Finite Group Databasebeta

← C127×#21003 #44584676

Subgroups — C127×#21003 #44584676

Order 32512.

1686 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575×C127C211yesmaximal, characteristic
2C22×#131×C127C211yesmaximal, characteristic, complemented
3C127×#2896C211yesmaximal, characteristic, complemented
4C2×#416×C127C211yesmaximal, characteristic, complemented
5C2×#413×C127C211yesmaximal, characteristic, complemented
6C127×#1908C214yesmaximal, complemented
7C127×#1907C214yesmaximal, complemented
8C127×#1742C211yesmaximal, characteristic, complemented
9C2×#341×C127C211yesmaximal, characteristic, complemented
10C22×#41×C127C2211yescharacteristic
11C22×#41×C127C2211yescharacteristic
12#575×C127—24no—
13#575×C127C418yescomplemented
14C2×#137×C127—21no—
15C2×#137×C127C2212yes—
16C2×AGL(1,ℤ/8)×C127—21no—
17C2×AGL(1,ℤ/8)×C127C2212yes—
18C2×#135×C127—22no—
19C2×#131×C127—24no—
20C2×#131×C127—24no—
21C2×#131×C127—22no—
22C2×#131×C127C2214yes—
23C22×M16×C127—21no—
24C22×M16×C127C2211yescharacteristic
25C22×M16×C127C2211yescharacteristic
26C22×M16×C127C2211yescharacteristic
27C23×C8×C127—21no—
28C23×C8×C127C2211yescharacteristic
29C2×#117×C127C2211yescharacteristic
30#419×C127—21no—
31#418×C127—21no—
32#417×C127C2214yes—
33#416×C127—22no—
34#416×C127—21no—
35#416×C127—21no—
36#416×C127—22no—
37C2×#103×C127C2212yescomplemented
38C2×#102×C127C2212yescomplemented
39#413×C127—24no—
40C2×#101×C127C2212yescomplemented
41C2×#100×C127C2212yescomplemented
42#343×C127C2214yescomplemented
43#341×C127C2214yescomplemented
44C2×#41×C127—22no—
45C2×#41×C127—21no—
46C2×#41×C127—22no—
47C2×#41×C127—22no—
48C2×#41×C127—21no—
49C2×#41×C127—21no—
50C2×#41×C127—21no—
51C2×#41×C127—22no—
52C2×#41×C127—21no—
53C2×#41×C127—21no—
54C2×#41×C127—22no—
55C2×#41×C127C2×C414yes—
56C2×#41×C127C2×C414yes—
57C2×#41×C127D414yes—
58C2×#41×C127D412yes—
59C2×#41×C127C2312yes—
60C22×Q8×C127D411yescharacteristic
61C22×Q8×C127C2311yescharacteristic
62C22×D4×C127—21no—
63C22×D4×C127—21no—
64C22×D4×C127D411yescharacteristic
65C22×D4×C127C2311yescharacteristic
66C23×C4×C127—21no—
67C23×C4×C127—21no—
68C23×C4×C127C2311yescharacteristic
69#137×C127—24no—
70#137×C127—28no—
71#137×C127C2×C418yes—
72AGL(1,ℤ/8)×C127—24no—
73AGL(1,ℤ/8)×C127—28no—
74AGL(1,ℤ/8)×C127C2×C418yes—
75#135×C127—216no—
76C2×Q16×C127—22no—
77C2×SD16×C127—22no—
78C2×SD16×C127—22no—
79C2×D8×C127—22no—
80#131×C127—216no—
81#131×C127—28no—
82#131×C127—28no—
83C2×M16×C127—28no—
84C2×M16×C127—22no—
85C2×M16×C127—21no—
86C2×M16×C127—21no—
87C2×M16×C127—22no—
88C2×M16×C127—21no—
89C2×M16×C127—24no—
90C2×M16×C127—22no—
91C2×M16×C127—22no—
92C2×M16×C127C2×C414yes—
93C2×M16×C127D414yes—
94C2×M16×C127C2312yes—
95C2×M16×C127C2312yes—
96C2×M16×C127C2312yes—
97C22×C8×C127—28no—
98C22×C8×C127—21no—
99C22×C8×C127—22no—
100C22×C8×C127—22no—
101C22×C8×C127—21no—
102C22×C8×C127—24no—
103C22×C8×C127—21no—
104C22×C8×C127D411yescharacteristic
105C22×C8×C127D411yescharacteristic
106C22×C8×C127D412yes—
107C22×C8×C127C2312yes—
108#117×C127—21no—
109#117×C127—21no—
110#117×C127—21no—
111#117×C127—21no—
112C2×C4⋊C4×C127C2312yes—
