Finite Group Databasebeta

← C127×#21775 #44585448

Subgroups — C127×#21775 #44585448

Order 32512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3449C211yesmaximal, characteristic
2C127×#2879C211yesmaximal, characteristic, complemented
3C127×#2297C211yesmaximal, characteristic, complemented
4C127×#1897C212yesmaximal, complemented
5C127×#1896C212yesmaximal, complemented
6C127×#1871C214yesmaximal, complemented
7C127×#1826C212yesmaximal, complemented
8C127×#1789C212yesmaximal, complemented
9C22×#41×C127C2211yescharacteristic
10C22×#41×C127C2211yescharacteristic
11C22×M16×C127C2212yescomplemented
12#537×C127C414yescomplemented
13#536×C127—21no—
14#536×C127C412yescomplemented
15#535×C127—21no—
16#535×C127C412yescomplemented
17#534×C127—21no—
18#534×C127—21no—
19#532×C127—22no—
20#525×C127—22no—
21#525×C127C2212yes—
22#525×C127C2212yes—
23C2×#117×C127C2211yescharacteristic
24C2×#117×C127C2211yescharacteristic, complemented
25C22×C42×C127C2211yescharacteristic, complemented
26C2×#104×C127C2214yes—
27C2×#104×C127C2214yes—
28#408×C127—22no—
29#408×C127—22no—
30#407×C127C2212yes—
31#407×C127C2212yescomplemented
32#391×C127—21no—
33#390×C127—21no—
34#386×C127—21no—
35#386×C127—21no—
36C4×C22⋊C4×C127C2212yes—
37C4×C22⋊C4×C127C2212yescomplemented
38#339×C127C2214yescomplemented
39#339×C127C2214yescomplemented
40C2×#41×C127—21no—
41C2×#41×C127—21no—
42C2×#41×C127—42no—
43C2×#41×C127—42no—
44C2×#41×C127—42no—
45C2×#41×C127—42no—
46C2×#41×C127C2×C412yes—
47C2×#41×C127C2×C412yes—
48C2×#41×C127C2×C412yescomplemented
49C2×#41×C127D412yes—
50C2×#41×C127D412yes—
51C2×#41×C127C2312yes—
52C22×Q8×C127C2×C411yescharacteristic, complemented
53C22×Q8×C127D411yescharacteristic
54C22×D4×C127—21no—
55C22×D4×C127C2×C411yescharacteristic, complemented
56C22×D4×C127D411yescharacteristic
57C22×D4×C127D411yescharacteristic
58C22×D4×C127D411yescharacteristic
59C23×C4×C127—21no—
60C23×C4×C127—21no—
61C23×C4×C127C2311yescharacteristic
62C23×C4×C127C2311yescharacteristic
63C23×C4×C127C2311yescharacteristic
64C2×M16×C127—22no—
65C2×M16×C127—42no—
66C2×M16×C127—42no—
67C2×M16×C127C2314yes—
68C22×C8×C127—22no—
69C4⋊Q8×C127—22no—
70#127×C127—22no—
71#124×C127—22no—
72#124×C127—22no—
73#123×C127—22no—
74#123×C127—22no—
75#123×C127—22no—
76#123×C127—22no—
77C22⋊Q8×C127—24no—
78C22⋊Q8×C127C2×C414yes—
79C22⋊Q8×C127C2×C414yescomplemented
80#121×C127—24no—
81#121×C127—24no—
82#121×C127—24no—
83#121×C127C2×C414yes—
84#121×C127C2×C414yescomplemented
85#120×C127—22no—
86#120×C127—22no—
87#120×C127—22no—
88#120×C127—22no—
89C4×D4×C127—24no—
90C4×D4×C127—24no—
91#117×C127—21no—
92#117×C127—21no—
93#117×C127—42no—
94#117×C127C2×C412yescomplemented
95#117×C127C2312yes—
96C2×C4⋊C4×C127—21no—
97C2×C4⋊C4×C127C2×C412yescomplemented
98C2×C22⋊C4×C127—21no—
99C2×C22⋊C4×C127—21no—
100C2×C22⋊C4×C127—22no—
101C2×C22⋊C4×C127—22no—
102C2×C42×C127—21no—
103C2×C42×C127—21no—
104C2×C42×C127—42no—
