Finite Group Databasebeta

← C127×#16987 #44580660

Subgroups — C127×#16987 #44580660

Order 32512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#385×C127C212yesmaximal
2C127×#1444C212yesmaximal, complemented
3C127×#1442C212yesmaximal, complemented
4C127×#1441C212yesmaximal, complemented
5C127×#1441C212yesmaximal, complemented
6C127×#1439C211yesmaximal, characteristic
7C127×#1439C212yesmaximal
8C127×#1439C211yesmaximal, characteristic
9C127×C22⋊#95C211yesmaximal, characteristic
10C24×C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2211yescharacteristic
19C22×C22⋊C4×C127C2211yescharacteristic
20#385×C127—28no—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2211yescharacteristic
32C25×C127C2311yescharacteristic
33C23×C4×C127—42no—
34C23×C4×C127—42no—
35C23×C4×C127—41no—
36C23×C4×C127—41no—
37C23×C4×C127D414yes—
38C23×C4×C127C2312yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2311yescharacteristic
42C23×C4×C127C2312yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2312yes—
45C23×C4×C127C2312yes—
46C2×C4⋊C4×C127—24no—
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—24no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—42no—
52C2×C4⋊C4×C127—42no—
53C2×C4⋊C4×C127—42no—
54C2×C4⋊C4×C127—42no—
55C2×C4⋊C4×C127—42no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127—42no—
58C2×C4⋊C4×C127—41no—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127—41no—
64C2×C22⋊C4×C127—41no—
65C2×C22⋊C4×C127—41no—
66C2×C22⋊C4×C127—41no—
67C2×C22⋊C4×C127—42no—
68C2×C22⋊C4×C127—42no—
69C2×C22⋊C4×C127Q814yes—
70C2×C22⋊C4×C127Q814yes—
71#95×C127—24no—
72#95×C127—24no—
73#95×C127—24no—
74#95×C127—24no—
75#95×C127—42no—
76#95×C127—42no—
77#95×C127—42no—
78#95×C127—41no—
79#95×C127—42no—
80#95×C127—41no—
81#95×C127—42no—
82C24×C127—22no—
83C24×C127—22no—
84C24×C127—21no—
85C24×C127—22no—
86C24×C127—22no—
87C24×C127—21no—
88C24×C127—21no—
89C24×C127—21no—
90C24×C127C2×D412yes—
91C24×C127C2×Q812yes—
92C24×C127C2×Q812yes—
93C24×C127C2411yescharacteristic
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
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—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—44no—
141C22×C4×C127—48no—
142C22×C4×C127—44no—
143C22×C4×C127—48no—
144C22×C4×C127—44no—
145C22×C4×C127C2×D414yes—
146C22×C4×C127C2×Q812yes—
147C22×C4×C127C2×Q814yes—
148C22×C4×C127C2×Q812yes—
149C22×C4×C127#4114yes—
150C22×C4×C127#4114yes—
151C4⋊C4×C127—44no—
152C4⋊C4×C127—48no—
153C4⋊C4×C127—48no—
154C4⋊C4×C127—48no—
155C4⋊C4×C127—44no—
156C22⋊C4×C127—216no—
157C22⋊C4×C127—44no—
158C22⋊C4×C127—44no—
159C22⋊C4×C127—48no—
160C23×C127—24no—
161C23×C127—24no—
162C23×C127—22no—
163C23×C127—22no—
164C23×C127—22no—
165C23×C127—22no—
166C23×C127—22no—
167C23×C127—22no—
168C23×C127—24no—
169C23×C127—24no—
170C23×C127—24no—
171C23×C127—24no—
172C23×C127—42no—
173C23×C127—42no—
174C23×C127—42no—
175C23×C127—42no—
176C23×C127—41no—
177C23×C127—42no—
178C23×C127—41no—
179C23×C127—41no—
180C23×C127—42no—
181C23×C127—41no—
182C23×C127C4⋊Q814yes—
183C23×C127C22×D411yescharacteristic
184C23×C127C22×Q811yescharacteristic
185C23×C127C22×Q811yescharacteristic
186C23×C127C2×#4112yes—
187C23×C127C2×#4112yes—
188C23×C127H(5,𝔽2)12yes—
189C23×C127H(5,𝔽2)12yes—
190C23×C127H(5,𝔽2)11yescharacteristic
191C23×C127H(5,𝔽2)11yescharacteristic
