Finite Group Databasebeta

← C127×#20918 #44584591

Subgroups — C127×#20918 #44584591

Order 32512.

1714 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×#575×C127C211yesmaximal, characteristic
2C2×#568×C127C211yesmaximal, characteristic, complemented
3C22×#104×C127C211yesmaximal, characteristic, complemented
4C22×#104×C127C211yesmaximal, characteristic, complemented
5C127×#2879C211yesmaximal, characteristic, complemented
6C127×#1876C218yesmaximal, complemented
7C127×#1738C211yesmaximal, characteristic, complemented
8C127×#1728C211yesmaximal, characteristic, complemented
9C22×#41×C127C2211yescharacteristic
10C22×#41×C127C2211yescharacteristic
11#575×C127—24no—
12#575×C127—24no—
13C2×#137×C127C412yescomplemented
14C2×#137×C127C412yescomplemented
15C2×AGL(1,ℤ/8)×C127C412yescomplemented
16C2×AGL(1,ℤ/8)×C127C412yescomplemented
17C2×#135×C127C2214yes—
18#568×C127—24no—
19#568×C127—24no—
20C2×#131×C127—22no—
21C2×#131×C127C2214yescomplemented
22C22×M16×C127—22no—
23C22×M16×C127C2211yescharacteristic
24C22×M16×C127C2211yescharacteristic
25C22×M16×C127C2211yescharacteristic
26C22×M16×C127C2211yescharacteristic
27C22×C42×C127C2211yescharacteristic
28C2×#104×C127—24no—
29C2×#104×C127—24no—
30C2×#104×C127C2214yescomplemented
31C2×#104×C127C2214yescomplemented
32#408×C127—22no—
33#408×C127C2214yescomplemented
34#407×C127—22no—
35#341×C127C2214yescomplemented
36#336×C127C2214yescomplemented
37C2×#41×C127—21no—
38C2×#41×C127—22no—
39C2×#41×C127—21no—
40C2×#41×C127—21no—
41C2×#41×C127—22no—
42C2×#41×C127—21no—
43C2×#41×C127—22no—
44C2×#41×C127—22no—
45C2×#41×C127—22no—
46C2×#41×C127—22no—
47C2×#41×C127C2×C412yes—
48C2×#41×C127C2×C412yes—
49C2×#41×C127D414yes—
50C2×#41×C127D414yes—
51C2×#41×C127C2312yes—
52C2×#41×C127C2312yes—
53C22×Q8×C127C2×C411yescharacteristic
54C22×Q8×C127C2×C411yescharacteristic
55C22×D4×C127—21no—
56C22×D4×C127—21no—
57C22×D4×C127C2×C411yescharacteristic
58C22×D4×C127C2×C411yescharacteristic
59C23×C4×C127—21no—
60C23×C4×C127—21no—
61C23×C4×C127—21no—
62C23×C4×C127C2311yescharacteristic
63#137×C127—28no—
64#137×C127—28no—
65AGL(1,ℤ/8)×C127—28no—
66AGL(1,ℤ/8)×C127—28no—
67#135×C127—28no—
68#135×C127—28no—
69C2×Q16×C127C2×C414yescomplemented
70C2×SD16×C127C2×C414yescomplemented
71C2×SD16×C127C2×C414yescomplemented
72C2×D8×C127C2×C414yescomplemented
73#131×C127—216no—
74#131×C127—28no—
75#131×C127—28no—
76C2×M16×C127—216no—
77C2×M16×C127—21no—
78C2×M16×C127—22no—
79C2×M16×C127—22no—
80C2×M16×C127—22no—
81C2×M16×C127—22no—
82C2×M16×C127—22no—
83C2×M16×C127—21no—
84C2×M16×C127—24no—
85C2×M16×C127—22no—
86C2×M16×C127—22no—
87C2×M16×C127C2×C412yes—
88C2×M16×C127C2×C412yes—
89C2×M16×C127D414yes—
90C2×M16×C127D412yes—
91C2×M16×C127D412yes—
92C2×M16×C127C2312yes—
93C2×M16×C127C2312yes—
94C22×C8×C127—21no—
95C22×C8×C127—24no—
96C22×C8×C127—21no—
97C22×C8×C127C2312yes—
98C22×C8×C127C2312yes—
99C2×C42×C127—22no—
100C2×C42×C127—22no—
101C2×C42×C127—21no—
102C2×C42×C127C2312yes—
103#104×C127—28no—