105C2×C42×C127C2312yes—
106C2×C42×C127C2312yes—
107#104×C127—44no—
108#104×C127—44no—
109C22⋊C8×C127—22no—
110C22⋊C8×C127—22no—
111C22⋊C8×C127—24no—
112#95×C127—22no—
113#95×C127—22no—
114C24×C127—21no—
115C24×C127C2×D411yescharacteristic
116C24×C127C2×D411yescharacteristic
117C24×C127C2×D411yescharacteristic
118#41×C127—24no—
119#41×C127—24no—
120#41×C127—44no—
121#41×C127—44no—
122#41×C127—42no—
123#41×C127—42no—
124#41×C127—42no—
125#41×C127—42no—
126#41×C127—44no—
127#41×C127—42no—
128#41×C127—42no—
129#41×C127—44no—
130C2×Q8×C127—21no—
131C2×Q8×C127—41no—
132C2×Q8×C127—41no—
133C2×Q8×C127—42no—
134C2×Q8×C127C22⋊C412yes—
135C2×Q8×C127C22⋊C412yes—
136C2×Q8×C127C22×C412yes—
137C2×D4×C127—22no—
138C2×D4×C127—22no—
139C2×D4×C127—24no—
140C2×D4×C127—24no—
141C2×D4×C127—21no—
142C2×D4×C127—42no—
143C2×D4×C127—41no—
144C2×D4×C127—41no—
145C2×D4×C127—41no—
146C2×D4×C127—41no—
147C2×D4×C127—41no—
148C2×D4×C127—41no—
149C2×D4×C127—42no—
150C2×D4×C127—42no—
151C2×D4×C127C22⋊C412yes—
152C2×D4×C127C22⋊C412yes—
153C2×D4×C127C22⋊C412yes—
154C2×D4×C127C22⋊C412yes—
155C2×D4×C127C22⋊C412yes—
156C2×D4×C127C22⋊C412yes—
157C2×D4×C127C22×C412yes—
158C22×C4×C127—22no—
159C22×C4×C127—22no—
160C22×C4×C127—22no—
161C22×C4×C127—22no—
162C22×C4×C127—22no—
163C22×C4×C127—22no—
164C22×C4×C127—21no—
165C22×C4×C127—22no—
166C22×C4×C127—22no—
167C22×C4×C127—22no—
168C22×C4×C127—22no—
169C22×C4×C127—22no—
170C22×C4×C127—44no—
171C22×C4×C127—44no—
172C22×C4×C127—42no—
173C22×C4×C127C22×C411yescharacteristic
174C22×C4×C127C22×C411yescharacteristic
175C22×C4×C127C22×C412yes—
176C22×C4×C127C22×C412yes—
177C22×C4×C127C2×D411yescharacteristic
178C22×C4×C127C2×D411yescharacteristic
179C22×C4×C127C2×D411yescharacteristic
180C22×C4×C127C2×D411yescharacteristic
181C22×C4×C127C2×D411yescharacteristic
182C22×C4×C127C2×D412yes—
183C22×C4×C127C2×D412yes—
184C22×C4×C127C2411yescharacteristic
185M16×C127—24no—
186M16×C127—24no—
187M16×C127—44no—
188C2×C8×C127—24no—
189C2×C8×C127—44no—
190C4⋊C4×C127—22no—
191C4⋊C4×C127—24no—
192C4⋊C4×C127—24no—
193C4⋊C4×C127—42no—
194C4⋊C4×C127C22×C414yes—
195C22⋊C4×C127—22no—
196C22⋊C4×C127—24no—
197C22⋊C4×C127—24no—
198C22⋊C4×C127—22no—
199C22⋊C4×C127—24no—
200C22⋊C4×C127—24no—
201C22⋊C4×C127—44no—
202C22⋊C4×C127—44no—
203C22⋊C4×C127—42no—
204C42×C127—22no—
205C42×C127—22no—
206C42×C127—24no—
207C42×C127—24no—
208C42×C127—42no—
209C23×C127—21no—
210C23×C127—21no—
211C23×C127—21no—
212C23×C127—21no—
213C23×C127—21no—
214C23×C127—21no—
215C23×C127—22no—
216C23×C127—42no—
217C23×C127—42no—
218C23×C127—42no—
219C23×C127—42no—
220C23×C127—82no—
221C23×C127C2×C22⋊C411yescharacteristic
222C23×C127C2×C22⋊C412yes—
223C23×C127C2×C22⋊C412yes—
224C23×C127C2×C22⋊C411yescharacteristic
225C23×C127C22×D411yescharacteristic
226Q8×C127—24no—
227Q8×C127—42no—
228Q8×C127—42no—
229Q8×C127—42no—
230D4×C127—24no—
231D4×C127—42no—
232D4×C127—42no—
233D4×C127—42no—
234D4×C127—42no—
