Finite Group Databasebeta

← C127×#20618 #44584291

Subgroups — C127×#20618 #44584291

Order 32512.

3690 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585×C127C211yesmaximal, characteristic
2C2×#421×C127C212yesmaximal, complemented
3C2×#417×C127C212yesmaximal, complemented
4C2×#413×C127C211yesmaximal, characteristic, complemented
5C127×#2311C211yesmaximal, characteristic, complemented
6C127×#1793C214yesmaximal, complemented
7C127×#1792C214yesmaximal, complemented
8#585×C127—28no—
9#585×C127—24no—
10#585×C127C418yescomplemented
11C2×2−1+4×C127—22no—
12C2×2−1+4×C127C2212yes—
13C2×H(5,𝔽2)×C127—22no—
14C2×H(5,𝔽2)×C127—22no—
15C2×H(5,𝔽2)×C127C2212yes—
16C22×#41×C127—24no—
17C22×#41×C127—22no—
18C22×#41×C127C2212yes—
19C22×#41×C127C2211yescharacteristic
20C22×M16×C127C2212yescomplemented
21C2×#117×C127C2211yescharacteristic, complemented
22C22×C4⋊C4×C127C2211yescharacteristic, complemented
23#421×C127—28no—
24C2×#104×C127C2218yescomplemented
25#417×C127—24no—
26#417×C127C2218yescomplemented
27C2×#103×C127—22no—
28C2×#102×C127—22no—
29#413×C127—24no—
30C2×#101×C127C2212yescomplemented
31C2×#100×C127C2212yescomplemented
32#391×C127C2212yescomplemented
33#390×C127C2212yescomplemented
34#386×C127—22no—
35#380×C127—22no—
362−1+4×C127—216no—
372−1+4×C127—28no—
382−1+4×C127—216no—
392−1+4×C127—24no—
402−1+4×C127C2×C418yescomplemented
41H(5,𝔽2)×C127—216no—
42H(5,𝔽2)×C127—216no—
43H(5,𝔽2)×C127—28no—
44H(5,𝔽2)×C127—216no—
45H(5,𝔽2)×C127—216no—
46H(5,𝔽2)×C127—24no—
47H(5,𝔽2)×C127C2×C418yescomplemented
48C2×#41×C127—24no—
49C2×#41×C127—28no—
50C2×#41×C127—24no—
51C2×#41×C127—232no—
52C2×#41×C127—216no—
53C2×#41×C127—232no—
54C2×#41×C127—24no—
55C2×#41×C127—24no—
56C2×#41×C127—216no—
57C2×#41×C127—22no—
58C2×#41×C127—22no—
59C2×#41×C127—24no—
60C2×#41×C127—22no—
61C2×#41×C127—22no—
62C2×#41×C127—24no—
63C2×#41×C127—24no—
64C2×#41×C127—24no—
65C2×#41×C127C2×C418yes—
66C2×#41×C127C2×C414yes—
67C2×#41×C127D418yes—
68C2×#41×C127D414yes—
69C2×#41×C127D414yes—
70C2×#41×C127C2314yes—
71C22×Q8×C127—22no—
72C22×Q8×C127—24no—
73C22×Q8×C127D412yes—
74C22×Q8×C127C2311yescharacteristic
75C22×D4×C127—28no—
76C22×D4×C127—22no—
77C22×D4×C127—24no—
78C22×D4×C127—22no—
79C22×D4×C127—24no—
80C22×D4×C127D412yes—
81C22×D4×C127D412yes—
82C22×D4×C127C2311yescharacteristic
83C23×C4×C127—24no—
84C23×C4×C127—22no—
85C23×C4×C127—21no—
86C23×C4×C127D412yes—
87C23×C4×C127C2311yescharacteristic
88C2×M16×C127—22no—
89C2×M16×C127—24no—
90C2×M16×C127—24no—
91C2×M16×C127C2314yes—
92C22×C8×C127—22no—
93#117×C127—24no—
94C2×C4⋊C4×C127—22no—
95C2×C4⋊C4×C127—22no—
96C2×C4⋊C4×C127—21no—
97C2×C4⋊C4×C127—21no—
98C2×C4⋊C4×C127C2312yes—
99C2×C22⋊C4×C127—21no—
100C2×C22⋊C4×C127—24no—
101C2×C42×C127C2312yes—
102#104×C127—216no—
103#103×C127—28no—
104#102×C127—28no—
105#101×C127—24no—
106#100×C127—24no—
107#95×C127—24no—
108C24×C127—24no—
109C24×C127—22no—
110C24×C127C2×D412yes—
111C24×C127C2×D411yescharacteristic
112#41×C127—232no—
