Finite Group Databasebeta

← C127×#15020 #44578693

Subgroups — C127×#15020 #44578693

Order 32512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C127C211yesmaximal, characteristic, complemented
2C2×#399×C127C211yesmaximal, characteristic
3C2×#398×C127C211yesmaximal, characteristic, complemented
4C2×#397×C127C211yesmaximal, characteristic
5C2×#388×C127C211yesmaximal, characteristic
6C2×#382×C127C211yesmaximal, characteristic, complemented
7C127×#1445C212yesmaximal, complemented
8C127×#1439C212yesmaximal
9C127×#1437C212yesmaximal
10C127×C4⋊#95C212yesmaximal, complemented
11C4×#95×C127C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2212yes—
16C22×C4⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2211yescharacteristic
20C22×C22⋊C4×C127C2211yescharacteristic
21C22×C22⋊C4×C127C2212yes—
22C22×C42×C127C2212yes—
23C22×C42×C127C2211yescharacteristic
24#400×C127—24no—
25#399×C127—24no—
26#398×C127—24no—
27#397×C127—24no—
28#388×C127—24no—
29#382×C127—24no—
30C2×#95×C127C2212yes—
31C2×#95×C127C2212yes—
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C2×#95×C127C2211yescharacteristic
36C2×#95×C127C2212yes—
37C2×#95×C127C2212yes—
38C2×#95×C127C2211yescharacteristic
39C2×#95×C127C2211yescharacteristic
40C2×#95×C127C2211yescharacteristic
41C25×C127C2311yescharacteristic
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2312yes—
44C23×C4×C127C2312yes—
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2312yes—
48C23×C4×C127C2311yescharacteristic
49C23×C4×C127C2311yescharacteristic
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2312yes—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127—24no—
57C2×C4⋊C4×C127—22no—
58C2×C4⋊C4×C127—22no—
59C2×C4⋊C4×C127—24no—
60C2×C4⋊C4×C127—22no—
61C2×C4⋊C4×C127—22no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—24no—
64C2×C4⋊C4×C127—24no—
65C2×C4⋊C4×C127—24no—
66C2×C4⋊C4×C127—42no—
67C2×C22⋊C4×C127—22no—
68C2×C22⋊C4×C127—22no—
69C2×C22⋊C4×C127—22no—
70C2×C22⋊C4×C127—24no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—22no—
73C2×C22⋊C4×C127—24no—
74C2×C22⋊C4×C127—22no—
75C2×C22⋊C4×C127—24no—
76C2×C22⋊C4×C127—22no—
77C2×C22⋊C4×C127—24no—
78C2×C22⋊C4×C127—24no—
79C2×C22⋊C4×C127—24no—
80C2×C22⋊C4×C127—42no—
81C2×C42×C127—24no—
82C2×C42×C127—24no—
83C2×C42×C127—24no—
84C2×C42×C127—42no—
85C2×C42×C127D414yes—
86#95×C127—24no—
87#95×C127—24no—
88#95×C127—22no—
89#95×C127—22no—
90#95×C127—24no—
91#95×C127—24no—
92#95×C127—24no—
93#95×C127—24no—
94#95×C127—24no—
95#95×C127D414yes—
96#95×C127D414yes—
97C24×C127—21no—
98C24×C127—21no—
99C24×C127—21no—
100C24×C127—21no—
101C24×C127—22no—
102C24×C127—22no—
103C24×C127—22no—
104C24×C127—21no—
105C24×C127—21no—
106C24×C127—22no—
107C24×C127#4112yes—
108C24×C127C2411yescharacteristic
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—21no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—22no—
118C22×C4×C127—21no—
119C22×C4×C127—21no—
120C22×C4×C127—21no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—21no—
127C22×C4×C127—21no—
