82no—
301C4×C127—82no—
302#14641C12711yesmaximal, characteristic, direct product factor, Sylow
303C2×C127—84no—
304C2×C127#233311yescharacteristic
305C2×C127#233512yes—
306C2×C127#237511yescharacteristic
307C2×C127#237712yes—
308C2×C127#237911yescharacteristic
309C2×C127#282412yes—
310C2×C127#282812yes—
311C2×C127#282912yes—
312C2×C127#283712yes—
313C2×#394C2×C12711yescharacteristic
314#1445C2×C12711yescharacteristic, complemented
315#1442C2×C12712yescomplemented
316#1440C2×C12712yescomplemented
317#1439C2×C12712yescomplemented
318#1439C2×C12712yescomplemented
319#1438C2×C12712yes—
320#1437C2×C12712yescomplemented
321#1436C2×C12711yescharacteristic, complemented
322C127#1464111yescharacteristic, direct product factor, Sylow
323C23×D4C22×C12711yescharacteristic
324C22×C4⋊C4C22×C12712yes—
325C22×C4⋊C4C22×C12711yescharacteristic
326C22×C22⋊C4C22×C12712yes—
327C22×C22⋊C4C22×C12712yes—
328C22×C22⋊C4C22×C12712yes—
329C22×C22⋊C4C22×C12712yes—
330C22×C22⋊C4C22×C12712yes—
331C22×C22⋊C4C22×C12712yes—
332C22×C42C22×C12711yescharacteristic
333#394C4×C12718yescomplemented
334C2×#95C22×C12712yes—
335C2×#95C22×C12712yes—
336C2×#95C22×C12711yescharacteristic
337C2×#95C22×C12712yes—
338C2×#95C22×C12712yes—
339C2×#95C22×C12712yes—
340C2×#95C22×C12712yes—
341C2×#95C22×C12711yescharacteristic
342C2×#95C22×C12712yes—
343C2×#95C22×C12711yescharacteristic
344C2×#95C22×C12711yescharacteristic
345C25C23×C12712yes—
346C22×D4—42no—
347C22×D4—42no—
348C22×D4—41no—
349C22×D4—41no—
350C22×D4C2×C4×C12714yes—
351C23×C4C23×C12712yes—
352C23×C4C23×C12712yes—
353C23×C4C23×C12712yes—
354C23×C4C23×C12711yescharacteristic
355C23×C4C23×C12712yes—
356C23×C4C23×C12711yescharacteristic
357C23×C4C23×C12711yescharacteristic
358C23×C4C23×C12712yes—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—42no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4C2×C4×C12714yes—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—42no—
371C2×C22⋊C4—42no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4C2×C4×C12718yes—
382C2×C22⋊C4C2×C4×C12718yes—
383C2×C42—42no—
384C2×C42—41no—
385#95—24no—
386#95—42no—
387#95—42no—
388#95—42no—
389#95—42no—
390#95—42no—
391#95—41no—
392#95—41no—
393#95—41no—
394#95—42no—
395#95C2×C4×C12714yes—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24C22×C4×C12714yes—
411C24C24×C12711yescharacteristic
412C2×D4—44no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
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×C4C22×C4×C12714yes—
463C22×C4C22×C4×C12712yes—
464C22×C4C22×C4×C12712yes—
465C22×C4C22×C4×C12712yes—
466C22×C4#41×C12714yes—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C42—44no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
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—
511C23C23×C4×C12711yescharacteristic
512C23C2×#41×C12712yes—
513C23H(5,𝔽2)×C12711yescharacteristic
514C23H(5,𝔽2)×C12712yes—
515C23H(5,𝔽2)×C12712yes—
516C23H(5,𝔽2)×C12711yescharacteristic
517C23H(5,𝔽2)×C12712yes—
518C23H(5,𝔽2)×C12712yes—
519C232−1+4×C12712yes—
520D4—416no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
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—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C4×#41×C12711yescharacteristic
587C22#518×C12711yescharacteristic
588C22#518×C12711yescharacteristic
589C22#518×C12712yes—
590C22#520×C12712yes—
591C22#538×C12712yes—
592C22#538×C12712yes—
593C22#539×C12712yes—
594C22#540×C12712yes—
595C22#540×C12712yes—
596C22#542×C12712yes—
597C22#560×C12712yes—
598C22#560×C12712yes—
599C22#560×C12712yes—
600C22#560×C12711yescharacteristic
601C22H(3,𝔽4)×C12711yescharacteristic
602C22H(3,𝔽4)×C12711yescharacteristic
603C22#562×C12712yes—
604C22#562×C12712yes—
605C22#562×C12712yes—
606C22#563×C12711yescharacteristic
607C4—44no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—82no—
613C4—82no—
614C4—82no—
615C2—84no—
616C2C127×#233311yescharacteristic
617C2C127×#233512yes—
618C2C127×#237511yescharacteristic
619C2C127×#237712yes—
620C2C127×#237911yescharacteristic
621C2C127×#282412yes—
622C2C127×#282812yes—
623C2C127×#282912yes—
624C2C127×#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. C127×#1445C2semidirect
3. C127×#1442C2semidirect
4. C127×#1440C2semidirect
5. C127×#1439C2semidirect
6. C127×#1439C2semidirect
8. C127×#1437C2semidirect
9. C127×#1436C2semidirect
20. #394×C127C4semidirect
302. #14641C127direct (N × H)
314. #1445C2×C127semidirect
315. #1442C2×C127semidirect
316. #1440C2×C127semidirect
317. #1439C2×C127semidirect
318. #1439C2×C127semidirect
320. #1437C2×C127semidirect
321. #1436C2×C127semidirect
322. C127#14641direct (N × H)
333. #394C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#394×C1272
C127×#14452
C127×#14422
C127×#14402
C127×#14392
C127×#14382
C127×#14372
C127×#14362
#14641127