192C23×C127H(5,𝔽2)11yescharacteristic
193C23×C1272−1+411yescharacteristic
194C2×C4×C127—24no—
195C2×C4×C127—24no—
196C2×C4×C127—24no—
197C2×C4×C127—24no—
198C2×C4×C127—24no—
199C2×C4×C127—24no—
200C2×C4×C127—24no—
201C2×C4×C127—22no—
202C2×C4×C127—24no—
203C2×C4×C127—22no—
204C2×C4×C127—24no—
205C2×C4×C127—24no—
206C2×C4×C127—24no—
207C2×C4×C127—24no—
208C2×C4×C127—24no—
209C2×C4×C127—24no—
210C2×C4×C127—416no—
211C2×C4×C127—42no—
212C2×C4×C127—42no—
213C2×C4×C127—42no—
214C2×C4×C127—42no—
215C2×C4×C127—42no—
216C2×C4×C127—42no—
217C2×C4×C127—42no—
218C2×C4×C127—42no—
219C2×C4×C127—42no—
220C2×C4×C127—42no—
221C2×C4×C127—41no—
222C2×C4×C127—42no—
223C2×C4×C127—42no—
224C2×C4×C127—42no—
225C2×C4×C127—41no—
226C2×C4×C127—41no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—41no—
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—
247C22×C127—28no—
248C22×C127—24no—
249C22×C127—42no—
250C22×C127—44no—
251C22×C127—42no—
252C22×C127—42no—
253C22×C127—44no—
254C22×C127—44no—
255C22×C127—42no—
256C22×C127—42no—
257C22×C127—42no—
258C22×C127#52512yes—
259C22×C127C2×C4⋊Q811yescharacteristic
260C22×C127#53412yes—
261C22×C127#53411yescharacteristic
262C22×C127#53711yescharacteristic
263C22×C127#53812yes—
264C22×C127#53912yes—
265C22×C127#54012yes—
266C22×C127#54212yes—
267C22×C127#54311yescharacteristic
268C22×C127#54311yescharacteristic
269C22×C127#54411yescharacteristic
270C22×C127#54411yescharacteristic
271C22×C127#55111yescharacteristic
272C22×C127#55312yes—
273C22×C127#55412yes—
274C22×C127#55412yes—
275C22×C127#55412yes—
276C22×C127#55412yes—
277C22×C127#55911yescharacteristic
278C22×C127#56011yescharacteristic
279C22×C127#56012yes—
280C22×C127H(3,𝔽4)11yescharacteristic
281C4×C127—44no—
282C4×C127—44no—
283C4×C127—42no—
284C4×C127—44no—
285C4×C127—42no—
286C4×C127—44no—
287C4×C127—82no—
288C4×C127—82no—
289#16987C12711yesmaximal, characteristic, direct product factor, Sylow
290C2×C127—44no—
291C2×C127#258312yes—
292C2×C127#260011yescharacteristic
293C2×C127#263012yes—
294C2×C127#265911yescharacteristic
295C2×C127#274811yescharacteristic
296C2×C127#276311yescharacteristic
297C2×C127#278012yes—
298C2×C127#279012yes—
299C2×C127#280411yescharacteristic
300C2×C127#280711yescharacteristic
301C2×C127#281211yescharacteristic
302C2×#385C2×C12712yes—
303#1444C2×C12712yescomplemented
304#1442C2×C12712yescomplemented
305#1441C2×C12712yescomplemented
306#1441C2×C12712yescomplemented
307#1439C2×C12711yescharacteristic
308#1439C2×C12712yes—
309#1439C2×C12711yescharacteristic
310C22⋊#95C2×C12711yescharacteristic
311C127#1698711yescharacteristic, direct product factor, Sylow
312C24×C4C22×C12711yescharacteristic
313C22×C4⋊C4C22×C12711yescharacteristic
314C22×C4⋊C4C22×C12712yes—
315C22×C4⋊C4C22×C12712yes—
316C22×C4⋊C4C22×C12712yes—
317C22×C4⋊C4C22×C12711yescharacteristic
318C22×C22⋊C4C22×C12712yes—
319C22×C22⋊C4C22×C12712yes—
320C22×C22⋊C4C22×C12711yescharacteristic
321C22×C22⋊C4C22×C12711yescharacteristic
322#385—28no—
323C2×#95C22×C12712yes—
324C2×#95C22×C12712yes—
325C2×#95C22×C12712yes—
326C2×#95C22×C12712yes—
327C2×#95C22×C12712yes—
328C2×#95C22×C12712yes—
329C2×#95C22×C12712yes—
330C2×#95C22×C12711yescharacteristic
331C2×#95C22×C12712yes—
332C2×#95C22×C12712yes—
333C2×#95C22×C12711yescharacteristic
334C25C23×C12711yescharacteristic
