Finite Group Databasebeta

← C127×#21476 #44585149

Subgroups — C127×#21476 #44585149

Order 32512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C127C211yesmaximal, characteristic
2C2×#456×C127C211yesmaximal, characteristic
3C4×#137×C127C212yesmaximal, complemented
4C127×#2327C211yesmaximal, characteristic, complemented
5C127×#1886C211yesmaximal, characteristic, complemented
6C127×#1882C212yesmaximal, complemented
7C127×#1860C212yesmaximal, complemented
8C127×#1857C211yesmaximal, characteristic, complemented
9C127×#1793C211yesmaximal, characteristic
10C127×#1759C212yesmaximal, complemented
11C4×#101×C127C211yesmaximal, characteristic, complemented
12C2×2−1+4×C127C2211yescharacteristic
13C2×#137×C127C2212yes—
14C2×C4⋊Q8×C127C2211yescharacteristic
15#520×C127—22no—
16#520×C127—22no—
17#520×C127C2212yes—
18#519×C127—22no—
19#519×C127C2211yescharacteristic
20C4×#41×C127—22no—
21C4×#41×C127—22no—
22C4×#41×C127C2212yes—
23C2×C4×Q8×C127—22no—
24C2×C4×Q8×C127C2211yescharacteristic
25C2×C4×Q8×C127C2211yescharacteristic
26#456×C127C418yescomplemented
27#441×C127—22no—
28#440×C127—22no—
29C4×Q16×C127—22no—
30C4×SD16×C127—22no—
31#428×C127C2212yes—
32C2×#104×C127C2212yes—
33C2×#104×C127C2212yes—
34#419×C127C2212yes—
35#417×C127C2212yes—
36C2×#101×C127C2211yescharacteristic
37C2×#101×C127C2211yescharacteristic
38C4×M16×C127C2212yes—
39#391×C127C2211yescharacteristic
40#391×C127C2211yescharacteristic
41#387×C127—22no—
42#387×C127—22no—
43#382×C127C2211yescharacteristic
44#376×C127C2211yescharacteristic
45#343×C127C2212yes—
46#339×C127C2212yes—
47#339×C127C2212yes—
482−1+4×C127—24no—
492−1+4×C127—22no—
502−1+4×C127C2×C414yes—
51C2×#41×C127—22no—
52C2×#41×C127—22no—
53C2×#41×C127C2312yes—
54C22×Q8×C127—22no—
55C22×Q8×C127C2311yescharacteristic
56C22×Q8×C127C2311yescharacteristic
57#137×C127—24no—
58#137×C127C2×C418yescomplemented
59C2×Q16×C127—22no—
60C2×SD16×C127—22no—
61C2×M16×C127C2312yes—
62C2×M16×C127C2312yes—
63C4⋊Q8×C127—22no—
64C4⋊Q8×C127C2×C414yescomplemented
65C4×Q8×C127—22no—
66C4×Q8×C127—24no—
67C4×Q8×C127—22no—
68C4×Q8×C127—22no—
69C4×Q8×C127—22no—
70C4×Q8×C127—22no—
71C4×Q8×C127—22no—
72C4×Q8×C127—22no—
73C4×Q8×C127—22no—
74C4×Q8×C127—22no—
75C4×Q8×C127D412yes—
76C4×Q8×C127D412yes—
77C4×D4×C127—22no—
78C4×D4×C127—24no—
79C4×D4×C127—22no—
80C4×D4×C127—22no—
81C4×D4×C127—22no—
82C4×D4×C127—24no—
83C4×D4×C127—22no—
84C4×D4×C127D412yes—
85C4×D4×C127D412yes—
86#117×C127—24no—
87#117×C127—24no—
88#117×C127—22no—
89#117×C127—22no—
90#117×C127—22no—
91#117×C127C2312yes—
92C2×C4⋊C4×C127—24no—
93C2×C4⋊C4×C127—22no—
94C2×C4⋊C4×C127—22no—
95C2×C4⋊C4×C127—21no—
96C2×C4⋊C4×C127D412yes—
97C2×C4⋊C4×C127C2311yescharacteristic
98C2×C4⋊C4×C127C2311yescharacteristic
99C2×C42×C127—24no—
100C2×C42×C127—22no—
101C2×C42×C127—21no—
102C2×C42×C127D412yes—
103C2×C42×C127C2311yescharacteristic
104C2×C42×C127C2311yescharacteristic
105C2×C42×C127C2311yescharacteristic
106#107×C127—22no—
107#106×C127—22no—
108#104×C127—24no—
109#104×C127C2×C418yescomplemented
110#103×C127—22no—
111#103×C127—22no—
112#103×C127—22no—
113#102×C127—22no—
114#101×C127—22no—
