Finite Group Databasebeta

← C4×#454×C127 #44585144

Subgroups — C4×#454×C127 #44585144

Order 32512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C127C211yesmaximal, characteristic, complemented
2C2×#454×C127C211yesmaximal, characteristic
3C4×AGL(1,ℤ/8)×C127C212yesmaximal, complemented
4C4×#124×C127C211yesmaximal, characteristic, complemented
5C127×#1888C211yesmaximal, characteristic, complemented
6C127×#1883C212yesmaximal, complemented
7C127×#1860C212yesmaximal, complemented
8C127×#1857C211yesmaximal, characteristic, complemented
9C127×#1793C211yesmaximal, characteristic, complemented
10C4×#104×C127C212yesmaximal, complemented
11C4×#101×C127C211yesmaximal, characteristic, complemented
12C2×2−1+4×C127C2211yescharacteristic
13C2×AGL(1,ℤ/8)×C127C2212yes—
14C2×#124×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×D4×C127C2211yescharacteristic
26#454×C127C418yesdirect product factor
27#442×C127—22no—
28#440×C127—22no—
29C4×SD16×C127—22no—
30C4×D8×C127—22no—
31#428×C127C2212yes—
32C2×#104×C127C2212yes—
33C2×#104×C127C2212yes—
34#418×C127C2212yes—
35#417×C127C2212yes—
36C2×#101×C127C2211yescharacteristic
37C2×#101×C127C2211yescharacteristic
38C4×M16×C127C2212yes—
39#391×C127C2211yescharacteristic
40#390×C127C2211yescharacteristic
41#388×C127—22no—
42#382×C127C2211yescharacteristic
43C4×C22⋊C4×C127—22no—
44C43×C127C2211yescharacteristic
45#343×C127C2212yes—
46#339×C127C2212yes—
47#339×C127C2212yes—
482−1+4×C127—24no—
492−1+4×C127—22no—
502−1+4×C127C2×C414yescomplemented
51C2×#41×C127—22no—
52C2×#41×C127—22no—
53C2×#41×C127C2312yes—
54C22×Q8×C127—22no—
55C22×Q8×C127C2311yescharacteristic
56C22×D4×C127C2311yescharacteristic
57C23×C4×C127—21no—
58AGL(1,ℤ/8)×C127—24no—
59AGL(1,ℤ/8)×C127C2×C418yescomplemented
60C2×SD16×C127—22no—
61C2×D8×C127—22no—
62C2×M16×C127C2312yes—
63C2×M16×C127C2312yes—
64#124×C127—22no—
65#124×C127C2×C414yescomplemented
66C4×Q8×C127—22no—
67C4×Q8×C127—24no—
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×C127D412yes—
75C4×Q8×C127D412yes—
76C4×D4×C127—22no—
77C4×D4×C127—24no—
78C4×D4×C127—22no—
79C4×D4×C127—22no—
80C4×D4×C127—22no—
81C4×D4×C127—24no—
82C4×D4×C127—22no—
83C4×D4×C127—22no—
84C4×D4×C127—22no—
85C4×D4×C127D412yes—
86C4×D4×C127D412yes—
87#117×C127—24no—
88#117×C127—24no—
89#117×C127—22no—
90#117×C127—22no—
91#117×C127—22no—
92#117×C127C2312yes—
93C2×C4⋊C4×C127—24no—
94C2×C4⋊C4×C127—22no—
95C2×C4⋊C4×C127D412yes—
96C2×C4⋊C4×C127C2311yescharacteristic
97C2×C4⋊C4×C127C2311yescharacteristic
98C2×C22⋊C4×C127—22no—
99C2×C22⋊C4×C127—21no—
100C2×C22⋊C4×C127—22no—
101C2×C42×C127—24no—
102C2×C42×C127—22no—
103C2×C42×C127—21no—
104C2×C42×C127—21no—
105C2×C42×C127D412yes—
106C2×C42×C127C2311yescharacteristic
107C2×C42×C127C2311yescharacteristic
108C2×C42×C127C2311yescharacteristic
109#107×C127—22no—
110#106×C127—22no—
111#104×C127—24no—
112#104×C127C2×C418yescomplemented
113#103×C127—22no—
114#102×C127—22no—
115#102×C127—22no—
116#102×C127—22no—
117#101×C127—22no—
118#101×C127C2×C414yescomplemented
119#97×C127—22no—
120C4×C8×C127—22no—
121#95×C127—22no—
122#95×C127—22no—
123C24×C127—21no—
124#41×C127—28no—
125#41×C127—28no—
126#41×C127—28no—
127#41×C127—28no—
128#41×C127—22no—
129#41×C127—22no—
130#41×C127C22×C414yes—
131#41×C127#4114yes—
132C2×Q8×C127—22no—
133C2×Q8×C127—28no—
134C2×Q8×C127—28no—