104#104×C127—28no—
105C22⋊C8×C127—28no—
106C24×C127—21no—
107C24×C127—21no—
108C24×C127#4111yescharacteristic
109#41×C127—24no—
110#41×C127—24no—
111#41×C127—24no—
112#41×C127—24no—
113#41×C127—24no—
114#41×C127—22no—
115#41×C127—22no—
116#41×C127—24no—
117#41×C127—24no—
118#41×C127—22no—
119#41×C127—22no—
120#41×C127—22no—
121#41×C127—22no—
122#41×C127—42no—
123#41×C127—42no—
124#41×C127—42no—
125#41×C127—42no—
126#41×C127C22⋊C418yes—
127#41×C127C2×D414yes—
128#41×C127C2×D414yes—
129C2×Q8×C127—21no—
130C2×Q8×C127—22no—
131C2×Q8×C127—21no—
132C2×Q8×C127—22no—
133C2×Q8×C127—22no—
134C2×Q8×C127C22⋊C414yes—
135C2×Q8×C127C22×C412yes—
136C2×Q8×C127C22×C412yes—
137C2×D4×C127—22no—
138C2×D4×C127—24no—
139C2×D4×C127—24no—
140C2×D4×C127—22no—
141C2×D4×C127—22no—
142C2×D4×C127—22no—
143C2×D4×C127—22no—
144C2×D4×C127—22no—
145C2×D4×C127—21no—
146C2×D4×C127—22no—
147C2×D4×C127—21no—
148C2×D4×C127—42no—
149C2×D4×C127—42no—
150C2×D4×C127C22⋊C414yes—
151C2×D4×C127C22×C412yes—
152C2×D4×C127C22×C412yes—
153C22×C4×C127—21no—
154C22×C4×C127—22no—
155C22×C4×C127—24no—
156C22×C4×C127—24no—
157C22×C4×C127—22no—
158C22×C4×C127—21no—
159C22×C4×C127—22no—
160C22×C4×C127—22no—
161C22×C4×C127—22no—
162C22×C4×C127—44no—
163C22×C4×C127—42no—
164C22×C4×C127—42no—
165C22×C4×C127—42no—
166C22×C4×C127C22×C411yescharacteristic
167C22×C4×C127C22×C411yescharacteristic
168C22×C4×C127C2×D412yes—
169C22×C4×C127C2×D412yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127C2×D411yescharacteristic
172C22×C4×C127C2×D411yescharacteristic
173C22×C4×C127#4111yescharacteristic
174C22×C4×C127#4112yes—
175C22×C4×C127C2411yescharacteristic
176Q16×C127—28no—
177SD16×C127—28no—
178SD16×C127—28no—
179D8×C127—28no—
180M16×C127—216no—
181M16×C127—24no—
182M16×C127—24no—
183M16×C127—24no—
184M16×C127—24no—
185M16×C127—24no—
186M16×C127—22no—
187M16×C127—22no—
188M16×C127—22no—
189M16×C127—42no—
190M16×C127C2×D414yes—
191C2×C8×C127—216no—
192C2×C8×C127—24no—
193C2×C8×C127—22no—
194C2×C8×C127—22no—
195C2×C8×C127—22no—
196C2×C8×C127—22no—
197C2×C8×C127—22no—
198C2×C8×C127—22no—
199C2×C8×C127—42no—
200C2×C8×C127C22×C414yes—
201C2×C8×C127C2×D414yes—
202C42×C127—22no—
203C42×C127—22no—
204C42×C127—42no—
205C23×C127—22no—
206C23×C127—24no—
207C23×C127—22no—
208C23×C127—42no—
209C23×C127—42no—
210C23×C127—82no—
211C23×C127—82no—
212C23×C127C4×D411yescharacteristic
213C23×C127C4×D411yescharacteristic
214C23×C127#12112yes—
215C23×C127#12312yes—
216C23×C127C2×#4111yescharacteristic
217Q8×C127—22no—
218Q8×C127—24no—
219Q8×C127—22no—
220Q8×C127—22no—
221Q8×C127—42no—
222Q8×C127C2×C22⋊C414yes—
223D4×C127—24no—
224D4×C127—24no—
225D4×C127—28no—
226D4×C127—22no—
227D4×C127—22no—
228D4×C127—22no—
229D4×C127—24no—
230D4×C127—42no—
231D4×C127—42no—
232D4×C127—44no—
233D4×C127—42no—
234D4×C127C2×C22⋊C414yes—
235C2×C4×C127—22no—
236C2×C4×C127—24no—
237C2×C4×C127—24no—
238C2×C4×C127—24no—
239C2×C4×C127—28no—
240C2×C4×C127—24no—
241C2×C4×C127—22no—
242C2×C4×C127—42no—