128C22×C4×C127—21no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—21no—
132C22×C4×C127—22no—
133C22×C4×C127—21no—
134C22×C4×C127—21no—
135C22×C4×C127—21no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—21no—
140C22×C4×C127—21no—
141C22×C4×C127—21no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—22no—
145C22×C4×C127—21no—
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×C127C2×D414yes—
159C22×C4×C127C2×D412yes—
160C22×C4×C127C2×D412yes—
161C22×C4×C127C2×D414yes—
162C22×C4×C127C2×D412yes—
163C22×C4×C127C2×D412yes—
164C22×C4×C127C2×D412yes—
165C22×C4×C127#4112yes—
166C22×C4×C127#4114yes—
167C22×C4×C127#4114yes—
168C22×C4×C127#4114yes—
169C4⋊C4×C127—48no—
170C4⋊C4×C127—48no—
171C4⋊C4×C127—44no—
172C4⋊C4×C127—44no—
173C4⋊C4×C127—44no—
174C22⋊C4×C127—28no—
175C22⋊C4×C127—48no—
176C22⋊C4×C127—44no—
177C22⋊C4×C127—48no—
178C22⋊C4×C127—44no—
179C42×C127—28no—
180C42×C127—48no—
181C23×C127—22no—
182C23×C127—24no—
183C23×C127—22no—
184C23×C127—22no—
185C23×C127—24no—
186C23×C127—41no—
187C23×C127—42no—
188C23×C127—42no—
189C23×C127—41no—
190C23×C127—41no—
191C23×C127—41no—
192C23×C127—42no—
193C23×C127—41no—
194C23×C127—41no—
195C23×C127—42no—
196C23×C127—42no—
197C23×C127—41no—
198C23×C127—42no—
199C23×C127—42no—
200C23×C127—41no—
201C23×C127—42no—
202C23×C127—42no—
203C23×C127—42no—
204C23×C127C22×D411yescharacteristic
205C23×C127C22×D411yescharacteristic
206C23×C127C22×D411yescharacteristic
207C23×C127C2×#4112yes—
208C23×C127C2×#4111yescharacteristic
209C23×C127C2×#4111yescharacteristic
210C23×C127C2×#4112yes—
211C23×C127C2×#4112yes—
212C23×C127H(5,𝔽2)11yescharacteristic
213C23×C1272−1+412yes—
214C23×C1272−1+411yescharacteristic
215C2×C4×C127—22no—
216C2×C4×C127—24no—
217C2×C4×C127—22no—
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—24no—
227C2×C4×C127—24no—
228C2×C4×C127—24no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—22no—
232C2×C4×C127—22no—
233C2×C4×C127—24no—
234C2×C4×C127—24no—
235C2×C4×C127—22no—
236C2×C4×C127—24no—
237C2×C4×C127—22no—
238C2×C4×C127—42no—
239C2×C4×C127—42no—
240C2×C4×C127—42no—
241C2×C4×C127—41no—
242C2×C4×C127—41no—
243C2×C4×C127—42no—
244C2×C4×C127—41no—
245C2×C4×C127—41no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—41no—
249C2×C4×C127—42no—
250C2×C4×C127—42no—
251C2×C4×C127—42no—
252C2×C4×C127—42no—
253C2×C4×C127—41no—
254C2×C4×C127—42no—
255C2×C4×C127—41no—
256C2×C4×C127—42no—
257C2×C4×C127—41no—
258C2×C4×C127—42no—
259C2×C4×C127—41no—
260C2×C4×C127—42no—
261C2×C4×C127—42no—
262C2×C4×C127—42no—
263C2×C4×C127—42no—
264C2×C4×C127—41no—
265C2×C4×C127—42no—
266C2×C4×C127—42no—
267C2×C4×C127—42no—
268C2×C4×C127—42no—
269C2×C4×C127—42no—
270C2×C4×C127—42no—
271C2×C4×C127—42no—
272C2×C4×C127#12314yes—
273C2×C4×C127#12418yes—
274C2×C4×C127#12714yes—
275C22×C127—42no—
276C22×C127—42no—
277C22×C127—44no—
278C22×C127—42no—
279C22×C127—44no—
280C22×C127—81no—
281C22×C127—81no—
282C22×C127—82no—
283C22×C127—81no—
284C22×C127—81no—
285C22×C127—82no—
286C22×C127C2×#12311yescharacteristic
287C22×C127#52511yescharacteristic