115#101×C127C2×C414yescomplemented
116#97×C127—22no—
117C4×C8×C127—22no—
118#95×C127—21no—
119#95×C127—21no—
120#95×C127—22no—
121#95×C127—22no—
122#95×C127—22no—
123#41×C127—28no—
124#41×C127—28no—
125#41×C127—28no—
126#41×C127—28no—
127#41×C127—22no—
128#41×C127—22no—
129#41×C127C22×C414yes—
130#41×C127#4114yes—
131C2×Q8×C127—22no—
132C2×Q8×C127—28no—
133C2×Q8×C127—28no—
134C2×Q8×C127—22no—
135C2×Q8×C127—22no—
136C2×Q8×C127—21no—
137C2×Q8×C127—22no—
138C2×Q8×C127—22no—
139C2×Q8×C127C22×C412yes—
140C2×Q8×C127C22×C412yes—
141C2×Q8×C127C2×D412yes—
142C2×Q8×C127#4112yes—
143C2×D4×C127—22no—
144C2×D4×C127—22no—
145C2×D4×C127C2×D412yes—
146C22×C4×C127—24no—
147C22×C4×C127—22no—
148C22×C4×C127—21no—
149C22×C4×C127—21no—
150C22×C4×C127—24no—
151C22×C4×C127—22no—
152C22×C4×C127—22no—
153C22×C4×C127—21no—
154C22×C4×C127—21no—
155C22×C4×C127C2×D412yes—
156C22×C4×C127C2×D412yes—
157C22×C4×C127C2×D411yescharacteristic
158C22×C4×C127C2×D411yescharacteristic
159C22×C4×C127C2411yescharacteristic
160Q16×C127—28no—
161SD16×C127—28no—
162M16×C127—22no—
163M16×C127—22no—
164M16×C127—22no—
165M16×C127C22×C414yes—
166C2×C8×C127—22no—
167C2×C8×C127—22no—
168C4⋊C4×C127—24no—
169C4⋊C4×C127—22no—
170C4⋊C4×C127—22no—
171C4⋊C4×C127—28no—
172C4⋊C4×C127—22no—
173C4⋊C4×C127—24no—
174C4⋊C4×C127—22no—
175C4⋊C4×C127—22no—
176C4⋊C4×C127—22no—
177C4⋊C4×C127—41no—
178C4⋊C4×C127—41no—
179C4⋊C4×C127C2×D412yes—
180C4⋊C4×C127C2×D412yes—
181C22⋊C4×C127—24no—
182C22⋊C4×C127—22no—
183C22⋊C4×C127—22no—
184C22⋊C4×C127—22no—
185C42×C127—24no—
186C42×C127—24no—
187C42×C127—22no—
188C42×C127—22no—
189C42×C127—22no—
190C42×C127—22no—
191C42×C127—22no—
192C42×C127—21no—
193C42×C127—22no—
194C42×C127—42no—
195C42×C127C22×C412yes—
196C42×C127C2×D412yes—
197C42×C127C2×D412yes—
198C23×C127—22no—
199C23×C127C22×D411yescharacteristic
200Q8×C127—28no—
201Q8×C127—24no—
202Q8×C127—22no—
203Q8×C127—24no—
204Q8×C127—24no—
205Q8×C127—42no—
206Q8×C127C4×D414yes—
207D4×C127—28no—
208D4×C127—24no—
209D4×C127—24no—
210D4×C127—22no—
211D4×C127C4×D414yes—
212C2×C4×C127—216no—
213C2×C4×C127—22no—
214C2×C4×C127—22no—
215C2×C4×C127—22no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—22no—
219C2×C4×C127—22no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—22no—
223C2×C4×C127—41no—
224C2×C4×C127—41no—
225C2×C4×C127—44no—
226C2×C4×C127—41no—
227C2×C4×C127—41no—
228C2×C4×C127—41no—
229C2×C4×C127—41no—
230C2×C4×C127—42no—
231C2×C4×C127—41no—
232C2×C4×C127—41no—
233C2×C4×C127—42no—
234C2×C4×C127—41no—
235C2×C4×C127—41no—
236C2×C4×C127—82no—
237C2×C4×C127—84no—
238C2×C4×C127C4×D414yes—
239C2×C4×C127#12011yescharacteristic
240C2×C4×C127#12011yescharacteristic
241C2×C4×C127#12012yes—
242C2×C4×C127C23×C411yescharacteristic
243C2×C4×C127C22×D412yes—
244C2×C4×C127C2×#4112yes—
245C2×C4×C127C2×#4111yescharacteristic
246C8×C127—24no—
247C8×C127—42no—
248C22×C127—21no—
249C22×C127—21no—
250C22×C127—24no—
251C22×C127—42no—
252C22×C127—42no—
253C22×C127C2×C4×D411yescharacteristic
254C22×C127C2×#12011yescharacteristic
255C22×C127#52511yescharacteristic
256C4×C127—21no—
257C4×C127—21no—
258C4×C127—22no—
259C4×C127—24no—