113#41×C127—216no—
114#41×C127—264no—
115#41×C127—216no—
116#41×C127—216no—
117#41×C127—232no—
118#41×C127—216no—
119#41×C127—216no—
120#41×C127—28no—
121#41×C127—28no—
122#41×C127—216no—
123#41×C127—28no—
124#41×C127—28no—
125#41×C127—28no—
126#41×C127—28no—
127#41×C127—28no—
128#41×C127—28no—
129#41×C127—28no—
130#41×C127—24no—
131#41×C127—24no—
132#41×C127C22⋊C4116yescomplemented
133#41×C127C22×C418yes—
134#41×C127C2×D418yes—
135C2×Q8×C127—24no—
136C2×Q8×C127—216no—
137C2×Q8×C127—216no—
138C2×Q8×C127—216no—
139C2×Q8×C127—22no—
140C2×Q8×C127—24no—
141C2×Q8×C127—22no—
142C2×Q8×C127—22no—
143C2×Q8×C127C22⋊C418yescomplemented
144C2×Q8×C127C22×C412yes—
145C2×Q8×C127C2×D414yes—
146C2×Q8×C127C2×D412yes—
147C2×D4×C127—28no—
148C2×D4×C127—24no—
149C2×D4×C127—216no—
150C2×D4×C127—264no—
151C2×D4×C127—216no—
152C2×D4×C127—216no—
153C2×D4×C127—216no—
154C2×D4×C127—28no—
155C2×D4×C127—24no—
156C2×D4×C127—24no—
157C2×D4×C127—28no—
158C2×D4×C127—24no—
159C2×D4×C127—24no—
160C2×D4×C127—216no—
161C2×D4×C127—22no—
162C2×D4×C127—24no—
163C2×D4×C127—22no—
164C2×D4×C127—22no—
165C2×D4×C127C22⋊C418yes—
166C2×D4×C127C22⋊C418yescomplemented
167C2×D4×C127C22×C412yes—
168C2×D4×C127C2×D412yes—
169C2×D4×C127C2×D414yes—
170C22×C4×C127—24no—
171C22×C4×C127—28no—
172C22×C4×C127—232no—
173C22×C4×C127—22no—
174C22×C4×C127—28no—
175C22×C4×C127—24no—
176C22×C4×C127—22no—
177C22×C4×C127—28no—
178C22×C4×C127—22no—
179C22×C4×C127—24no—
180C22×C4×C127—24no—
181C22×C4×C127—22no—
182C22×C4×C127—22no—
183C22×C4×C127—42no—
184C22×C4×C127—82no—
185C22×C4×C127C22⋊C418yes—
186C22×C4×C127C22×C412yes—
187C22×C4×C127C2×D412yes—
188C22×C4×C127C2×D411yescharacteristic
189C22×C4×C127C2×D412yes—
190C22×C4×C127C2×D412yes—
191C22×C4×C127C2×D414yes—
192C22×C4×C127C2×D412yes—
193C22×C4×C127C2411yescharacteristic
194M16×C127—24no—
195M16×C127—24no—
196M16×C127—44no—
197C2×C8×C127—24no—
198C2×C8×C127—44no—
199C4⋊C4×C127—24no—
200C4⋊C4×C127—41no—
201C4⋊C4×C127—41no—
202C4⋊C4×C127—42no—
203C22⋊C4×C127—42no—
204C22⋊C4×C127—84no—
205C42×C127—24no—
206C23×C127—216no—
207C23×C127—22no—
208C23×C127—24no—
209C23×C127—22no—
210C23×C127—28no—
211C23×C127—41no—
212C23×C127—41no—
213C23×C127—42no—
214C23×C127—42no—
215C23×C127—82no—
216C23×C127—82no—
217C23×C127—84no—
218C23×C127—84no—
219C23×C127C2×C22⋊C414yes—
220C23×C127C2×C22⋊C412yes—
221C23×C127#12012yes—
222C23×C127#12012yes—
223C23×C127C22×D411yescharacteristic
224Q8×C127—216no—
225Q8×C127—24no—
226Q8×C127—24no—
227Q8×C127—24no—
228Q8×C127—28no—
229Q8×C127C2×C22⋊C418yes—
230D4×C127—232no—
231D4×C127—216no—
232D4×C127—28no—
233D4×C127—28no—
234D4×C127—216no—
235D4×C127—216no—
236D4×C127—24no—
237D4×C127—24no—
238D4×C127—24no—
239D4×C127—28no—
240D4×C127C2×C22⋊C418yes—
241C2×C4×C127—216no—
242C2×C4×C127—232no—
243C2×C4×C127—22no—
244C2×C4×C127—24no—
245C2×C4×C127—22no—
246C2×C4×C127—28no—
247C2×C4×C127—28no—
248C2×C4×C127—28no—
249C2×C4×C127—216no—
250C2×C4×C127—28no—