288C22×C127C2×#12412yes—
289C22×C127#52912yes—
290C22×C127#52912yes—
291C22×C127C2×#12711yescharacteristic
292C22×C127#53211yescharacteristic
293C22×C127#53211yescharacteristic
294C22×C127#53212yes—
295C22×C127#53611yescharacteristic
296C22×C127#53711yescharacteristic
297C22×C127#53912yes—
298C22×C127#54112yes—
299C22×C127#54611yescharacteristic
300C22×C127#54611yescharacteristic
301C22×C127#54712yes—
302C22×C127#54712yes—
303C22×C127#54811yescharacteristic
304C22×C127#54811yescharacteristic
305C22×C127#55112yes—
306C22×C127#55212yes—
307C22×C127#55212yes—
308C22×C127#55212yes—
309C4×C127—28no—
310C4×C127—24no—
311C4×C127—24no—
312C4×C127—42no—
313C4×C127—42no—
314C4×C127—44no—
315C4×C127—82no—
316C4×C127—81no—
317C4×C127—81no—
318C4×C127—82no—
319#15020C12711yesmaximal, characteristic, direct product factor, Sylow
320C2×C127—82no—
321C2×C127#239511yescharacteristic
322C2×C127#248812yes—
323C2×C127#248912yes—
324C2×C127#249811yescharacteristic
325C2×C127#250111yescharacteristic
326C2×C127#251912yes—
327C2×C127#254011yescharacteristic
328C2×C127#255312yes—
329C2×C127#261111yescharacteristic
330C2×C127#266411yescharacteristic
331C2×C127#285811yescharacteristic
332C2×#400C2×C12711yescharacteristic, complemented
333C2×#399C2×C12711yescharacteristic
334C2×#398C2×C12711yescharacteristic, complemented
335C2×#397C2×C12711yescharacteristic
336C2×#388C2×C12711yescharacteristic
337C2×#382C2×C12711yescharacteristic, complemented
338#1445C2×C12712yescomplemented
339#1439C2×C12712yes—
340#1437C2×C12712yes—
341C4⋊#95C2×C12712yescomplemented
342C4×#95C2×C12711yescharacteristic, complemented
343C127#1502011yescharacteristic, direct product factor, Sylow
344C22×C4⋊C4C22×C12711yescharacteristic
345C22×C4⋊C4C22×C12711yescharacteristic
346C22×C4⋊C4C22×C12711yescharacteristic
347C22×C4⋊C4C22×C12712yes—
348C22×C4⋊C4C22×C12712yes—
349C22×C22⋊C4C22×C12711yescharacteristic
350C22×C22⋊C4C22×C12712yes—
351C22×C22⋊C4C22×C12711yescharacteristic
352C22×C22⋊C4C22×C12711yescharacteristic
353C22×C22⋊C4C22×C12712yes—
354C22×C42C22×C12712yes—
355C22×C42C22×C12711yescharacteristic
356#400—24no—
357#399—24no—
358#398—24no—
359#397—24no—
360#388—24no—
361#382—24no—
362C2×#95C22×C12712yes—
363C2×#95C22×C12712yes—
364C2×#95C22×C12712yes—
365C2×#95C22×C12712yes—
366C2×#95C22×C12712yes—
367C2×#95C22×C12711yescharacteristic
368C2×#95C22×C12712yes—
369C2×#95C22×C12712yes—
370C2×#95C22×C12711yescharacteristic
371C2×#95C22×C12711yescharacteristic
372C2×#95C22×C12711yescharacteristic
373C25C23×C12711yescharacteristic
374C23×C4C23×C12711yescharacteristic
375C23×C4C23×C12712yes—
376C23×C4C23×C12712yes—
377C23×C4C23×C12711yescharacteristic
378C23×C4C23×C12711yescharacteristic
379C23×C4C23×C12712yes—
380C23×C4C23×C12711yescharacteristic
381C23×C4C23×C12711yescharacteristic
382C23×C4C23×C12712yes—
383C23×C4C23×C12711yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—42no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—42no—
413C2×C42—24no—
414C2×C42—24no—
415C2×C42—24no—
416C2×C42—42no—
