Finite Group Databasebeta

← C127×#21771 #44585444

Subgroups — C127×#21771 #44585444

Order 32512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C127C211yesmaximal, characteristic
2C2×#408×C127C211yesmaximal, characteristic
3C127×#2311C211yesmaximal, characteristic
4C127×#1894C212yesmaximal, complemented
5C127×#1892C212yesmaximal, complemented
6C127×#1873C212yesmaximal, complemented
7C127×#1872C212yesmaximal, complemented
8C127×#1825C212yesmaximal, complemented
9C127×#1787C212yesmaximal, complemented
10C22×#41×C127C2211yescharacteristic
11C23×Q8×C127C2211yescharacteristic
12C22×M16×C127C2211yescharacteristic
13C23×C8×C127C2211yescharacteristic
14#536×C127—24no—
15#536×C127C418yescomplemented
16C2×C4⋊Q8×C127—21no—
17C2×#124×C127—21no—
18C2×#123×C127—21no—
19C2×#123×C127—21no—
20C2×C22⋊Q8×C127C2212yes—
21C2×C22⋊Q8×C127C2212yes—
22C2×#117×C127C2211yescharacteristic
23C2×#117×C127C2211yescharacteristic
24C22×C4⋊C4×C127C2211yescharacteristic
25C2×#103×C127C2212yes—
26C2×#103×C127C2212yes—
27C2×#103×C127C2212yes—
28C2×#103×C127C2212yes—
29#408×C127—24no—
30C2×C22⋊C8×C127C2212yes—
31C2×C22⋊C8×C127C2212yes—
32#391×C127—21no—
33#390×C127—21no—
34#386×C127—21no—
35#386×C127—21no—
36#380×C127C2212yes—
37#380×C127C2212yes—
38#340×C127C2212yes—
39#340×C127C2212yes—
40#340×C127C2212yes—
41#340×C127C2212yes—
42C2×#41×C127—41no—
43C2×#41×C127—41no—
44C2×#41×C127—41no—
45C2×#41×C127—41no—
46C2×#41×C127C2×C414yes—
47C2×#41×C127D414yes—
48C22×Q8×C127—21no—
49C22×Q8×C127—22no—
50C22×Q8×C127—41no—
51C22×Q8×C127—41no—
52C22×Q8×C127—41no—
53C22×Q8×C127—41no—
54C22×Q8×C127C2×C414yescomplemented
55C22×Q8×C127D411yescharacteristic
56C22×Q8×C127C2312yes—
57C22×D4×C127D411yescharacteristic
58C22×D4×C127D411yescharacteristic
59C22×D4×C127D411yescharacteristic
60C23×C4×C127—21no—
61C23×C4×C127—21no—
62C23×C4×C127C2311yescharacteristic
63C23×C4×C127C2311yescharacteristic
64C23×C4×C127C2311yescharacteristic
65C2×M16×C127—22no—
66C2×M16×C127—41no—
67C2×M16×C127—41no—
68C22×C8×C127—22no—
69C22×C8×C127—41no—
70C22×C8×C127—41no—
71C22×C8×C127C2312yes—
72C22×C8×C127C2312yes—
73C4⋊Q8×C127—28no—
74#124×C127—28no—
75#123×C127—28no—
76#123×C127—28no—
77C22⋊Q8×C127—24no—
78C22⋊Q8×C127—24no—
79C22⋊Q8×C127C2×C418yes—
80C22⋊Q8×C127C2×C418yescomplemented
81#117×C127—22no—
82#117×C127—41no—
83#117×C127—41no—
84#117×C127C2×C414yescomplemented
85C2×C4⋊C4×C127—22no—
86C2×C4⋊C4×C127—22no—
87C2×C4⋊C4×C127—21no—
88C2×C4⋊C4×C127—41no—
89C2×C4⋊C4×C127—41no—
90C2×C4⋊C4×C127C2312yes—
91C2×C4⋊C4×C127C2312yes—
92C2×C4⋊C4×C127C2312yes—
93C2×C22⋊C4×C127—21no—
94C2×C22⋊C4×C127—22no—
95C2×C22⋊C4×C127—22no—
96C2×C22⋊C4×C127—21no—
97C2×C22⋊C4×C127—22no—
98C2×C22⋊C4×C127—22no—
99C2×C42×C127—21no—
100C2×C42×C127—21no—
101#103×C127—42no—
102#103×C127—42no—
103#103×C127—42no—
104#103×C127—42no—
105C22⋊C8×C127—24no—
106C22⋊C8×C127—24no—
107#95×C127—22no—
108#95×C127—22no—
109C24×C127C2×D411yescharacteristic
110C24×C127C2×D411yescharacteristic
111C24×C127C2×D411yescharacteristic
112#41×C127—44no—
113#41×C127—44no—
114#41×C127—44no—
115#41×C127—44no—
116C2×Q8×C127—24no—
117C2×Q8×C127—22no—
118C2×Q8×C127—22no—
119C2×Q8×C127—22no—
120C2×Q8×C127—22no—