135C2×Q8×C127—22no—
136C2×Q8×C127—22no—
137C2×Q8×C127C22×C412yes—
138C2×Q8×C127C2×D412yes—
139C2×Q8×C127#4112yes—
140C2×D4×C127—22no—
141C2×D4×C127—22no—
142C2×D4×C127—22no—
143C2×D4×C127—22no—
144C2×D4×C127—21no—
145C2×D4×C127C22×C412yes—
146C2×D4×C127C2×D412yes—
147C22×C4×C127—24no—
148C22×C4×C127—22no—
149C22×C4×C127—21no—
150C22×C4×C127—21no—
151C22×C4×C127—24no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127—21no—
155C22×C4×C127—44no—
156C22×C4×C127—41no—
157C22×C4×C127—41no—
158C22×C4×C127C2×D412yes—
159C22×C4×C127C2×D411yescharacteristic
160C22×C4×C127C2×D412yes—
161C22×C4×C127C2×D411yescharacteristic
162C22×C4×C127C2411yescharacteristic
163SD16×C127—28no—
164D8×C127—28no—
165M16×C127—22no—
166M16×C127—22no—
167M16×C127—22no—
168M16×C127C22×C414yes—
169C2×C8×C127—22no—
170C2×C8×C127—22no—
171C4⋊C4×C127—24no—
172C4⋊C4×C127—22no—
173C4⋊C4×C127—22no—
174C4⋊C4×C127—22no—
175C4⋊C4×C127—24no—
176C4⋊C4×C127—22no—
177C4⋊C4×C127—22no—
178C4⋊C4×C127—22no—
179C4⋊C4×C127C2×D412yes—
180C4⋊C4×C127C2×D412yes—
181C22⋊C4×C127—24no—
182C22⋊C4×C127—22no—
183C22⋊C4×C127—28no—
184C22⋊C4×C127—22no—
185C22⋊C4×C127—22no—
186C22⋊C4×C127—44no—
187C22⋊C4×C127—42no—
188C42×C127—24no—
189C42×C127—24no—
190C42×C127—22no—
191C42×C127—22no—
192C42×C127—22no—
193C42×C127—22no—
194C42×C127—22no—
195C42×C127—21no—
196C42×C127—22no—
197C42×C127—42no—
198C42×C127—42no—
199C42×C127C22×C412yes—
200C42×C127C2×D412yes—
201C42×C127C2×D412yes—
202C23×C127—22no—
203C23×C127—22no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—41no—
207C23×C127—41no—
208C23×C127C22×D411yescharacteristic
209Q8×C127—28no—
210Q8×C127—24no—
211Q8×C127—22no—
212Q8×C127—24no—
213Q8×C127C4×D414yes—
214D4×C127—28no—
215D4×C127—24no—
216D4×C127—24no—
217D4×C127—24no—
218D4×C127—22no—
219D4×C127—42no—
220D4×C127C4×D414yes—
221C2×C4×C127—216no—
222C2×C4×C127—22no—
223C2×C4×C127—22no—
224C2×C4×C127—22no—
225C2×C4×C127—24no—
226C2×C4×C127—24no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—22no—
232C2×C4×C127—22no—
233C2×C4×C127—41no—
234C2×C4×C127—41no—
235C2×C4×C127—41no—
236C2×C4×C127—41no—
237C2×C4×C127—41no—
238C2×C4×C127—44no—
239C2×C4×C127—42no—
240C2×C4×C127—42no—
241C2×C4×C127—41no—
242C2×C4×C127—82no—
243C2×C4×C127—82no—
244C2×C4×C127—84no—
245C2×C4×C127—82no—
246C2×C4×C127C4×D414yes—
247C2×C4×C127#12011yescharacteristic
248C2×C4×C127#12012yes—
249C2×C4×C127#12011yescharacteristic
250C2×C4×C127C23×C411yescharacteristic
251C2×C4×C127C22×D412yes—
252C2×C4×C127C2×#4112yes—
253C2×C4×C127C2×#4111yescharacteristic
254C8×C127—24no—
255C8×C127—42no—
256C22×C127—21no—
257C22×C127—21no—
258C22×C127—24no—
259C22×C127—44no—
260C22×C127—42no—
261C22×C127—42no—
262C22×C127—42no—
263C22×C127—82no—
264C22×C127—81no—
265C22×C127—81no—
266C22×C127C2×C4×D411yescharacteristic
267C22×C127C2×#12011yescharacteristic
268C22×C127#52511yescharacteristic
269C4×C127—21no—
270C4×C127—22no—
271C4×C127—24no—
272C4×C127—24no—
273C4×C127—42no—
274C4×C127—42no—
275C4×C127—42no—
276C4×C127—42no—
277C4×C127—42no—
278C4×C127—81no—
279C4×C127#45412yesdirect product factor
280C4×C127C2×C4×D412yes—
281C4×C127#52512yes—
282C4×#454C12711yesmaximal, characteristic, direct product factor, Sylow
283C2×C127—21no—
284C2×C127—21no—
285C2×C127—44no—
286C2×C127—82no—
287C2×C127C4×#12011yescharacteristic