243C2×C4×C127—48no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—44no—
249C2×C4×C127C2×C22⋊C412yes—
250C2×C4×C127C2×C22⋊C412yes—
251C2×C4×C127C4×D412yes—
252C2×C4×C127C4×D412yes—
253C2×C4×C127C4×D411yescharacteristic
254C2×C4×C127C4×D411yescharacteristic
255C2×C4×C127#12012yes—
256C2×C4×C127#12012yes—
257C2×C4×C127#12112yes—
258C2×C4×C127#12112yes—
259C2×C4×C127#12112yes—
260C2×C4×C127#12312yes—
261C2×C4×C127C23×C411yescharacteristic
262C2×C4×C127C22×D411yescharacteristic
263C2×C4×C127C22×D411yescharacteristic
264C2×C4×C127C22×D411yescharacteristic
265C2×C4×C127C22×D411yescharacteristic
266C2×C4×C127C2×#4111yescharacteristic
267C8×C127—24no—
268C8×C127—24no—
269C8×C127—24no—
270C8×C127—42no—
271C22×C127—22no—
272C22×C127—22no—
273C22×C127—22no—
274C22×C127—24no—
275C22×C127—42no—
276C22×C127—42no—
277C22×C127—44no—
278C22×C127—44no—
279C22×C127—42no—
280C22×C127—84no—
281C22×C127—84no—
282C22×C127#38612yes—
283C22×C127#38612yes—
284C22×C127C2×C4×D411yescharacteristic
285C22×C127C2×#12111yescharacteristic
286C22×C127C2×#12311yescharacteristic
287C4×C127—24no—
288C4×C127—44no—
289C4×C127—42no—
290C4×C127#38612yes—
291C4×C127#38612yes—
292C4×C127C22×C22⋊C411yescharacteristic
293C4×C127C2×C4×D411yescharacteristic
294C4×C127C2×#12011yescharacteristic
295C4×C127C2×#12111yescharacteristic
296#20918C12711yesmaximal, characteristic, direct product factor, Sylow
297C2×C127—21no—
298C2×C127—22no—
299C2×C127—22no—
300C2×C127—21no—
301C2×C127—44no—
302C2×C127—82no—
303C2×C127C2×#38611yescharacteristic
304C2×C127#295612yes—
305C2×#575C2×C12711yescharacteristic
306C2×#568C2×C12711yescharacteristic, complemented
307C22×#104C2×C12711yescharacteristic, complemented
308C22×#104C2×C12711yescharacteristic, complemented
309#2879C2×C12711yescharacteristic, complemented
310#1876C2×C12718yescomplemented
311#1738C2×C12711yescharacteristic, complemented
312#1728C2×C12711yescharacteristic, complemented
313C127#2091811yescharacteristic, direct product factor, Sylow
314C22×#41C22×C12711yescharacteristic
315C22×#41C22×C12711yescharacteristic
316#575—24no—
317#575—24no—
318C2×#137C4×C12712yescomplemented
319C2×#137C4×C12712yescomplemented
320C2×AGL(1,ℤ/8)C4×C12712yescomplemented
321C2×AGL(1,ℤ/8)C4×C12712yescomplemented
322C2×#135C22×C12714yes—
323#568—24no—
324#568—24no—
325C2×#131—22no—
326C2×#131C22×C12714yescomplemented
327C22×M16—22no—
328C22×M16C22×C12711yescharacteristic
329C22×M16C22×C12711yescharacteristic
330C22×M16C22×C12711yescharacteristic
331C22×M16C22×C12711yescharacteristic
332C22×C42C22×C12711yescharacteristic
333C2×#104—24no—
334C2×#104—24no—
335C2×#104C22×C12714yescomplemented
336C2×#104C22×C12714yescomplemented
337#408—22no—
338#408C22×C12714yescomplemented
339#407—22no—
340#341C22×C12714yescomplemented
341#336C22×C12714yescomplemented
342C2×#41—21no—
343C2×#41—22no—
344C2×#41—21no—
345C2×#41—21no—
346C2×#41—22no—
347C2×#41—21no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41C2×C4×C12712yes—
