Finite Group Databasebeta

← C127×#15783 #44579456

Subgroups — C127×#15783 #44579456

Order 32512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Q8×C127C211yesmaximal, characteristic
2C2×#400×C127C211yesmaximal, characteristic
3C2×#398×C127C211yesmaximal, characteristic
4C2×#391×C127C211yesmaximal, characteristic
5C127×#1446C212yesmaximal
6C127×#1446C211yesmaximal, characteristic
7C127×#1442C212yesmaximal
8C127×#1441C212yesmaximal
9C127×#1436C212yesmaximal
10C127×#1436C211yesmaximal, characteristic
11C127×#1436C211yesmaximal, characteristic
12C23×Q8×C127C2211yescharacteristic
13C23×Q8×C127C2211yescharacteristic
14C2×C4⋊Q8×C127—24no—
15C2×C4⋊Q8×C127—24no—
16C2×C4⋊Q8×C127—24no—
17C22×C4⋊C4×C127C2211yescharacteristic
18C22×C4⋊C4×C127C2212yes—
19C22×C4⋊C4×C127C2211yescharacteristic
20C22×C4⋊C4×C127C2211yescharacteristic
21C22×C4⋊C4×C127C2211yescharacteristic
22C22×C4⋊C4×C127C2211yescharacteristic
23C22×C42×C127C2211yescharacteristic
24C22×C42×C127C2211yescharacteristic
25#400×C127—24no—
26#398×C127—24no—
27#391×C127—24no—
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2211yescharacteristic
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C2×#95×C127C2212yes—
36C2×#95×C127C2212yes—
37C2×#95×C127C2212yes—
38C2×#95×C127C2212yes—
39C2×#95×C127C2212yes—
40C2×#95×C127C2212yes—
41C22×Q8×C127—22no—
42C22×Q8×C127—22no—
43C22×Q8×C127—22no—
44C22×Q8×C127—22no—
45C22×Q8×C127—22no—
46C22×Q8×C127—24no—
47C22×Q8×C127—24no—
48C22×Q8×C127—42no—
49C22×Q8×C127—42no—
50C22×Q8×C127D414yes—
51C23×C4×C127C2312yes—
52C23×C4×C127C2312yes—
53C23×C4×C127C2311yescharacteristic
54C23×C4×C127C2311yescharacteristic
55C23×C4×C127C2311yescharacteristic
56C23×C4×C127C2312yes—
57C23×C4×C127C2312yes—
58C23×C4×C127C2311yescharacteristic
59C23×C4×C127C2311yescharacteristic
60C23×C4×C127C2311yescharacteristic
61C23×C4×C127C2311yescharacteristic
62C4⋊Q8×C127—232no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127—22no—
65C2×C4⋊C4×C127—22no—
66C2×C4⋊C4×C127—22no—
67C2×C4⋊C4×C127—22no—
68C2×C4⋊C4×C127—22no—
69C2×C4⋊C4×C127—22no—
70C2×C4⋊C4×C127—22no—
71C2×C4⋊C4×C127—22no—
72C2×C4⋊C4×C127—22no—
73C2×C4⋊C4×C127—22no—
74C2×C4⋊C4×C127—22no—
75C2×C4⋊C4×C127—24no—
76C2×C4⋊C4×C127—22no—
77C2×C4⋊C4×C127—24no—
78C2×C4⋊C4×C127—24no—
79C2×C4⋊C4×C127D414yes—
80C2×C4⋊C4×C127D414yes—
81C2×C42×C127—22no—
82C2×C42×C127—22no—
83C2×C42×C127—22no—
84C2×C42×C127—22no—
85C2×C42×C127—42no—
86#95×C127—24no—
87#95×C127—24no—
88#95×C127—22no—
89#95×C127—24no—
90#95×C127—22no—
91#95×C127—24no—
92#95×C127—24no—
93#95×C127—24no—
94#95×C127—24no—
95#95×C127—42no—
96#95×C127—42no—
97#95×C127—42no—
98#95×C127Q818yescomplemented
99C24×C127C2411yescharacteristic
100C2×Q8×C127—28no—
101C2×Q8×C127—28no—
102C2×Q8×C127—28no—
103C2×Q8×C127—44no—
104C2×Q8×C127—48no—
105C2×Q8×C127—48no—
106C2×Q8×C127—48no—
107C2×Q8×C127—48no—
108C2×Q8×C127—44no—
109C2×Q8×C127—44no—
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127—21no—
113C22×C4×C127—21no—
114C22×C4×C127—21no—
115C22×C4×C127—21no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—21no—
124C22×C4×C127—21no—
125C22×C4×C127—21no—
126C22×C4×C127—21no—
127C22×C4×C127—21no—
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×C127—22no—
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×C127C2×D412yes—
160C22×C4×C127C2×D412yes—
161C22×C4×C127C2×D412yes—
162C22×C4×C127C2×D412yes—
163C22×C4×C127C2×D412yes—
164C22×C4×C127C2×D412yes—
165C22×C4×C127C2×D412yes—
166C22×C4×C127C2×D412yes—
167C22×C4×C127C2×D412yes—