288C2×C127C2×#45411yescharacteristic
289C2×C127#301811yescharacteristic
290C4×2−1+4C2×C12711yescharacteristic, complemented
291C2×#454C2×C12711yescharacteristic
292C4×AGL(1,ℤ/8)C2×C12712yescomplemented
293C4×#124C2×C12711yescharacteristic, complemented
294#1888C2×C12711yescharacteristic, complemented
295#1883C2×C12712yescomplemented
296#1860C2×C12712yescomplemented
297#1857C2×C12711yescharacteristic, complemented
298#1793C2×C12711yescharacteristic, complemented
299C4×#104C2×C12712yescomplemented
300C4×#101C2×C12711yescharacteristic, complemented
301C127C4×#45411yescharacteristic, direct product factor, Sylow
302C2×2−1+4C22×C12711yescharacteristic
303C2×AGL(1,ℤ/8)C22×C12712yes—
304C2×#124C22×C12711yescharacteristic
305#520—22no—
306#520—22no—
307#520C22×C12712yes—
308#519—22no—
309#519C22×C12711yescharacteristic
310C4×#41—22no—
311C4×#41—22no—
312C4×#41C22×C12712yes—
313C2×C4×Q8—22no—
314C2×C4×Q8C22×C12711yescharacteristic
315C2×C4×D4C22×C12711yescharacteristic
316#454C4×C12718yesdirect product factor
317#442—22no—
318#440—22no—
319C4×SD16—22no—
320C4×D8—22no—
321#428C22×C12712yes—
322C2×#104C22×C12712yes—
323C2×#104C22×C12712yes—
324#418C22×C12712yes—
325#417C22×C12712yes—
326C2×#101C22×C12711yescharacteristic
327C2×#101C22×C12711yescharacteristic
328C4×M16C22×C12712yes—
329#391C22×C12711yescharacteristic
330#390C22×C12711yescharacteristic
331#388—22no—
332#382C22×C12711yescharacteristic
333C4×C22⋊C4—22no—
334C43C22×C12711yescharacteristic
335#343C22×C12712yes—
336#339C22×C12712yes—
337#339C22×C12712yes—
3382−1+4—24no—
3392−1+4—22no—
3402−1+4C2×C4×C12714yescomplemented
341C2×#41—22no—
342C2×#41—22no—
343C2×#41C23×C12712yes—
344C22×Q8—22no—
345C22×Q8C23×C12711yescharacteristic
346C22×D4C23×C12711yescharacteristic
347C23×C4—21no—
348AGL(1,ℤ/8)—24no—
349AGL(1,ℤ/8)C2×C4×C12718yescomplemented
350C2×SD16—22no—
351C2×D8—22no—
352C2×M16C23×C12712yes—
353C2×M16C23×C12712yes—
354#124—22no—
355#124C2×C4×C12714yescomplemented
356C4×Q8—22no—
357C4×Q8—24no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8D4×C12712yes—
365C4×Q8D4×C12712yes—
366C4×D4—22no—
367C4×D4—24no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—24no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4D4×C12712yes—
376C4×D4D4×C12712yes—
377#117—24no—
378#117—24no—
379#117—22no—
380#117—22no—
381#117—22no—
382#117C23×C12712yes—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4D4×C12712yes—
386C2×C4⋊C4C23×C12711yescharacteristic
387C2×C4⋊C4C23×C12711yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C42—24no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42D4×C12712yes—
396C2×C42C23×C12711yescharacteristic
397C2×C42C23×C12711yescharacteristic
398C2×C42C23×C12711yescharacteristic
399#107—22no—
400#106—22no—
401#104—24no—
402#104C2×C4×C12718yescomplemented
403#103—22no—
404#102—22no—
405#102—22no—
406#102—22no—
407#101—22no—
408#101C2×C4×C12714yescomplemented
409#97—22no—
410C4×C8—22no—
411#95—22no—
412#95—22no—
413C24—21no—
414#41—28no—
415#41—28no—
416#41—28no—
417#41—28no—
418#41—22no—
419#41—22no—
420#41C22×C4×C12714yes—
421#41#41×C12714yes—
422C2×Q8—22no—
423C2×Q8—28no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8C22×C4×C12712yes—
428C2×Q8C2×D4×C12712yes—
429C2×Q8#41×C12712yes—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4C22×C4×C12712yes—