260C4×C127—24no—
261C4×C127—42no—
262C4×C127—42no—
263C4×C127—42no—
264C4×C127—42no—
265C4×C127—42no—
266C4×C127—81no—
267C4×C127—81no—
268C4×C127C2×C4×D412yes—
269C4×C127#52512yes—
270#21476C12711yesmaximal, characteristic, direct product factor, Sylow
271C2×C127—21no—
272C2×C127—21no—
273C2×C127—44no—
274C2×C127C4×#12011yescharacteristic
275C2×C127#302011yescharacteristic
276C2×C127#302211yescharacteristic
277C4×2−1+4C2×C12711yescharacteristic
278C2×#456C2×C12711yescharacteristic
279C4×#137C2×C12712yescomplemented
280#2327C2×C12711yescharacteristic, complemented
281#1886C2×C12711yescharacteristic, complemented
282#1882C2×C12712yescomplemented
283#1860C2×C12712yescomplemented
284#1857C2×C12711yescharacteristic, complemented
285#1793C2×C12711yescharacteristic
286#1759C2×C12712yescomplemented
287C4×#101C2×C12711yescharacteristic, complemented
288C127#2147611yescharacteristic, direct product factor, Sylow
289C2×2−1+4C22×C12711yescharacteristic
290C2×#137C22×C12712yes—
291C2×C4⋊Q8C22×C12711yescharacteristic
292#520—22no—
293#520—22no—
294#520C22×C12712yes—
295#519—22no—
296#519C22×C12711yescharacteristic
297C4×#41—22no—
298C4×#41—22no—
299C4×#41C22×C12712yes—
300C2×C4×Q8—22no—
301C2×C4×Q8C22×C12711yescharacteristic
302C2×C4×Q8C22×C12711yescharacteristic
303#456C4×C12718yescomplemented
304#441—22no—
305#440—22no—
306C4×Q16—22no—
307C4×SD16—22no—
308#428C22×C12712yes—
309C2×#104C22×C12712yes—
310C2×#104C22×C12712yes—
311#419C22×C12712yes—
312#417C22×C12712yes—
313C2×#101C22×C12711yescharacteristic
314C2×#101C22×C12711yescharacteristic
315C4×M16C22×C12712yes—
316#391C22×C12711yescharacteristic
317#391C22×C12711yescharacteristic
318#387—22no—
319#387—22no—
320#382C22×C12711yescharacteristic
321#376C22×C12711yescharacteristic
322#343C22×C12712yes—
323#339C22×C12712yes—
324#339C22×C12712yes—
3252−1+4—24no—
3262−1+4—22no—
3272−1+4C2×C4×C12714yes—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41C23×C12712yes—
331C22×Q8—22no—
332C22×Q8C23×C12711yescharacteristic
333C22×Q8C23×C12711yescharacteristic
334#137—24no—
335#137C2×C4×C12718yescomplemented
336C2×Q16—22no—
337C2×SD16—22no—
338C2×M16C23×C12712yes—
339C2×M16C23×C12712yes—
340C4⋊Q8—22no—
341C4⋊Q8C2×C4×C12714yescomplemented
342C4×Q8—22no—
343C4×Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8D4×C12712yes—
353C4×Q8D4×C12712yes—
354C4×D4—22no—
355C4×D4—24no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—24no—
360C4×D4—22no—
361C4×D4D4×C12712yes—
362C4×D4D4×C12712yes—
363#117—24no—
364#117—24no—
365#117—22no—
366#117—22no—
367#117—22no—
368#117C23×C12712yes—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4D4×C12712yes—
374C2×C4⋊C4C23×C12711yescharacteristic
375C2×C4⋊C4C23×C12711yescharacteristic
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42D4×C12712yes—
380C2×C42C23×C12711yescharacteristic
381C2×C42C23×C12711yescharacteristic
382C2×C42C23×C12711yescharacteristic
383#107—22no—
384#106—22no—
385#104—24no—
386#104C2×C4×C12718yescomplemented
387#103—22no—
388#103—22no—
389#103—22no—
390#102—22no—
391#101—22no—
392#101C2×C4×C12714yescomplemented
393#97—22no—