113C2×C22⋊C4×C127—21no—
114C2×C42×C127—21no—
115#103×C127—24no—
116#103×C127—22no—
117#103×C127—22no—
118#102×C127—24no—
119#102×C127—22no—
120#102×C127—22no—
121#101×C127—24no—
122#100×C127—24no—
123C24×C127—21no—
124C24×C127—21no—
125C24×C127#4111yescharacteristic
126#41×C127—24no—
127#41×C127—24no—
128#41×C127—24no—
129#41×C127—24no—
130#41×C127—24no—
131#41×C127—22no—
132#41×C127—22no—
133#41×C127—24no—
134#41×C127—24no—
135#41×C127—24no—
136#41×C127—24no—
137#41×C127—42no—
138#41×C127—42no—
139#41×C127—42no—
140#41×C127—42no—
141#41×C127C22⋊C418yes—
142#41×C127C22×C414yes—
143#41×C127C2×D414yes—
144C2×Q8×C127—21no—
145C2×Q8×C127—22no—
146C2×Q8×C127—21no—
147C2×Q8×C127—21no—
148C2×Q8×C127—21no—
149C2×Q8×C127—21no—
150C2×Q8×C127—21no—
151C2×Q8×C127C22⋊C414yes—
152C2×Q8×C127C22×C412yes—
153C2×Q8×C127C2×D412yes—
154C2×D4×C127—24no—
155C2×D4×C127—22no—
156C2×D4×C127—24no—
157C2×D4×C127—22no—
158C2×D4×C127—24no—
159C2×D4×C127—22no—
160C2×D4×C127—21no—
161C2×D4×C127—21no—
162C2×D4×C127—21no—
163C2×D4×C127—21no—
164C2×D4×C127—21no—
165C2×D4×C127—21no—
166C2×D4×C127—41no—
167C2×D4×C127—41no—
168C2×D4×C127—42no—
169C2×D4×C127C22⋊C414yes—
170C2×D4×C127C22×C412yes—
171C2×D4×C127C2×D412yes—
172C22×C4×C127—21no—
173C22×C4×C127—24no—
174C22×C4×C127—24no—
175C22×C4×C127—22no—
176C22×C4×C127—22no—
177C22×C4×C127—21no—
178C22×C4×C127—24no—
179C22×C4×C127—22no—
180C22×C4×C127—41no—
181C22×C4×C127—41no—
182C22×C4×C127—42no—
183C22×C4×C127C22×C412yes—
184C22×C4×C127C2×D412yes—
185C22×C4×C127C2×D411yescharacteristic
186C22×C4×C127C2×D411yescharacteristic
187C22×C4×C127C2×D411yescharacteristic
188C22×C4×C127C2×D411yescharacteristic
189C22×C4×C127C2×D412yes—
190C22×C4×C127#4112yes—
191C22×C4×C127#4111yescharacteristic
192C22×C4×C127C2411yescharacteristic
193Q16×C127—28no—
194SD16×C127—28no—
195SD16×C127—28no—
196D8×C127—28no—
197M16×C127—216no—
198M16×C127—24no—
199M16×C127—24no—
200M16×C127—22no—
201M16×C127—22no—
202M16×C127—22no—
203M16×C127—22no—
204M16×C127—42no—
205M16×C127C22×C414yes—
206M16×C127C2×D414yes—
207C2×C8×C127—216no—
208C2×C8×C127—24no—
209C2×C8×C127—22no—
210C2×C8×C127—24no—
211C2×C8×C127—22no—
212C2×C8×C127—24no—
213C2×C8×C127—22no—
214C2×C8×C127—22no—
215C2×C8×C127—22no—
216C2×C8×C127—22no—
217C2×C8×C127—22no—
218C2×C8×C127—42no—
219C2×C8×C127C2×D412yes—
220C2×C8×C127C2×D412yes—
221C4⋊C4×C127—22no—
222C4⋊C4×C127—22no—
223C22⋊C4×C127—41no—
224C22⋊C4×C127—41no—
225C42×C127—41no—
226C42×C127—41no—
227C23×C127—22no—
228C23×C127—24no—
229C23×C127—22no—
230C23×C127—42no—
231C23×C127—44no—
232C23×C127—41no—
233C23×C127—41no—
234C23×C127—82no—
235C23×C127C4×D412yes—
236C23×C127#12111yescharacteristic
237C23×C127#12111yescharacteristic
238C23×C127#12312yes—
239C23×C127C2×#4111yescharacteristic
240Q8×C127—22no—
241Q8×C127—24no—
242Q8×C127—22no—
243Q8×C127—22no—
244Q8×C127—41no—
245Q8×C127—41no—
246Q8×C127C2×C22⋊C414yes—
247D4×C127—24no—
248D4×C127—24no—
249D4×C127—28no—
250D4×C127—22no—
251D4×C127—22no—
252D4×C127—22no—
253D4×C127—24no—
254D4×C127—44no—
255D4×C127—44no—
256D4×C127—41no—
257D4×C127—41no—