251C2×C4×C127—28no—
252C2×C4×C127—42no—
253C2×C4×C127—44no—
254C2×C4×C127—42no—
255C2×C4×C127—42no—
256C2×C4×C127—42no—
257C2×C4×C127—84no—
258C2×C4×C127C2×C22⋊C414yes—
259C2×C4×C127C2×C22⋊C414yes—
260C2×C4×C127C2×C22⋊C414yes—
261C2×C4×C127C2×C22⋊C412yes—
262C2×C4×C127#12014yes—
263C2×C4×C127#12014yes—
264C2×C4×C127#12012yes—
265C2×C4×C127#12011yescharacteristic
266C2×C4×C127#12011yescharacteristic
267C2×C4×C127C23×C411yescharacteristic
268C2×C4×C127C22×D412yes—
269C2×C4×C127C22×D412yes—
270C2×C4×C127C22×D411yescharacteristic
271C8×C127—44no—
272C22×C127—21no—
273C22×C127—22no—
274C22×C127—21no—
275C22×C127—22no—
276C22×C127—24no—
277C22×C127—28no—
278C22×C127—42no—
279C22×C127—42no—
280C22×C127—44no—
281C22×C127—42no—
282C22×C127—44no—
283C22×C127—42no—
284C22×C127—44no—
285C22×C127—82no—
286C22×C127—88no—
287C22×C127—816no—
288C22×C127#37914yes—
289C22×C127C22×C22⋊C411yescharacteristic
290C22×C127C2×#12011yescharacteristic
291C22×C127C2×#12011yescharacteristic
292C4×C127—28no—
293C4×C127—24no—
294C4×C127—82no—
295C4×C127—82no—
296C4×C127#37912yes—
297C4×C127#37912yes—
298C4×C127C22×C22⋊C412yes—
299C4×C127C2×#12012yes—
300#20618C12711yesmaximal, characteristic, direct product factor, Sylow
301C2×C127—21no—
302C2×C127—21no—
303C2×C127—24no—
304C2×C127—44no—
305C2×C127—48no—
306C2×C127C2×#37911yescharacteristic
307C2×C127#302111yescharacteristic
308C2×C127#302211yescharacteristic
309C2×#585C2×C12711yescharacteristic
310C2×#421C2×C12712yescomplemented
311C2×#417C2×C12712yescomplemented
312C2×#413C2×C12711yescharacteristic, complemented
313#2311C2×C12711yescharacteristic, complemented
314#1793C2×C12714yescomplemented
315#1792C2×C12714yescomplemented
316C127#2061811yescharacteristic, direct product factor, Sylow
317#585—28no—
318#585—24no—
319#585C4×C12718yescomplemented
320C2×2−1+4—22no—
321C2×2−1+4C22×C12712yes—
322C2×H(5,𝔽2)—22no—
323C2×H(5,𝔽2)—22no—
324C2×H(5,𝔽2)C22×C12712yes—
325C22×#41—24no—
326C22×#41—22no—
327C22×#41C22×C12712yes—
328C22×#41C22×C12711yescharacteristic
329C22×M16C22×C12712yescomplemented
330C2×#117C22×C12711yescharacteristic, complemented
331C22×C4⋊C4C22×C12711yescharacteristic, complemented
332#421—28no—
333C2×#104C22×C12718yescomplemented
334#417—24no—
335#417C22×C12718yescomplemented
336C2×#103—22no—
337C2×#102—22no—
338#413—24no—
339C2×#101C22×C12712yescomplemented
340C2×#100C22×C12712yescomplemented
341#391C22×C12712yescomplemented
342#390C22×C12712yescomplemented
343#386—22no—
344#380—22no—
3452−1+4—216no—
3462−1+4—28no—
3472−1+4—216no—
3482−1+4—24no—
3492−1+4C2×C4×C12718yescomplemented
350H(5,𝔽2)—216no—
351H(5,𝔽2)—216no—
352H(5,𝔽2)—28no—
353H(5,𝔽2)—216no—
354H(5,𝔽2)—216no—
355H(5,𝔽2)—24no—
356H(5,𝔽2)C2×C4×C12718yescomplemented
357C2×#41—24no—
358C2×#41—28no—
359C2×#41—24no—
360C2×#41—232no—
361C2×#41—216no—
362C2×#41—232no—
363C2×#41—24no—
364C2×#41—24no—
365C2×#41—216no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—24no—
369C2×#41—22no—