168C22×C4×C127C2×Q814yes—
169C22×C4×C127C2×Q814yes—
170C22×C4×C127C2×Q814yes—
171C22×C4×C127#4114yes—
172C22×C4×C127#4114yes—
173C4⋊C4×C127—28no—
174C4⋊C4×C127—28no—
175C4⋊C4×C127—28no—
176C4⋊C4×C127—28no—
177C4⋊C4×C127—44no—
178C4⋊C4×C127—48no—
179C42×C127—28no—
180C42×C127—44no—
181C23×C127C22×D411yescharacteristic
182C23×C127C22×D411yescharacteristic
183C23×C127C22×D411yescharacteristic
184C23×C127C22×Q812yes—
185C23×C127C2×#4112yes—
186C23×C127C2×#4112yes—
187C23×C127H(5,𝔽2)12yes—
188C23×C1272−1+412yes—
189C23×C1272−1+412yes—
190Q8×C127—416no—
191Q8×C127—416no—
192C2×C4×C127—22no—
193C2×C4×C127—22no—
194C2×C4×C127—22no—
195C2×C4×C127—22no—
196C2×C4×C127—22no—
197C2×C4×C127—22no—
198C2×C4×C127—22no—
199C2×C4×C127—22no—
200C2×C4×C127—24no—
201C2×C4×C127—24no—
202C2×C4×C127—24no—
203C2×C4×C127—24no—
204C2×C4×C127—24no—
205C2×C4×C127—24no—
206C2×C4×C127—24no—
207C2×C4×C127—24no—
208C2×C4×C127—22no—
209C2×C4×C127—22no—
210C2×C4×C127—22no—
211C2×C4×C127—22no—
212C2×C4×C127—24no—
213C2×C4×C127—22no—
214C2×C4×C127—24no—
215C2×C4×C127—24no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—24no—
219C2×C4×C127—24no—
220C2×C4×C127—24no—
221C2×C4×C127—41no—
222C2×C4×C127—41no—
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—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#12014yes—
252C2×C4×C127#12314yes—
253C2×C4×C127#12414yes—
254C2×C4×C127C4⋊Q814yes—
255C22×C127C2×#12011yescharacteristic
256C22×C127C2×#12311yescharacteristic
257C22×C127C2×#12411yescharacteristic
258C22×C127C2×C4⋊Q811yescharacteristic
259C22×C127#53312yes—
260C22×C127#53411yescharacteristic
261C22×C127#53611yescharacteristic
262C22×C127#53611yescharacteristic
263C22×C127#54112yes—
264C22×C127#54212yes—
265C22×C127#54412yes—
266C22×C127#54612yes—
267C22×C127#54612yes—
268C22×C127#54812yes—
269C22×C127#54812yes—
270C22×C127D4×Q812yes—
271C22×C127D4×Q812yes—
272C22×C127#55412yes—
273C22×C127#55612yes—
274C22×C127#55712yes—
275C22×C127#56312yes—
276C4×C127—24no—
277C4×C127—24no—
278C4×C127—24no—
279C4×C127—24no—
280C4×C127—42no—
281C4×C127—42no—
282C4×C127—42no—
283C4×C127—44no—
284C4×C127—44no—
285C4×C127—82no—
286C4×C127—82no—
287#15783C12711yesmaximal, characteristic, direct product factor, Sylow
288C2×C127#240511yescharacteristic
289C2×C127#243411yescharacteristic
290C2×C127#243611yescharacteristic
291C2×C127#244612yes—
292C2×C127#249212yes—
293C2×C127#255012yes—
294C2×C127#271312yes—
295C2×C127#271612yes—
296C2×C127#272912yes—
297C22×C4⋊Q8C2×C12711yescharacteristic
298C2×#400C2×C12711yescharacteristic
299C2×#398C2×C12711yescharacteristic
300C2×#391C2×C12711yescharacteristic
301#1446C2×C12712yes—
302#1446C2×C12711yescharacteristic
303#1442C2×C12712yes—
304#1441C2×C12712yes—
305#1436C2×C12712yes—
306#1436C2×C12711yescharacteristic
307#1436C2×C12711yescharacteristic
308C127#1578311yescharacteristic, direct product factor, Sylow
309C23×Q8C22×C12711yescharacteristic
310C23×Q8C22×C12711yescharacteristic
311C2×C4⋊Q8—24no—
312C2×C4⋊Q8—24no—
313C2×C4⋊Q8—24no—
314C22×C4⋊C4C22×C12711yescharacteristic
315C22×C4⋊C4C22×C12712yes—
316C22×C4⋊C4C22×C12711yescharacteristic
317C22×C4⋊C4C22×C12711yescharacteristic
318C22×C4⋊C4C22×C12711yescharacteristic
319C22×C4⋊C4C22×C12711yescharacteristic
320C22×C42C22×C12711yescharacteristic
321C22×C42C22×C12711yescharacteristic
322#400—24no—
323#398—24no—
324#391—24no—
325C2×#95C22×C12711yescharacteristic
326C2×#95C22×C12712yes—
327C2×#95C22×C12712yes—
328C2×#95C22×C12711yescharacteristic