258D4×C127C2×C22⋊C414yes—
259C2×C4×C127—22no—
260C2×C4×C127—24no—
261C2×C4×C127—24no—
262C2×C4×C127—24no—
263C2×C4×C127—28no—
264C2×C4×C127—22no—
265C2×C4×C127—41no—
266C2×C4×C127—41no—
267C2×C4×C127—44no—
268C2×C4×C127—44no—
269C2×C4×C127—42no—
270C2×C4×C127—82no—
271C2×C4×C127C2×C22⋊C412yes—
272C2×C4×C127C2×C22⋊C412yes—
273C2×C4×C127C4×D412yes—
274C2×C4×C127C4×D412yes—
275C2×C4×C127C4×D412yes—
276C2×C4×C127#12012yes—
277C2×C4×C127#12012yes—
278C2×C4×C127#12112yes—
279C2×C4×C127#12112yes—
280C2×C4×C127#12111yescharacteristic
281C2×C4×C127#12111yescharacteristic
282C2×C4×C127#12312yes—
283C2×C4×C127C23×C411yescharacteristic
284C2×C4×C127C22×D411yescharacteristic
285C2×C4×C127C22×D411yescharacteristic
286C2×C4×C127C22×D411yescharacteristic
287C2×C4×C127C22×D411yescharacteristic
288C2×C4×C127C2×#4111yescharacteristic
289C8×C127—24no—
290C8×C127—22no—
291C8×C127—24no—
292C8×C127—22no—
293C8×C127—42no—
294C22×C127—22no—
295C22×C127—22no—
296C22×C127—22no—
297C22×C127—24no—
298C22×C127—42no—
299C22×C127—42no—
300C22×C127—48no—
301C22×C127—44no—
302C22×C127—42no—
303C22×C127—82no—
304C22×C127—82no—
305C22×C127—82no—
306C22×C127#38612yes—
307C22×C127#38612yes—
308C22×C127C2×C4×D411yescharacteristic
309C22×C127C2×#12111yescharacteristic
310C22×C127C2×#12311yescharacteristic
311C4×C127—24no—
312C4×C127—42no—
313C4×C127—82no—
314C4×C127#38612yes—
315C4×C127#38612yes—
316C4×C127C22×C22⋊C411yescharacteristic
317C4×C127C2×C4×D411yescharacteristic
318C4×C127C2×#12011yescharacteristic
319C4×C127C2×#12111yescharacteristic
320#21003C12711yesmaximal, characteristic, direct product factor, Sylow
321C2×C127—22no—
322C2×C127—21no—
323C2×C127—21no—
324C2×C127—22no—
325C2×C127—44no—
326C2×C127—82no—
327C2×C127C2×#38611yescharacteristic
328C2×C127#306712yes—
329C2×#575C2×C12711yescharacteristic
330C22×#131C2×C12711yescharacteristic, complemented
331#2896C2×C12711yescharacteristic, complemented
332C2×#416C2×C12711yescharacteristic, complemented
333C2×#413C2×C12711yescharacteristic, complemented
334#1908C2×C12714yescomplemented
335#1907C2×C12714yescomplemented
336#1742C2×C12711yescharacteristic, complemented
337C2×#341C2×C12711yescharacteristic, complemented
338C127#2100311yescharacteristic, direct product factor, Sylow
339C22×#41C22×C12711yescharacteristic
340C22×#41C22×C12711yescharacteristic
341#575—24no—
342#575C4×C12718yescomplemented
343C2×#137—21no—
344C2×#137C22×C12712yes—
345C2×AGL(1,ℤ/8)—21no—
346C2×AGL(1,ℤ/8)C22×C12712yes—
347C2×#135—22no—
348C2×#131—24no—
349C2×#131—24no—
350C2×#131—22no—
351C2×#131C22×C12714yes—
352C22×M16—21no—
353C22×M16C22×C12711yescharacteristic
354C22×M16C22×C12711yescharacteristic
355C22×M16C22×C12711yescharacteristic
356C23×C8—21no—
357C23×C8C22×C12711yescharacteristic
358C2×#117C22×C12711yescharacteristic
359#419—21no—
360#418—21no—
361#417C22×C12714yes—
362#416—22no—
363#416—21no—
364#416—21no—
365#416—22no—
366C2×#103C22×C12712yescomplemented
367C2×#102C22×C12712yescomplemented
368#413—24no—
369C2×#101C22×C12712yescomplemented
370C2×#100C22×C12712yescomplemented
371#343C22×C12714yescomplemented