436C2×D4C2×D4×C12712yes—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—44no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4C2×D4×C12712yes—
449C22×C4C2×D4×C12711yescharacteristic
450C22×C4C2×D4×C12712yes—
451C22×C4C2×D4×C12711yescharacteristic
452C22×C4C24×C12711yescharacteristic
453SD16—28no—
454D8—28no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16C22×C4×C12714yes—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4C2×D4×C12712yes—
470C4⋊C4C2×D4×C12712yes—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—28no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—22no—
487C42—42no—
488C42—42no—
489C42C22×C4×C12712yes—
490C42C2×D4×C12712yes—
491C42C2×D4×C12712yes—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23C22×D4×C12711yescharacteristic
499Q8—28no—
500Q8—24no—
501Q8—22no—
502Q8—24no—
503Q8C4×D4×C12714yes—
504D4—28no—
505D4—24no—
506D4—24no—
507D4—24no—
508D4—22no—
509D4—42no—
510D4C4×D4×C12714yes—
511C2×C4—216no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4C4×D4×C12714yes—
537C2×C4#120×C12711yescharacteristic
538C2×C4#120×C12712yes—
539C2×C4#120×C12711yescharacteristic
540C2×C4C23×C4×C12711yescharacteristic
541C2×C4C22×D4×C12712yes—
542C2×C4C2×#41×C12712yes—
543C2×C4C2×#41×C12711yescharacteristic
544C8—24no—
545C8—42no—
546C22—21no—
547C22—21no—
548C22—24no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—81no—
555C22—81no—
556C22C2×C4×D4×C12711yescharacteristic
557C22C2×#120×C12711yescharacteristic
558C22#525×C12711yescharacteristic
559C4—21no—
560C4—22no—
561C4—24no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4#454×C12712yesdirect product factor
570C4C2×C4×D4×C12712yes—
571C4#525×C12712yes—
572C2—21no—
573C2—21no—
574C2—44no—
575C2—82no—
576C2C4×#120×C12711yescharacteristic
577C2C2×#454×C12711yescharacteristic
578C2C127×#301811yescharacteristic
subgroupG/H
centerC2×C4×C127#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C2×#120
prime radicalC2×#454×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C4×2−1+4×C127C2semidirect
3. C4×AGL(1,ℤ/8)×C127C2semidirect
4. C4×#124×C127C2semidirect
5. C127×#1888C2semidirect
6. C127×#1883C2semidirect
7. C127×#1860C2semidirect
8. C127×#1857C2semidirect
9. C127×#1793C2semidirect
10. C4×#104×C127C2semidirect
11. C4×#101×C127C2semidirect
26. #454×C127C4direct (N × H)
50. 2−1+4×C127C2×C4semidirect
59. AGL(1,ℤ/8)×C127C2×C4semidirect
65. #124×C127C2×C4semidirect
112. #104×C127C2×C4semidirect
118. #101×C127C2×C4semidirect
279. C4×C127#454direct (N × H)
282. C4×#454C127direct (N × H)
290. C4×2−1+4C2×C127semidirect
292. C4×AGL(1,ℤ/8)C2×C127semidirect
293. C4×#124C2×C127semidirect
294. #1888C2×C127semidirect
295. #1883C2×C127semidirect
296. #1860C2×C127semidirect
297. #1857C2×C127semidirect
298. #1793C2×C127semidirect
299. C4×#104C2×C127semidirect
300. C4×#101C2×C127semidirect
301. C127C4×#454direct (N × H)
316. #454C4×C127direct (N × H)
340. 2−1+4C2×C4×C127semidirect
349. AGL(1,ℤ/8)C2×C4×C127semidirect
355. #124C2×C4×C127semidirect
402. #104C2×C4×C127semidirect
408. #101C2×C4×C127semidirect
569. C4#454×C127direct (N × H)

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×#454×C1272
C4×AGL(1,ℤ/8)×C1272
C4×#124×C1272
C127×#18882
C127×#18832
C127×#18602
C127×#18572
C127×#17932
C4×#104×C1272
C4×#101×C1272
C4×#454127