329C2×#95C22×C12712yes—
330C2×#95C22×C12712yes—
331C2×#95C22×C12712yes—
332C2×#95C22×C12712yes—
333C2×#95C22×C12712yes—
334C2×#95C22×C12712yes—
335C2×#95C22×C12712yes—
336C2×#95C22×C12712yes—
337C2×#95C22×C12712yes—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—24no—
344C22×Q8—24no—
345C22×Q8—42no—
346C22×Q8—42no—
347C22×Q8D4×C12714yes—
348C23×C4C23×C12712yes—
349C23×C4C23×C12712yes—
350C23×C4C23×C12711yescharacteristic
351C23×C4C23×C12711yescharacteristic
352C23×C4C23×C12711yescharacteristic
353C23×C4C23×C12712yes—
354C23×C4C23×C12712yes—
355C23×C4C23×C12711yescharacteristic
356C23×C4C23×C12711yescharacteristic
357C23×C4C23×C12711yescharacteristic
358C23×C4C23×C12711yescharacteristic
359C4⋊Q8—232no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4D4×C12714yes—
377C2×C4⋊C4D4×C12714yes—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—42no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—42no—
393#95—42no—
394#95—42no—
395#95Q8×C12718yescomplemented
396C24C24×C12711yescharacteristic
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—44no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—44no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
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—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×C4C2×D4×C12712yes—
457C22×C4C2×D4×C12712yes—
458C22×C4C2×D4×C12712yes—
459C22×C4C2×D4×C12712yes—
460C22×C4C2×D4×C12712yes—
461C22×C4C2×D4×C12712yes—
462C22×C4C2×D4×C12712yes—
463C22×C4C2×D4×C12712yes—
464C22×C4C2×D4×C12712yes—
465C22×C4C2×Q8×C12714yes—
466C22×C4C2×Q8×C12714yes—
467C22×C4C2×Q8×C12714yes—
468C22×C4#41×C12714yes—
469C22×C4#41×C12714yes—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C42—28no—
477C42—44no—
478C23C22×D4×C12711yescharacteristic
479C23C22×D4×C12711yescharacteristic
480C23C22×D4×C12711yescharacteristic
481C23C22×Q8×C12712yes—
482C23C2×#41×C12712yes—
483C23C2×#41×C12712yes—
484C23H(5,𝔽2)×C12712yes—
485C232−1+4×C12712yes—
486C232−1+4×C12712yes—
487Q8—416no—
488Q8—416no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
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#120×C12714yes—
549C2×C4#123×C12714yes—
550C2×C4#124×C12714yes—
551C2×C4C4⋊Q8×C12714yes—
552C22C2×#120×C12711yescharacteristic
553C22C2×#123×C12711yescharacteristic
554C22C2×#124×C12711yescharacteristic
555C22C2×C4⋊Q8×C12711yescharacteristic
556C22#533×C12712yes—
557C22#534×C12711yescharacteristic
558C22#536×C12711yescharacteristic
559C22#536×C12711yescharacteristic
560C22#541×C12712yes—
561C22#542×C12712yes—
562C22#544×C12712yes—
563C22#546×C12712yes—
564C22#546×C12712yes—
565C22#548×C12712yes—
566C22#548×C12712yes—
567C22D4×Q8×C12712yes—
568C22D4×Q8×C12712yes—
569C22#554×C12712yes—
570C22#556×C12712yes—
571C22#557×C12712yes—
572C22#563×C12712yes—
573C4—24no—
574C4—24no—
575C4—24no—
576C4—24no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—44no—
581C4—44no—
582C4—82no—
583C4—82no—
584C2C127×#240511yescharacteristic
585C2C127×#243411yescharacteristic
586C2C127×#243611yescharacteristic
587C2C127×#244612yes—
588C2C127×#249212yes—
589C2C127×#255012yes—
590C2C127×#271312yes—
591C2C127×#271612yes—
592C2C127×#272912yes—
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
98. #95×C127Q8semidirect
287. #15783C127direct (N × H)
308. C127#15783direct (N × H)
395. #95Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q8×C1272
C2×#400×C1272
C2×#398×C1272
C2×#391×C1272
C127×#14462
C127×#14422
C127×#14412
C127×#14362
#15783127