121C2×Q8×C127—22no—
122C2×Q8×C127—41no—
123C2×Q8×C127—41no—
124C2×Q8×C127—44no—
125C2×Q8×C127—44no—
126C2×Q8×C127—42no—
127C2×Q8×C127—42no—
128C2×Q8×C127—44no—
129C2×Q8×C127—44no—
130C2×Q8×C127C22⋊C414yes—
131C2×Q8×C127C22×C414yes—
132C2×D4×C127—41no—
133C2×D4×C127—41no—
134C2×D4×C127—41no—
135C2×D4×C127—41no—
136C2×D4×C127—41no—
137C2×D4×C127—41no—
138C2×D4×C127C22⋊C414yes—
139C2×D4×C127C22⋊C414yes—
140C2×D4×C127C22⋊C414yes—
141C22×C4×C127—21no—
142C22×C4×C127—21no—
143C22×C4×C127—21no—
144C22×C4×C127—21no—
145C22×C4×C127—21no—
146C22×C4×C127—21no—
147C22×C4×C127—21no—
148C22×C4×C127—21no—
149C22×C4×C127—21no—
150C22×C4×C127—21no—
151C22×C4×C127—21no—
152C22×C4×C127—21no—
153C22×C4×C127—24no—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127—42no—
159C22×C4×C127—42no—
160C22×C4×C127—42no—
161C22×C4×C127—82no—
162C22×C4×C127C22×C412yes—
163C22×C4×C127C22×C412yes—
164C22×C4×C127C22×C412yes—
165C22×C4×C127C2×D411yescharacteristic
166C22×C4×C127C2×D411yescharacteristic
167C22×C4×C127C2×D411yescharacteristic
168C22×C4×C127C2×D412yes—
169C22×C4×C127C2×D412yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127C2411yescharacteristic
172M16×C127—44no—
173C2×C8×C127—22no—
174C2×C8×C127—22no—
175C2×C8×C127—22no—
176C2×C8×C127—22no—
177C2×C8×C127—22no—
178C2×C8×C127—22no—
179C2×C8×C127—44no—
180C4⋊C4×C127—28no—
181C4⋊C4×C127—28no—
182C4⋊C4×C127—22no—
183C4⋊C4×C127—22no—
184C4⋊C4×C127—22no—
185C4⋊C4×C127—22no—
186C4⋊C4×C127—22no—
187C4⋊C4×C127—42no—
188C4⋊C4×C127C22×C414yes—
189C22⋊C4×C127—24no—
190C22⋊C4×C127—28no—
191C22⋊C4×C127—28no—
192C22⋊C4×C127—42no—
193C22⋊C4×C127—82no—
194C22⋊C4×C127—82no—
195C42×C127—24no—
196C42×C127—42no—
197C23×C127—21no—
198C23×C127—21no—
199C23×C127—21no—
200C23×C127—21no—
201C23×C127—21no—
202C23×C127—21no—
203C23×C127—22no—
204C23×C127—22no—
205C23×C127—22no—
206C23×C127—22no—
207C23×C127—22no—
208C23×C127—22no—
209C23×C127C2×C22⋊C412yes—
210C23×C127C2×C22⋊C412yes—
211C23×C127C2×C22⋊C412yes—
212C23×C127C22×D411yescharacteristic
213Q8×C127—24no—
214Q8×C127—24no—
215Q8×C127—44no—
216Q8×C127—48no—
217Q8×C127—42no—
218Q8×C127—42no—
219D4×C127—44no—
220D4×C127—44no—
221D4×C127—44no—
222C2×C4×C127—21no—
223C2×C4×C127—21no—
224C2×C4×C127—21no—
225C2×C4×C127—21no—
226C2×C4×C127—21no—
227C2×C4×C127—21no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—22no—
232C2×C4×C127—22no—
233C2×C4×C127—22no—
234C2×C4×C127—22no—
235C2×C4×C127—22no—
236C2×C4×C127—22no—
237C2×C4×C127—22no—
238C2×C4×C127—22no—
239C2×C4×C127—22no—
240C2×C4×C127—24no—
241C2×C4×C127—24no—
242C2×C4×C127—48no—
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—84no—
254C2×C4×C127C2×C22⋊C412yes—
255C2×C4×C127C2×C22⋊C412yes—
256C2×C4×C127C2×C22⋊C412yes—
257C2×C4×C127C23×C411yescharacteristic
258C2×C4×C127C22×D411yescharacteristic
259C2×C4×C127H(5,𝔽2)11yescharacteristic
260C2×C4×C127H(5,𝔽2)11yescharacteristic
261C2×C4×C127H(5,𝔽2)11yescharacteristic
262C2×C4×C127H(5,𝔽2)11yescharacteristic
263C8×C127—42no—
264C8×C127—42no—
265C22×C127—22no—
266C22×C127—22no—
267C22×C127—22no—
268C22×C127—24no—
269C22×C127—24no—
270C22×C127—24no—
271C22×C127—41no—
272C22×C127—41no—
273C22×C127—41no—
274C22×C127—42no—