235D4×C127—42no—
236D4×C127—42no—
237D4×C127—42no—
238D4×C127—44no—
239D4×C127—42no—
240D4×C127—42no—
241D4×C127—84no—
242D4×C127—84no—
243C2×C4×C127—21no—
244C2×C4×C127—21no—
245C2×C4×C127—21no—
246C2×C4×C127—21no—
247C2×C4×C127—21no—
248C2×C4×C127—21no—
249C2×C4×C127—22no—
250C2×C4×C127—24no—
251C2×C4×C127—24no—
252C2×C4×C127—22no—
253C2×C4×C127—22no—
254C2×C4×C127—22no—
255C2×C4×C127—22no—
256C2×C4×C127—22no—
257C2×C4×C127—22no—
258C2×C4×C127—22no—
259C2×C4×C127—24no—
260C2×C4×C127—24no—
261C2×C4×C127—22no—
262C2×C4×C127—44no—
263C2×C4×C127—42no—
264C2×C4×C127—42no—
265C2×C4×C127—48no—
266C2×C4×C127—42no—
267C2×C4×C127—44no—
268C2×C4×C127—42no—
269C2×C4×C127—42no—
270C2×C4×C127—42no—
271C2×C4×C127—82no—
272C2×C4×C127C2×C22⋊C411yescharacteristic
273C2×C4×C127C2×C22⋊C412yes—
274C2×C4×C127C2×C22⋊C412yes—
275C2×C4×C127C2×C22⋊C411yescharacteristic
276C2×C4×C127C23×C411yescharacteristic
277C2×C4×C127C22×D411yescharacteristic
278C2×C4×C127H(5,𝔽2)12yes—
279C2×C4×C127H(5,𝔽2)12yes—
280C8×C127—44no—
281C22×C127—22no—
282C22×C127—21no—
283C22×C127—21no—
284C22×C127—22no—
285C22×C127—22no—
286C22×C127—22no—
287C22×C127—24no—
288C22×C127—24no—
289C22×C127—22no—
290C22×C127—42no—
291C22×C127—42no—
292C22×C127—42no—
293C22×C127—44no—
294C22×C127—81no—
295C22×C127—81no—
296C22×C127—81no—
297C22×C127—81no—
298C22×C127—84no—
299C22×C127C22×C22⋊C411yescharacteristic
300C22×C127#53411yescharacteristic
301C22×C127#53411yescharacteristic
302C4×C127—21no—
303C4×C127—21no—
304C4×C127—22no—
305C4×C127—22no—
306C4×C127—44no—
307C4×C127—42no—
308C4×C127—44no—
309C4×C127—82no—
310C4×C127#51812yes—
311C4×C127#53412yes—
312#21775C12711yesmaximal, characteristic, direct product factor, Sylow
313C2×C127—21no—
314C2×C127—21no—
315C2×C127—42no—
316C2×C127—41no—
317C2×C127—42no—
318C2×C127—41no—
319C2×C127—82no—
320C2×C127#230811yescharacteristic
321C2×C127#290112yes—
322#3449C2×C12711yescharacteristic
323#2879C2×C12711yescharacteristic, complemented
324#2297C2×C12711yescharacteristic, complemented
325#1897C2×C12712yescomplemented
326#1896C2×C12712yescomplemented
327#1871C2×C12714yescomplemented
328#1826C2×C12712yescomplemented
329#1789C2×C12712yescomplemented
330C127#2177511yescharacteristic, direct product factor, Sylow
331C22×#41C22×C12711yescharacteristic
332C22×#41C22×C12711yescharacteristic
333C22×M16C22×C12712yescomplemented
334#537C4×C12714yescomplemented
335#536—21no—
336#536C4×C12712yescomplemented
337#535—21no—
338#535C4×C12712yescomplemented
339#534—21no—
340#534—21no—
341#532—22no—
342#525—22no—
343#525C22×C12712yes—
344#525C22×C12712yes—
345C2×#117C22×C12711yescharacteristic
346C2×#117C22×C12711yescharacteristic, complemented
347C22×C42C22×C12711yescharacteristic, complemented
348C2×#104C22×C12714yes—
349C2×#104C22×C12714yes—
350#408—22no—
351#408—22no—
352#407C22×C12712yes—
353#407C22×C12712yescomplemented
354#391—21no—
355#390—21no—
356#386—21no—
357#386—21no—