372#341C22×C12714yescomplemented
373C2×#41—22no—
374C2×#41—21no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—22no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—22no—
384C2×#41C2×C4×C12714yes—
385C2×#41C2×C4×C12714yes—
386C2×#41D4×C12714yes—
387C2×#41D4×C12712yes—
388C2×#41C23×C12712yes—
389C22×Q8D4×C12711yescharacteristic
390C22×Q8C23×C12711yescharacteristic
391C22×D4—21no—
392C22×D4—21no—
393C22×D4D4×C12711yescharacteristic
394C22×D4C23×C12711yescharacteristic
395C23×C4—21no—
396C23×C4—21no—
397C23×C4C23×C12711yescharacteristic
398#137—24no—
399#137—28no—
400#137C2×C4×C12718yes—
401AGL(1,ℤ/8)—24no—
402AGL(1,ℤ/8)—28no—
403AGL(1,ℤ/8)C2×C4×C12718yes—
404#135—216no—
405C2×Q16—22no—
406C2×SD16—22no—
407C2×SD16—22no—
408C2×D8—22no—
409#131—216no—
410#131—28no—
411#131—28no—
412C2×M16—28no—
413C2×M16—22no—
414C2×M16—21no—
415C2×M16—21no—
416C2×M16—22no—
417C2×M16—21no—
418C2×M16—24no—
419C2×M16—22no—
420C2×M16—22no—
421C2×M16C2×C4×C12714yes—
422C2×M16D4×C12714yes—
423C2×M16C23×C12712yes—
424C2×M16C23×C12712yes—
425C2×M16C23×C12712yes—
426C22×C8—28no—
427C22×C8—21no—
428C22×C8—22no—
429C22×C8—22no—
430C22×C8—21no—
431C22×C8—24no—
432C22×C8—21no—
433C22×C8D4×C12711yescharacteristic
434C22×C8D4×C12711yescharacteristic
435C22×C8D4×C12712yes—
436C22×C8C23×C12712yes—
437#117—21no—
438#117—21no—
439#117—21no—
440#117—21no—
441C2×C4⋊C4C23×C12712yes—
442C2×C22⋊C4—21no—
443C2×C42—21no—
444#103—24no—
445#103—22no—
446#103—22no—
447#102—24no—
448#102—22no—
449#102—22no—
450#101—24no—
451#100—24no—
452C24—21no—
453C24—21no—
454C24#41×C12711yescharacteristic
455#41—24no—
456#41—24no—
457#41—24no—
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41—24no—
463#41—24no—
464#41—24no—
465#41—24no—
466#41—42no—
467#41—42no—
468#41—42no—
469#41—42no—
470#41C22⋊C4×C12718yes—
471#41C22×C4×C12714yes—
472#41C2×D4×C12714yes—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C22⋊C4×C12714yes—
481C2×Q8C22×C4×C12712yes—
482C2×Q8C2×D4×C12712yes—
483C2×D4—24no—
484C2×D4—22no—
485C2×D4—24no—
486C2×D4—22no—
487C2×D4—24no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—41no—
496C2×D4—41no—
497C2×D4—42no—
498C2×D4C22⋊C4×C12714yes—
499C2×D4C22×C4×C12712yes—
500C2×D4C2×D4×C12712yes—
501C22×C4—21no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4—42no—
512C22×C4C22×C4×C12712yes—
513C22×C4C2×D4×C12712yes—
514C22×C4C2×D4×C12711yescharacteristic
515C22×C4C2×D4×C12711yescharacteristic
516C22×C4C2×D4×C12711yescharacteristic
517C22×C4C2×D4×C12711yescharacteristic
518C22×C4C2×D4×C12712yes—
519C22×C4#41×C12711yescharacteristic
520C22×C4#41×C12712yes—
521C22×C4C24×C12711yescharacteristic
522Q16—28no—
523SD16—28no—
524SD16—28no—
525D8—28no—
526M16—216no—
527M16—24no—
528M16—24no—
529M16—22no—
530M16—22no—
531M16—22no—
532M16—22no—
533M16—42no—
534M16C22×C4×C12714yes—
535M16C2×D4×C12714yes—
536C2×C8—216no—
537C2×C8—24no—
538C2×C8—22no—
539C2×C8—24no—
540C2×C8—22no—
541C2×C8—24no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—42no—
548C2×C8C2×D4×C12712yes—
549C2×C8C2×D4×C12712yes—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C42—41no—