335C23×C4—42no—
336C23×C4—42no—
337C23×C4—41no—
338C23×C4—41no—
339C23×C4D4×C12714yes—
340C23×C4C23×C12712yes—
341C23×C4C23×C12712yes—
342C23×C4C23×C12712yes—
343C23×C4C23×C12711yescharacteristic
344C23×C4C23×C12712yes—
345C23×C4C23×C12711yescharacteristic
346C23×C4C23×C12712yes—
347C23×C4C23×C12712yes—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—41no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4—41no—
367C2×C22⋊C4—41no—
368C2×C22⋊C4—41no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—42no—
371C2×C22⋊C4Q8×C12714yes—
372C2×C22⋊C4Q8×C12714yes—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—42no—
378#95—42no—
379#95—42no—
380#95—41no—
381#95—42no—
382#95—41no—
383#95—42no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24C2×D4×C12712yes—
393C24C2×Q8×C12712yes—
394C24C2×Q8×C12712yes—
395C24C24×C12711yescharacteristic
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4C2×D4×C12714yes—
448C22×C4C2×Q8×C12712yes—
449C22×C4C2×Q8×C12714yes—
450C22×C4C2×Q8×C12712yes—
451C22×C4#41×C12714yes—
452C22×C4#41×C12714yes—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C22⋊C4—216no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23C4⋊Q8×C12714yes—
485C23C22×D4×C12711yescharacteristic
486C23C22×Q8×C12711yescharacteristic
487C23C22×Q8×C12711yescharacteristic
488C23C2×#41×C12712yes—
489C23C2×#41×C12712yes—
490C23H(5,𝔽2)×C12712yes—
491C23H(5,𝔽2)×C12712yes—
492C23H(5,𝔽2)×C12711yescharacteristic
493C23H(5,𝔽2)×C12711yescharacteristic
494C23H(5,𝔽2)×C12711yescharacteristic
495C232−1+4×C12711yescharacteristic
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—416no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C22—28no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#525×C12712yes—
561C22C2×C4⋊Q8×C12711yescharacteristic
562C22#534×C12712yes—
563C22#534×C12711yescharacteristic
564C22#537×C12711yescharacteristic
565C22#538×C12712yes—
566C22#539×C12712yes—
567C22#540×C12712yes—
568C22#542×C12712yes—
569C22#543×C12711yescharacteristic
570C22#543×C12711yescharacteristic
571C22#544×C12711yescharacteristic
572C22#544×C12711yescharacteristic
573C22#551×C12711yescharacteristic
574C22#553×C12712yes—
575C22#554×C12712yes—
576C22#554×C12712yes—
577C22#554×C12712yes—
578C22#554×C12712yes—
579C22#559×C12711yescharacteristic
580C22#560×C12711yescharacteristic
581C22#560×C12712yes—
582C22H(3,𝔽4)×C12711yescharacteristic
583C4—44no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—44no—
589C4—82no—
590C4—82no—
591C2—44no—
592C2C127×#258312yes—
593C2C127×#260011yescharacteristic
594C2C127×#263012yes—
595C2C127×#265911yescharacteristic
596C2C127×#274811yescharacteristic
597C2C127×#276311yescharacteristic
598C2C127×#278012yes—
599C2C127×#279012yes—
600C2C127×#280411yescharacteristic
601C2C127×#280711yescharacteristic
602C2C127×#281211yescharacteristic
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
2. C127×#1444C2semidirect
3. C127×#1442C2semidirect
4. C127×#1441C2semidirect
5. C127×#1441C2semidirect
289. #16987C127direct (N × H)
303. #1444C2×C127semidirect
304. #1442C2×C127semidirect
305. #1441C2×C127semidirect
306. #1441C2×C127semidirect
311. C127#16987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#385×C1272
C127×#14442
C127×#14422
C127×#14412
C127×#14392
C127×C22⋊#952
#16987127