370C2×#41—22no—
371C2×#41—24no—
372C2×#41—24no—
373C2×#41—24no—
374C2×#41C2×C4×C12718yes—
375C2×#41C2×C4×C12714yes—
376C2×#41D4×C12718yes—
377C2×#41D4×C12714yes—
378C2×#41D4×C12714yes—
379C2×#41C23×C12714yes—
380C22×Q8—22no—
381C22×Q8—24no—
382C22×Q8D4×C12712yes—
383C22×Q8C23×C12711yescharacteristic
384C22×D4—28no—
385C22×D4—22no—
386C22×D4—24no—
387C22×D4—22no—
388C22×D4—24no—
389C22×D4D4×C12712yes—
390C22×D4D4×C12712yes—
391C22×D4C23×C12711yescharacteristic
392C23×C4—24no—
393C23×C4—22no—
394C23×C4—21no—
395C23×C4D4×C12712yes—
396C23×C4C23×C12711yescharacteristic
397C2×M16—22no—
398C2×M16—24no—
399C2×M16—24no—
400C2×M16C23×C12714yes—
401C22×C8—22no—
402#117—24no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4C23×C12712yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—24no—
410C2×C42C23×C12712yes—
411#104—216no—
412#103—28no—
413#102—28no—
414#101—24no—
415#100—24no—
416#95—24no—
417C24—24no—
418C24—22no—
419C24C2×D4×C12712yes—
420C24C2×D4×C12711yescharacteristic
421#41—232no—
422#41—216no—
423#41—264no—
424#41—216no—
425#41—216no—
426#41—232no—
427#41—216no—
428#41—216no—
429#41—28no—
430#41—28no—
431#41—216no—
432#41—28no—
433#41—28no—
434#41—28no—
435#41—28no—
436#41—28no—
437#41—28no—
438#41—28no—
439#41—24no—
440#41—24no—
441#41C22⋊C4×C127116yescomplemented
442#41C22×C4×C12718yes—
443#41C2×D4×C12718yes—
444C2×Q8—24no—
445C2×Q8—216no—
446C2×Q8—216no—
447C2×Q8—216no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8C22⋊C4×C12718yescomplemented
453C2×Q8C22×C4×C12712yes—
454C2×Q8C2×D4×C12714yes—
455C2×Q8C2×D4×C12712yes—
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—216no—
459C2×D4—264no—
460C2×D4—216no—
461C2×D4—216no—
462C2×D4—216no—
463C2×D4—28no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—28no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—216no—
470C2×D4—22no—
471C2×D4—24no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4C22⋊C4×C12718yes—
475C2×D4C22⋊C4×C12718yescomplemented
476C2×D4C22×C4×C12712yes—
477C2×D4C2×D4×C12712yes—
478C2×D4C2×D4×C12714yes—
479C22×C4—24no—
480C22×C4—28no—
481C22×C4—232no—
482C22×C4—22no—
483C22×C4—28no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—28no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—82no—
494C22×C4C22⋊C4×C12718yes—
495C22×C4C22×C4×C12712yes—
496C22×C4C2×D4×C12712yes—
497C22×C4C2×D4×C12711yescharacteristic
498C22×C4C2×D4×C12712yes—
499C22×C4C2×D4×C12712yes—
500C22×C4C2×D4×C12714yes—
501C22×C4C2×D4×C12712yes—
502C22×C4C24×C12711yescharacteristic
503M16—24no—
504M16—24no—
505M16—44no—
506C2×C8—24no—
507C2×C8—44no—
508C4⋊C4—24no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C42—24no—
515C23—216no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—28no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—82no—
525C23—82no—
526C23—84no—
527C23—84no—
528C23C2×C22⋊C4×C12714yes—
529C23C2×C22⋊C4×C12712yes—
530C23#120×C12712yes—
531C23#120×C12712yes—