353C2×#41C2×C4×C12712yes—
354C2×#41D4×C12714yes—
355C2×#41D4×C12714yes—
356C2×#41C23×C12712yes—
357C2×#41C23×C12712yes—
358C22×Q8C2×C4×C12711yescharacteristic
359C22×Q8C2×C4×C12711yescharacteristic
360C22×D4—21no—
361C22×D4—21no—
362C22×D4C2×C4×C12711yescharacteristic
363C22×D4C2×C4×C12711yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—21no—
367C23×C4C23×C12711yescharacteristic
368#137—28no—
369#137—28no—
370AGL(1,ℤ/8)—28no—
371AGL(1,ℤ/8)—28no—
372#135—28no—
373#135—28no—
374C2×Q16C2×C4×C12714yescomplemented
375C2×SD16C2×C4×C12714yescomplemented
376C2×SD16C2×C4×C12714yescomplemented
377C2×D8C2×C4×C12714yescomplemented
378#131—216no—
379#131—28no—
380#131—28no—
381C2×M16—216no—
382C2×M16—21no—
383C2×M16—22no—
384C2×M16—22no—
385C2×M16—22no—
386C2×M16—22no—
387C2×M16—22no—
388C2×M16—21no—
389C2×M16—24no—
390C2×M16—22no—
391C2×M16—22no—
392C2×M16C2×C4×C12712yes—
393C2×M16C2×C4×C12712yes—
394C2×M16D4×C12714yes—
395C2×M16D4×C12712yes—
396C2×M16D4×C12712yes—
397C2×M16C23×C12712yes—
398C2×M16C23×C12712yes—
399C22×C8—21no—
400C22×C8—24no—
401C22×C8—21no—
402C22×C8C23×C12712yes—
403C22×C8C23×C12712yes—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42C23×C12712yes—
408#104—28no—
409#104—28no—
410C22⋊C8—28no—
411C24—21no—
412C24—21no—
413C24#41×C12711yescharacteristic
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41—22no—
420#41—22no—
421#41—24no—
422#41—24no—
423#41—22no—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—42no—
428#41—42no—
429#41—42no—
430#41—42no—
431#41C22⋊C4×C12718yes—
432#41C2×D4×C12714yes—
433#41C2×D4×C12714yes—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C22⋊C4×C12714yes—
440C2×Q8C22×C4×C12712yes—
441C2×Q8C22×C4×C12712yes—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4C22⋊C4×C12714yes—
456C2×D4C22×C4×C12712yes—
457C2×D4C22×C4×C12712yes—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4C22×C4×C12711yescharacteristic
472C22×C4C22×C4×C12711yescharacteristic
473C22×C4C2×D4×C12712yes—
474C22×C4C2×D4×C12712yes—
475C22×C4C2×D4×C12712yes—
476C22×C4C2×D4×C12711yescharacteristic
477C22×C4C2×D4×C12711yescharacteristic
478C22×C4#41×C12711yescharacteristic
479C22×C4#41×C12712yes—
480C22×C4C24×C12711yescharacteristic
481Q16—28no—
482SD16—28no—
483SD16—28no—
484D8—28no—
485M16—216no—
486M16—24no—
487M16—24no—
488M16—24no—
489M16—24no—
490M16—24no—
491M16—22no—
492M16—22no—
493M16—22no—
494M16—42no—
495M16C2×D4×C12714yes—
496C2×C8—216no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—42no—
505C2×C8C22×C4×C12714yes—
506C2×C8C2×D4×C12714yes—
507C42—22no—
508C42—22no—
509C42—42no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—82no—
516C23—82no—
517C23C4×D4×C12711yescharacteristic
518C23C4×D4×C12711yescharacteristic
519C23#121×C12712yes—
520C23#123×C12712yes—
521C23C2×#41×C12711yescharacteristic
522Q8—22no—
523Q8—24no—
524Q8—22no—
525Q8—22no—
526Q8—42no—
527Q8C2×C22⋊C4×C12714yes—
528D4—24no—
529D4—24no—
530D4—28no—
531D4—22no—
532D4—22no—