394C4×C8—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#41—28no—
401#41—28no—
402#41—28no—
403#41—28no—
404#41—22no—
405#41—22no—
406#41C22×C4×C12714yes—
407#41#41×C12714yes—
408C2×Q8—22no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8C22×C4×C12712yes—
417C2×Q8C22×C4×C12712yes—
418C2×Q8C2×D4×C12712yes—
419C2×Q8#41×C12712yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4C2×D4×C12712yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C2×D4×C12712yes—
433C22×C4C2×D4×C12712yes—
434C22×C4C2×D4×C12711yescharacteristic
435C22×C4C2×D4×C12711yescharacteristic
436C22×C4C24×C12711yescharacteristic
437Q16—28no—
438SD16—28no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16C22×C4×C12714yes—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—28no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4C2×D4×C12712yes—
457C4⋊C4C2×D4×C12712yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—22no—
471C42—42no—
472C42C22×C4×C12712yes—
473C42C2×D4×C12712yes—
474C42C2×D4×C12712yes—
475C23—22no—
476C23C22×D4×C12711yescharacteristic
477Q8—28no—
478Q8—24no—
479Q8—22no—
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8C4×D4×C12714yes—
484D4—28no—
485D4—24no—
486D4—24no—
487D4—22no—
488D4C4×D4×C12714yes—
489C2×C4—216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4C4×D4×C12714yes—
516C2×C4#120×C12711yescharacteristic
517C2×C4#120×C12711yescharacteristic
518C2×C4#120×C12712yes—
519C2×C4C23×C4×C12711yescharacteristic
520C2×C4C22×D4×C12712yes—
521C2×C4C2×#41×C12712yes—
522C2×C4C2×#41×C12711yescharacteristic
523C8—24no—
524C8—42no—
525C22—21no—
526C22—21no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22C2×C4×D4×C12711yescharacteristic
531C22C2×#120×C12711yescharacteristic
532C22#525×C12711yescharacteristic
533C4—21no—
534C4—21no—
535C4—22no—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—81no—
544C4—81no—
545C4C2×C4×D4×C12712yes—
546C4#525×C12712yes—
547C2—21no—
548C2—21no—
549C2—44no—
550C2C4×#120×C12711yescharacteristic
551C2C127×#302011yescharacteristic
552C2C127×#302211yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×2−1+4×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#137×C127C2semidirect
4. C127×#2327C2semidirect
5. C127×#1886C2semidirect
6. C127×#1882C2semidirect
7. C127×#1860C2semidirect
8. C127×#1857C2semidirect
10. C127×#1759C2semidirect
11. C4×#101×C127C2semidirect
26. #456×C127C4semidirect
58. #137×C127C2×C4semidirect
64. C4⋊Q8×C127C2×C4semidirect
109. #104×C127C2×C4semidirect
115. #101×C127C2×C4semidirect
270. #21476C127direct (N × H)
279. C4×#137C2×C127semidirect
280. #2327C2×C127semidirect
281. #1886C2×C127semidirect
282. #1882C2×C127semidirect
283. #1860C2×C127semidirect
284. #1857C2×C127semidirect
286. #1759C2×C127semidirect
287. C4×#101C2×C127semidirect
288. C127#21476direct (N × H)
303. #456C4×C127semidirect
335. #137C2×C4×C127semidirect
341. C4⋊Q8C2×C4×C127semidirect
386. #104C2×C4×C127semidirect
392. #101C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4×C1272
C2×#456×C1272
C4×#137×C1272
C127×#23272
C127×#18862
C127×#18822
C127×#18602
C127×#18572
C127×#17932
C127×#17592
C4×#101×C1272
#21476127