532C23C22×D4×C12711yescharacteristic
533Q8—216no—
534Q8—24no—
535Q8—24no—
536Q8—24no—
537Q8—28no—
538Q8C2×C22⋊C4×C12718yes—
539D4—232no—
540D4—216no—
541D4—28no—
542D4—28no—
543D4—216no—
544D4—216no—
545D4—24no—
546D4—24no—
547D4—24no—
548D4—28no—
549D4C2×C22⋊C4×C12718yes—
550C2×C4—216no—
551C2×C4—232no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—216no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4C2×C22⋊C4×C12714yes—
568C2×C4C2×C22⋊C4×C12714yes—
569C2×C4C2×C22⋊C4×C12714yes—
570C2×C4C2×C22⋊C4×C12712yes—
571C2×C4#120×C12714yes—
572C2×C4#120×C12714yes—
573C2×C4#120×C12711yescharacteristic
574C2×C4#120×C12711yescharacteristic
575C2×C4#120×C12712yes—
576C2×C4C23×C4×C12711yescharacteristic
577C2×C4C22×D4×C12712yes—
578C2×C4C22×D4×C12712yes—
579C2×C4C22×D4×C12711yescharacteristic
580C8—44no—
581C22—21no—
582C22—22no—
583C22—21no—
584C22—22no—
585C22—24no—
586C22—28no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—82no—
595C22—88no—
596C22—816no—
597C22#379×C12714yes—
598C22C22×C22⋊C4×C12711yescharacteristic
599C22C2×#120×C12711yescharacteristic
600C22C2×#120×C12711yescharacteristic
601C4—28no—
602C4—24no—
603C4—82no—
604C4—82no—
605C4#379×C12712yes—
606C4#379×C12712yes—
607C4C22×C22⋊C4×C12712yes—
608C4C2×#120×C12712yes—
609C2—21no—
610C2—21no—
611C2—24no—
612C2—44no—
613C2—48no—
614C2C2×#379×C12711yescharacteristic
615C2C127×#302111yescharacteristic
616C2C127×#302211yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×#585×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#421×C127C2semidirect
3. C2×#417×C127C2semidirect
4. C2×#413×C127C2semidirect
5. C127×#2311C2semidirect
6. C127×#1793C2semidirect
7. C127×#1792C2semidirect
10. #585×C127C4semidirect
20. C22×M16×C127C22semidirect
21. C2×#117×C127C22semidirect
22. C22×C4⋊C4×C127C22semidirect
24. C2×#104×C127C22semidirect
26. #417×C127C22semidirect
30. C2×#101×C127C22semidirect
31. C2×#100×C127C22semidirect
32. #391×C127C22semidirect
33. #390×C127C22semidirect
40. 2−1+4×C127C2×C4semidirect
47. H(5,𝔽2)×C127C2×C4semidirect
132. #41×C127C22⋊C4semidirect
143. C2×Q8×C127C22⋊C4semidirect
166. C2×D4×C127C22⋊C4semidirect
300. #20618C127direct (N × H)
310. C2×#421C2×C127semidirect
311. C2×#417C2×C127semidirect
312. C2×#413C2×C127semidirect
313. #2311C2×C127semidirect
314. #1793C2×C127semidirect
315. #1792C2×C127semidirect
316. C127#20618direct (N × H)
319. #585C4×C127semidirect
329. C22×M16C22×C127semidirect
330. C2×#117C22×C127semidirect
331. C22×C4⋊C4C22×C127semidirect
333. C2×#104C22×C127semidirect
335. #417C22×C127semidirect
339. C2×#101C22×C127semidirect
340. C2×#100C22×C127semidirect
341. #391C22×C127semidirect
342. #390C22×C127semidirect
349. 2−1+4C2×C4×C127semidirect
356. H(5,𝔽2)C2×C4×C127semidirect
441. #41C22⋊C4×C127semidirect
452. C2×Q8C22⋊C4×C127semidirect
475. C2×D4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#585×C1272
C2×#421×C1272
C2×#417×C1272
C2×#413×C1272
C127×#23112
C127×#17932
C127×#17922
#20618127