Finite Group Databasebeta

← C127×#18469 #44582142

Subgroups — C127×#18469 #44582142

Order 32512.

814 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3563C211yesmaximal, characteristic
2C127×#3236C211yesmaximal, characteristic, complemented
3C127×#3232C211yesmaximal, characteristic
4C127×#2822C211yesmaximal, characteristic, complemented
5C127×#1617C212yesmaximal, complemented
6C127×#1615C212yesmaximal
7C127×#1611C212yesmaximal, complemented
8C127×#1609C212yesmaximal
9C127×#1518C211yesmaximal, characteristic, complemented
10C127×#1512C211yesmaximal, characteristic
11C127×#1470C211yesmaximal, characteristic, complemented
12#564×C127—21no—
13#563×C127—21no—
14#557×C127—21no—
15#554×C127—21no—
16#552×C127—21no—
17#552×C127—22no—
18#552×C127—21no—
19#544×C127—21no—
20#544×C127C2212yes—
21#541×C127—21no—
22#541×C127—22no—
23#539×C127C2212yes—
24#533×C127C2211yescharacteristic
25#529×C127C2211yescharacteristic
26C2×#125×C127C2211yescharacteristic
27C2×#125×C127C2211yescharacteristic
28#501×C127—21no—
29#499×C127—21no—
30#491×C127—21no—
31#488×C127—21no—
32#484×C127C2212yes—
33#484×C127C2211yescharacteristic
34#483×C127C2212yes—
35#483×C127C2211yescharacteristic
36#481×C127C2211yescharacteristic
37#480×C127C2211yescharacteristic
38#479×C127—21no—
39#479×C127—22no—
40#478×C127—21no—
41#432×C127C2211yescharacteristic
42#431×C127C2211yescharacteristic
43#400×C127—21no—
44#400×C127C2212yes—
45#398×C127—21no—
46#398×C127—22no—
47#398×C127C2212yes—
48#382×C127C2211yescharacteristic
49#328×C127C2212yes—
50#328×C127C2212yes—
51#328×C127C2212yes—
52#327×C127C2212yes—
53#324×C127C2212yes—
54#324×C127C2212yes—
55C4⋊Q8×C127—21no—
56C4⋊Q8×C127—22no—
57C4⋊Q8×C127—22no—
58C4⋊Q8×C127D411yescharacteristic
59C4⋊Q8×C127D411yescharacteristic
60#126×C127—22no—
61#126×C127—22no—
62#126×C127—22no—
63#126×C127—22no—
64#126×C127—21no—
65#125×C127—22no—
66#125×C127—22no—
67#125×C127—22no—
68#125×C127—22no—
69#125×C127—21no—
70#125×C127—22no—
71#125×C127—22no—
72#125×C127—22no—
73#125×C127—22no—
74#125×C127—21no—
75#125×C127—21no—
76#125×C127—21no—
77#125×C127—21no—
78#125×C127—21no—
79#125×C127—21no—
80#125×C127—21no—
81#125×C127—21no—
82#125×C127—21no—
83#125×C127D411yescharacteristic
84#124×C127D411yescharacteristic
85#123×C127—22no—
86#123×C127—21no—
87#123×C127—22no—
88C22⋊Q8×C127—21no—
89C22⋊Q8×C127—21no—
90C22⋊Q8×C127—22no—
91C22⋊Q8×C127—22no—
92C22⋊Q8×C127—22no—
93C22⋊Q8×C127—21no—
94C22⋊Q8×C127—22no—
95C22⋊Q8×C127C2311yescharacteristic
96C22⋊Q8×C127C2312yes—
97#121×C127C2311yescharacteristic
98C4×Q8×C127—21no—
99C4×Q8×C127—22no—
100C4×Q8×C127—21no—
101C4×Q8×C127—21no—