555C42—41no—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—42no—
560C23—44no—
561C23—41no—
562C23—41no—
563C23—82no—
564C23C4×D4×C12712yes—
565C23#121×C12711yescharacteristic
566C23#121×C12711yescharacteristic
567C23#123×C12712yes—
568C23C2×#41×C12711yescharacteristic
569Q8—22no—
570Q8—24no—
571Q8—22no—
572Q8—22no—
573Q8—41no—
574Q8—41no—
575Q8C2×C22⋊C4×C12714yes—
576D4—24no—
577D4—24no—
578D4—28no—
579D4—22no—
580D4—22no—
581D4—22no—
582D4—24no—
583D4—44no—
584D4—44no—
585D4—41no—
586D4—41no—
587D4C2×C22⋊C4×C12714yes—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4C2×C22⋊C4×C12712yes—
601C2×C4C2×C22⋊C4×C12712yes—
602C2×C4C4×D4×C12712yes—
603C2×C4C4×D4×C12712yes—
604C2×C4C4×D4×C12712yes—
605C2×C4#120×C12712yes—
606C2×C4#120×C12712yes—
607C2×C4#121×C12712yes—
608C2×C4#121×C12712yes—
609C2×C4#121×C12711yescharacteristic
610C2×C4#121×C12711yescharacteristic
611C2×C4#123×C12712yes—
612C2×C4C23×C4×C12711yescharacteristic
613C2×C4C22×D4×C12711yescharacteristic
614C2×C4C22×D4×C12711yescharacteristic
615C2×C4C22×D4×C12711yescharacteristic
616C2×C4C22×D4×C12711yescharacteristic
617C2×C4C2×#41×C12711yescharacteristic
618C8—24no—
619C8—22no—
620C8—24no—
621C8—22no—
622C8—42no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—48no—
630C22—44no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#386×C12712yes—
636C22#386×C12712yes—
637C22C2×C4×D4×C12711yescharacteristic
638C22C2×#121×C12711yescharacteristic
639C22C2×#123×C12711yescharacteristic
640C4—24no—
641C4—42no—
642C4—82no—
643C4#386×C12712yes—
644C4#386×C12712yes—
645C4C22×C22⋊C4×C12711yescharacteristic
646C4C2×C4×D4×C12711yescharacteristic
647C4C2×#120×C12711yescharacteristic
648C4C2×#121×C12711yescharacteristic
649C2—22no—
650C2—21no—
651C2—21no—
652C2—22no—
653C2—44no—
654C2—82no—
655C2C2×#386×C12711yescharacteristic
656C2C127×#306712yes—
subgroupG/H
centerC2×C4×C127#121
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C2×#121
prime radicalC2×#575×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#131×C127C2semidirect
3. C127×#2896C2semidirect
4. C2×#416×C127C2semidirect
5. C2×#413×C127C2semidirect
6. C127×#1908C2semidirect
7. C127×#1907C2semidirect
8. C127×#1742C2semidirect
9. C2×#341×C127C2semidirect
13. #575×C127C4semidirect
37. C2×#103×C127C22semidirect
38. C2×#102×C127C22semidirect
40. C2×#101×C127C22semidirect
41. C2×#100×C127C22semidirect
42. #343×C127C22semidirect
43. #341×C127C22semidirect
320. #21003C127direct (N × H)
330. C22×#131C2×C127semidirect
331. #2896C2×C127semidirect
332. C2×#416C2×C127semidirect
333. C2×#413C2×C127semidirect
334. #1908C2×C127semidirect
335. #1907C2×C127semidirect
336. #1742C2×C127semidirect
337. C2×#341C2×C127semidirect
338. C127#21003direct (N × H)
342. #575C4×C127semidirect
366. C2×#103C22×C127semidirect
367. C2×#102C22×C127semidirect
369. C2×#101C22×C127semidirect
370. C2×#100C22×C127semidirect
371. #343C22×C127semidirect
372. #341C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#575×C1272
C22×#131×C1272
C127×#28962
C2×#416×C1272
C2×#413×C1272
C127×#19082
C127×#19072
C127×#17422
C2×#341×C1272
#21003127