275C22×C127—42no—
276C22×C127—42no—
277C22×C127—41no—
278C22×C127—41no—
279C22×C127—41no—
280C22×C127C22×C22⋊C411yescharacteristic
281C22×C127#53411yescharacteristic
282C22×C127#53411yescharacteristic
283C22×C127#57712yes—
284C22×C127#57812yes—
285C4×C127—22no—
286C4×C127—22no—
287C4×C127—22no—
288C4×C127—44no—
289C4×C127—44no—
290C4×C127—82no—
291C4×C127—82no—
292C4×C127#51811yescharacteristic
293C4×C127#51811yescharacteristic
294C4×C127#53411yescharacteristic
295C4×C127#53411yescharacteristic
296#21771C12711yesmaximal, characteristic, direct product factor, Sylow
297C2×C127—42no—
298C2×C127—42no—
299C2×C127—42no—
300C2×C127#230811yescharacteristic
301C2×C127#289812yes—
302C2×C127#320011yescharacteristic
303C2×C127#320112yes—
304C2×C127#320211yescharacteristic
305C2×#536C2×C12711yescharacteristic
306C2×#408C2×C12711yescharacteristic
307#2311C2×C12711yescharacteristic
308#1894C2×C12712yescomplemented
309#1892C2×C12712yescomplemented
310#1873C2×C12712yescomplemented
311#1872C2×C12712yescomplemented
312#1825C2×C12712yescomplemented
313#1787C2×C12712yescomplemented
314C127#2177111yescharacteristic, direct product factor, Sylow
315C22×#41C22×C12711yescharacteristic
316C23×Q8C22×C12711yescharacteristic
317C22×M16C22×C12711yescharacteristic
318C23×C8C22×C12711yescharacteristic
319#536—24no—
320#536C4×C12718yescomplemented
321C2×C4⋊Q8—21no—
322C2×#124—21no—
323C2×#123—21no—
324C2×#123—21no—
325C2×C22⋊Q8C22×C12712yes—
326C2×C22⋊Q8C22×C12712yes—
327C2×#117C22×C12711yescharacteristic
328C2×#117C22×C12711yescharacteristic
329C22×C4⋊C4C22×C12711yescharacteristic
330C2×#103C22×C12712yes—
331C2×#103C22×C12712yes—
332C2×#103C22×C12712yes—
333C2×#103C22×C12712yes—
334#408—24no—
335C2×C22⋊C8C22×C12712yes—
336C2×C22⋊C8C22×C12712yes—
337#391—21no—
338#390—21no—
339#386—21no—
340#386—21no—
341#380C22×C12712yes—
342#380C22×C12712yes—
343#340C22×C12712yes—
344#340C22×C12712yes—
345#340C22×C12712yes—
346#340C22×C12712yes—
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41—41no—
351C2×#41C2×C4×C12714yes—
352C2×#41D4×C12714yes—
353C22×Q8—21no—
354C22×Q8—22no—
355C22×Q8—41no—
356C22×Q8—41no—
357C22×Q8—41no—
358C22×Q8—41no—
359C22×Q8C2×C4×C12714yescomplemented
360C22×Q8D4×C12711yescharacteristic
361C22×Q8C23×C12712yes—
362C22×D4D4×C12711yescharacteristic
363C22×D4D4×C12711yescharacteristic
364C22×D4D4×C12711yescharacteristic
365C23×C4—21no—
366C23×C4—21no—
367C23×C4C23×C12711yescharacteristic
368C23×C4C23×C12711yescharacteristic
369C23×C4C23×C12711yescharacteristic
370C2×M16—22no—
371C2×M16—41no—
372C2×M16—41no—
373C22×C8—22no—
374C22×C8—41no—
375C22×C8—41no—
376C22×C8C23×C12712yes—
377C22×C8C23×C12712yes—
378C4⋊Q8—28no—
379#124—28no—
380#123—28no—
381#123—28no—
382C22⋊Q8—24no—
383C22⋊Q8—24no—
384C22⋊Q8C2×C4×C12718yes—
385C22⋊Q8C2×C4×C12718yescomplemented
386#117—22no—
387#117—41no—
388#117—41no—
389#117C2×C4×C12714yescomplemented
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—41no—
394C2×C4⋊C4—41no—
395C2×C4⋊C4C23×C12712yes—
396C2×C4⋊C4C23×C12712yes—
397C2×C4⋊C4C23×C12712yes—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C42—21no—
405C2×C42—21no—
406#103—42no—
407#103—42no—
408#103—42no—