533D4—22no—
534D4—24no—
535D4—42no—
536D4—42no—
537D4—44no—
538D4—42no—
539D4C2×C22⋊C4×C12714yes—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4C2×C22⋊C4×C12712yes—
555C2×C4C2×C22⋊C4×C12712yes—
556C2×C4C4×D4×C12712yes—
557C2×C4C4×D4×C12712yes—
558C2×C4C4×D4×C12711yescharacteristic
559C2×C4C4×D4×C12711yescharacteristic
560C2×C4#120×C12712yes—
561C2×C4#120×C12712yes—
562C2×C4#121×C12712yes—
563C2×C4#121×C12712yes—
564C2×C4#121×C12712yes—
565C2×C4#123×C12712yes—
566C2×C4C23×C4×C12711yescharacteristic
567C2×C4C22×D4×C12711yescharacteristic
568C2×C4C22×D4×C12711yescharacteristic
569C2×C4C22×D4×C12711yescharacteristic
570C2×C4C22×D4×C12711yescharacteristic
571C2×C4C2×#41×C12711yescharacteristic
572C8—24no—
573C8—24no—
574C8—24no—
575C8—42no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—84no—
586C22—84no—
587C22#386×C12712yes—
588C22#386×C12712yes—
589C22C2×C4×D4×C12711yescharacteristic
590C22C2×#121×C12711yescharacteristic
591C22C2×#123×C12711yescharacteristic
592C4—24no—
593C4—44no—
594C4—42no—
595C4#386×C12712yes—
596C4#386×C12712yes—
597C4C22×C22⋊C4×C12711yescharacteristic
598C4C2×C4×D4×C12711yescharacteristic
599C4C2×#120×C12711yescharacteristic
600C4C2×#121×C12711yescharacteristic
601C2—21no—
602C2—22no—
603C2—22no—
604C2—21no—
605C2—44no—
606C2—82no—
607C2C2×#386×C12711yescharacteristic
608C2C127×#295612yes—
subgroupG/H
centerC2×C4×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C2×C4×D4
prime radicalC2×#575×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#568×C127C2semidirect
3. C22×#104×C127C2semidirect
4. C22×#104×C127C2semidirect
5. C127×#2879C2semidirect
6. C127×#1876C2semidirect
7. C127×#1738C2semidirect
8. C127×#1728C2semidirect
13. C2×#137×C127C4semidirect
14. C2×#137×C127C4semidirect
15. C2×AGL(1,ℤ/8)×C127C4semidirect
16. C2×AGL(1,ℤ/8)×C127C4semidirect
21. C2×#131×C127C22semidirect
30. C2×#104×C127C22semidirect
31. C2×#104×C127C22semidirect
33. #408×C127C22semidirect
35. #341×C127C22semidirect
36. #336×C127C22semidirect
69. C2×Q16×C127C2×C4semidirect
70. C2×SD16×C127C2×C4semidirect
71. C2×SD16×C127C2×C4semidirect
72. C2×D8×C127C2×C4semidirect
296. #20918C127direct (N × H)
306. C2×#568C2×C127semidirect
307. C22×#104C2×C127semidirect
308. C22×#104C2×C127semidirect
309. #2879C2×C127semidirect
310. #1876C2×C127semidirect
311. #1738C2×C127semidirect
312. #1728C2×C127semidirect
313. C127#20918direct (N × H)
318. C2×#137C4×C127semidirect
319. C2×#137C4×C127semidirect
320. C2×AGL(1,ℤ/8)C4×C127semidirect
321. C2×AGL(1,ℤ/8)C4×C127semidirect
326. C2×#131C22×C127semidirect
335. C2×#104C22×C127semidirect
336. C2×#104C22×C127semidirect
338. #408C22×C127semidirect
340. #341C22×C127semidirect
341. #336C22×C127semidirect
374. C2×Q16C2×C4×C127semidirect
375. C2×SD16C2×C4×C127semidirect
376. C2×SD16C2×C4×C127semidirect
377. C2×D8C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#575×C1272
C2×#568×C1272
C22×#104×C1272
C127×#28792
C127×#18762
C127×#17382
C127×#17282
#20918127