417C2×C42D4×C12714yes—
418#95—24no—
419#95—24no—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95D4×C12714yes—
428#95D4×C12714yes—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24#41×C12712yes—
440C24C24×C12711yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4C2×D4×C12714yes—
491C22×C4C2×D4×C12712yes—
492C22×C4C2×D4×C12712yes—
493C22×C4C2×D4×C12714yes—
494C22×C4C2×D4×C12712yes—
495C22×C4C2×D4×C12712yes—
496C22×C4C2×D4×C12712yes—
497C22×C4#41×C12712yes—
498C22×C4#41×C12714yes—
499C22×C4#41×C12714yes—
500C22×C4#41×C12714yes—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C42—28no—
512C42—48no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C22×D4×C12711yescharacteristic
537C23C22×D4×C12711yescharacteristic
538C23C22×D4×C12711yescharacteristic
539C23C2×#41×C12712yes—
540C23C2×#41×C12711yescharacteristic
541C23C2×#41×C12711yescharacteristic
542C23C2×#41×C12712yes—
543C23C2×#41×C12712yes—
544C23H(5,𝔽2)×C12711yescharacteristic
545C232−1+4×C12712yes—
546C232−1+4×C12711yescharacteristic
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4#123×C12714yes—
605C2×C4#124×C12718yes—
606C2×C4#127×C12714yes—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—81no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22C2×#123×C12711yescharacteristic
619C22#525×C12711yescharacteristic
620C22C2×#124×C12712yes—
621C22#529×C12712yes—
622C22#529×C12712yes—
623C22C2×#127×C12711yescharacteristic
624C22#532×C12711yescharacteristic
625C22#532×C12711yescharacteristic
626C22#532×C12712yes—
627C22#536×C12711yescharacteristic
628C22#537×C12711yescharacteristic
629C22#539×C12712yes—
630C22#541×C12712yes—
631C22#546×C12711yescharacteristic
632C22#546×C12711yescharacteristic
633C22#547×C12712yes—
634C22#547×C12712yes—
635C22#548×C12711yescharacteristic
636C22#548×C12711yescharacteristic
637C22#551×C12712yes—
638C22#552×C12712yes—
639C22#552×C12712yes—
640C22#552×C12712yes—
641C4—28no—
642C4—24no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—82no—
648C4—81no—
649C4—81no—
650C4—82no—
651C2—82no—
652C2C127×#239511yescharacteristic
653C2C127×#248812yes—
654C2C127×#248912yes—
655C2C127×#249811yescharacteristic
656C2C127×#250111yescharacteristic
657C2C127×#251912yes—
658C2C127×#254011yescharacteristic
659C2C127×#255312yes—
660C2C127×#261111yescharacteristic
661C2C127×#266411yescharacteristic
662C2C127×#285811yescharacteristic
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
1. C2×#400×C127C2semidirect
3. C2×#398×C127C2semidirect
6. C2×#382×C127C2semidirect
7. C127×#1445C2semidirect
10. C127×C4⋊#95C2semidirect
11. C4×#95×C127C2semidirect
319. #15020C127direct (N × H)
332. C2×#400C2×C127semidirect
334. C2×#398C2×C127semidirect
337. C2×#382C2×C127semidirect
338. #1445C2×C127semidirect
341. C4⋊#95C2×C127semidirect
342. C4×#95C2×C127semidirect
343. C127#15020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C1272
C2×#399×C1272
C2×#398×C1272
C2×#397×C1272
C2×#388×C1272
C2×#382×C1272
C127×#14452
C127×#14392
C127×#14372
C127×C4⋊#952
C4×#95×C1272
#15020127