409#103—42no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412#95—22no—
413#95—22no—
414C24C2×D4×C12711yescharacteristic
415C24C2×D4×C12711yescharacteristic
416C24C2×D4×C12711yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8C22⋊C4×C12714yes—
436C2×Q8C22×C4×C12714yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4C22⋊C4×C12714yes—
444C2×D4C22⋊C4×C12714yes—
445C2×D4C22⋊C4×C12714yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—82no—
467C22×C4C22×C4×C12712yes—
468C22×C4C22×C4×C12712yes—
469C22×C4C22×C4×C12712yes—
470C22×C4C2×D4×C12711yescharacteristic
471C22×C4C2×D4×C12711yescharacteristic
472C22×C4C2×D4×C12711yescharacteristic
473C22×C4C2×D4×C12712yes—
474C22×C4C2×D4×C12712yes—
475C22×C4C2×D4×C12712yes—
476C22×C4C24×C12711yescharacteristic
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4C22×C4×C12714yes—
494C22⋊C4—24no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—42no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C42—24no—
501C42—42no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23C2×C22⋊C4×C12712yes—
515C23C2×C22⋊C4×C12712yes—
516C23C2×C22⋊C4×C12712yes—
517C23C22×D4×C12711yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—44no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—48no—
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—84no—
559C2×C4C2×C22⋊C4×C12712yes—
560C2×C4C2×C22⋊C4×C12712yes—
561C2×C4C2×C22⋊C4×C12712yes—
562C2×C4C23×C4×C12711yescharacteristic
563C2×C4C22×D4×C12711yescharacteristic
564C2×C4H(5,𝔽2)×C12711yescharacteristic
565C2×C4H(5,𝔽2)×C12711yescharacteristic
566C2×C4H(5,𝔽2)×C12711yescharacteristic
567C2×C4H(5,𝔽2)×C12711yescharacteristic
568C8—42no—
569C8—42no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×C22⋊C4×C12711yescharacteristic
586C22#534×C12711yescharacteristic
587C22#534×C12711yescharacteristic
588C22#577×C12712yes—
589C22#578×C12712yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—44no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4#518×C12711yescharacteristic
598C4#518×C12711yescharacteristic
599C4#534×C12711yescharacteristic
600C4#534×C12711yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2C127×#230811yescharacteristic
605C2C127×#289812yes—
606C2C127×#320011yescharacteristic
607C2C127×#320112yes—
608C2C127×#320211yescharacteristic
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×#41×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1894C2semidirect
5. C127×#1892C2semidirect
6. C127×#1873C2semidirect
7. C127×#1872C2semidirect
8. C127×#1825C2semidirect
9. C127×#1787C2semidirect
15. #536×C127C4semidirect
54. C22×Q8×C127C2×C4semidirect
80. C22⋊Q8×C127C2×C4semidirect
84. #117×C127C2×C4semidirect
296. #21771C127direct (N × H)
308. #1894C2×C127semidirect
309. #1892C2×C127semidirect
310. #1873C2×C127semidirect
311. #1872C2×C127semidirect
312. #1825C2×C127semidirect
313. #1787C2×C127semidirect
314. C127#21771direct (N × H)
320. #536C4×C127semidirect
359. C22×Q8C2×C4×C127semidirect
385. C22⋊Q8C2×C4×C127semidirect
389. #117C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1272
C2×#408×C1272
C127×#23112
C127×#18942
C127×#18922
C127×#18732
C127×#18722
C127×#18252
C127×#17872
#21771127