358C4×C22⋊C4C22×C12712yes—
359C4×C22⋊C4C22×C12712yescomplemented
360#339C22×C12714yescomplemented
361#339C22×C12714yescomplemented
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—42no—
365C2×#41—42no—
366C2×#41—42no—
367C2×#41—42no—
368C2×#41C2×C4×C12712yes—
369C2×#41C2×C4×C12712yes—
370C2×#41C2×C4×C12712yescomplemented
371C2×#41D4×C12712yes—
372C2×#41D4×C12712yes—
373C2×#41C23×C12712yes—
374C22×Q8C2×C4×C12711yescharacteristic, complemented
375C22×Q8D4×C12711yescharacteristic
376C22×D4—21no—
377C22×D4C2×C4×C12711yescharacteristic, complemented
378C22×D4D4×C12711yescharacteristic
379C22×D4D4×C12711yescharacteristic
380C22×D4D4×C12711yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4C23×C12711yescharacteristic
384C23×C4C23×C12711yescharacteristic
385C23×C4C23×C12711yescharacteristic
386C2×M16—22no—
387C2×M16—42no—
388C2×M16—42no—
389C2×M16C23×C12714yes—
390C22×C8—22no—
391C4⋊Q8—22no—
392#127—22no—
393#124—22no—
394#124—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399C22⋊Q8—24no—
400C22⋊Q8C2×C4×C12714yes—
401C22⋊Q8C2×C4×C12714yescomplemented
402#121—24no—
403#121—24no—
404#121—24no—
405#121C2×C4×C12714yes—
406#121C2×C4×C12714yescomplemented
407#120—22no—
408#120—22no—
409#120—22no—
410#120—22no—
411C4×D4—24no—
412C4×D4—24no—
413#117—21no—
414#117—21no—
415#117—42no—
416#117C2×C4×C12712yescomplemented
417#117C23×C12712yes—
418C2×C4⋊C4—21no—
419C2×C4⋊C4C2×C4×C12712yescomplemented
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—42no—
427C2×C42C23×C12712yes—
428C2×C42C23×C12712yes—
429#104—44no—
430#104—44no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—24no—
434#95—22no—
435#95—22no—
436C24—21no—
437C24C2×D4×C12711yescharacteristic
438C24C2×D4×C12711yescharacteristic
439C24C2×D4×C12711yescharacteristic
440#41—24no—
441#41—24no—
442#41—44no—
443#41—44no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—44no—
449#41—42no—
450#41—42no—
451#41—44no—
452C2×Q8—21no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8C22⋊C4×C12712yes—
457C2×Q8C22⋊C4×C12712yes—
458C2×Q8C22×C4×C12712yes—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4C22⋊C4×C12712yes—
474C2×D4C22⋊C4×C12712yes—
475C2×D4C22⋊C4×C12712yes—
476C2×D4C22⋊C4×C12712yes—
477C2×D4C22⋊C4×C12712yes—
478C2×D4C22⋊C4×C12712yes—
479C2×D4C22×C4×C12712yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4C22×C4×C12711yescharacteristic
496C22×C4C22×C4×C12711yescharacteristic
497C22×C4C22×C4×C12712yes—
498C22×C4C22×C4×C12712yes—
499C22×C4C2×D4×C12711yescharacteristic
500C22×C4C2×D4×C12711yescharacteristic
501C22×C4C2×D4×C12711yescharacteristic
502C22×C4C2×D4×C12711yescharacteristic
503C22×C4C2×D4×C12711yescharacteristic
504C22×C4C2×D4×C12712yes—
505C22×C4C2×D4×C12712yes—
506C22×C4C24×C12711yescharacteristic
507M16—24no—
508M16—24no—
509M16—44no—
510C2×C8—24no—
511C2×C8—44no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—42no—
516C4⋊C4C22×C4×C12714yes—
517C22⋊C4—22no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—42no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—82no—
543C23C2×C22⋊C4×C12711yescharacteristic
544C23C2×C22⋊C4×C12712yes—
545C23C2×C22⋊C4×C12712yes—
546C23C2×C22⋊C4×C12711yescharacteristic
547C23C22×D4×C12711yescharacteristic
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552D4—24no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—44no—
561D4—42no—
562D4—42no—
563D4—84no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4C2×C22⋊C4×C12711yescharacteristic
595C2×C4C2×C22⋊C4×C12712yes—
596C2×C4C2×C22⋊C4×C12712yes—
597C2×C4C2×C22⋊C4×C12711yescharacteristic
598C2×C4C23×C4×C12711yescharacteristic
599C2×C4C22×D4×C12711yescharacteristic
600C2×C4H(5,𝔽2)×C12712yes—
601C2×C4H(5,𝔽2)×C12712yes—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—84no—
621C22C22×C22⋊C4×C12711yescharacteristic
622C22#534×C12711yescharacteristic
623C22#534×C12711yescharacteristic
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—82no—
632C4#518×C12712yes—
633C4#534×C12712yes—
634C2—21no—
635C2—21no—
636C2—42no—
637C2—41no—
638C2—41no—
639C2—42no—
640C2—82no—
641C2C127×#230811yescharacteristic
642C2C127×#290112yes—
subgroupG/H
centerC2×C4×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC127×#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2879C2semidirect
3. C127×#2297C2semidirect
4. C127×#1897C2semidirect
5. C127×#1896C2semidirect
6. C127×#1871C2semidirect
7. C127×#1826C2semidirect
8. C127×#1789C2semidirect
11. C22×M16×C127C22semidirect
12. #537×C127C4semidirect
14. #536×C127C4semidirect
16. #535×C127C4semidirect
24. C2×#117×C127C22semidirect
25. C22×C42×C127C22semidirect
31. #407×C127C22semidirect
37. C4×C22⋊C4×C127C22semidirect
38. #339×C127C22semidirect
39. #339×C127C22semidirect
48. C2×#41×C127C2×C4semidirect
52. C22×Q8×C127C2×C4semidirect
55. C22×D4×C127C2×C4semidirect
79. C22⋊Q8×C127C2×C4semidirect
84. #121×C127C2×C4semidirect
94. #117×C127C2×C4semidirect
97. C2×C4⋊C4×C127C2×C4semidirect
312. #21775C127direct (N × H)
323. #2879C2×C127semidirect
324. #2297C2×C127semidirect
325. #1897C2×C127semidirect
326. #1896C2×C127semidirect
327. #1871C2×C127semidirect
328. #1826C2×C127semidirect
329. #1789C2×C127semidirect
330. C127#21775direct (N × H)
333. C22×M16C22×C127semidirect
334. #537C4×C127semidirect
336. #536C4×C127semidirect
338. #535C4×C127semidirect
346. C2×#117C22×C127semidirect
347. C22×C42C22×C127semidirect
353. #407C22×C127semidirect
359. C4×C22⋊C4C22×C127semidirect
360. #339C22×C127semidirect
361. #339C22×C127semidirect
370. C2×#41C2×C4×C127semidirect
374. C22×Q8C2×C4×C127semidirect
377. C22×D4C2×C4×C127semidirect
401. C22⋊Q8C2×C4×C127semidirect
406. #121C2×C4×C127semidirect
416. #117C2×C4×C127semidirect
419. C2×C4⋊C4C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34492
C127×#28792
C127×#22972
C127×#18972
C127×#18962
C127×#18712
C127×#18262
C127×#17892
#21775127