102C4×Q8×C127D411yescharacteristic
103C4×Q8×C127D412yes—
104C4×D4×C127—21no—
105C4×D4×C127—22no—
106C4×D4×C127D411yescharacteristic
107#117×C127—22no—
108#117×C127—21no—
109#117×C127—21no—
110#117×C127—22no—
111#117×C127—21no—
112C2×C4⋊C4×C127—21no—
113C2×C4⋊C4×C127—21no—
114C2×C4⋊C4×C127—21no—
115C2×C4⋊C4×C127—21no—
116C2×C4⋊C4×C127—22no—
117C2×C4⋊C4×C127D411yescharacteristic
118C2×C4⋊C4×C127D412yes—
119C2×C4⋊C4×C127D411yescharacteristic
120C2×C4⋊C4×C127C2311yescharacteristic
121C2×C4⋊C4×C127C2311yescharacteristic
122C2×C4⋊C4×C127C2312yes—
123C2×C42×C127C2311yescharacteristic
124#107×C127—21no—
125#107×C127—21no—
126#107×C127—22no—
127#106×C127—21no—
128#106×C127—21no—
129#106×C127—22no—
130C4⋊C8×C127—21no—
131C4⋊C8×C127—21no—
132#103×C127—21no—
133#103×C127—22no—
134#103×C127—21no—
135#103×C127—22no—
136#102×C127—21no—
137#102×C127—21no—
138C22⋊C8×C127C2312yes—
139C22⋊C8×C127C2311yescharacteristic
140C22⋊C8×C127C2311yescharacteristic
141#97×C127—21no—
142C4×C8×C127—21no—
143#95×C127—21no—
144#95×C127—22no—
145#95×C127—22no—
146#95×C127—22no—
147#95×C127C2312yes—
148C2×Q8×C127—22no—
149C2×Q8×C127—21no—
150C2×Q8×C127C2×D412yes—
151C2×Q8×C127C2×D411yescharacteristic
152C2×D4×C127—21no—
153C2×D4×C127C2×D411yescharacteristic
154C22×C4×C127—22no—
155C22×C4×C127—21no—
156C22×C4×C127—22no—
157C22×C4×C127—21no—
158C22×C4×C127—21no—
159C22×C4×C127C2×D412yes—
160C22×C4×C127C2×D411yescharacteristic
161C22×C4×C127C2×D412yes—
162C22×C4×C127C2×D411yescharacteristic
163C22×C4×C127C2411yescharacteristic
164C2×C8×C127—21no—
165C2×C8×C127—21no—
166C2×C8×C127—22no—
167C4⋊C4×C127—22no—
168C4⋊C4×C127—22no—
169C4⋊C4×C127—22no—
170C4⋊C4×C127—22no—
171C4⋊C4×C127—22no—
172C4⋊C4×C127—22no—
173C4⋊C4×C127—22no—
174C4⋊C4×C127—22no—
175C4⋊C4×C127—21no—
176C4⋊C4×C127—21no—
177C4⋊C4×C127—21no—
178C4⋊C4×C127—22no—
179C4⋊C4×C127—21no—
180C4⋊C4×C127—22no—
181C4⋊C4×C127—21no—
182C4⋊C4×C127—21no—
183C4⋊C4×C127—21no—
184C4⋊C4×C127—21no—
185C4⋊C4×C127—21no—
186C4⋊C4×C127—21no—
187C4⋊C4×C127—22no—
188C4⋊C4×C127—22no—
189C4⋊C4×C127—22no—
190C4⋊C4×C127—21no—
191C4⋊C4×C127—22no—
192C4⋊C4×C127—21no—
193C4⋊C4×C127—21no—
194C4⋊C4×C127—21no—
195C4⋊C4×C127—21no—
196C4⋊C4×C127—41no—
197C4⋊C4×C127—41no—
198C4⋊C4×C127—41no—
199C4⋊C4×C127—41no—
200C4⋊C4×C127—81no—
201C4⋊C4×C127—82no—
202C4⋊C4×C127C2×D411yescharacteristic
203C4⋊C4×C127C2×D411yescharacteristic
204C4⋊C4×C127C2×D412yes—
205C22⋊C4×C127—22no—
206C22⋊C4×C127—22no—
207C22⋊C4×C127—22no—
208C22⋊C4×C127—22no—
209C22⋊C4×C127—21no—
210C22⋊C4×C127—21no—
211C22⋊C4×C127—21no—
212C22⋊C4×C127—21no—
213C22⋊C4×C127—21no—
214C42×C127—22no—
215C42×C127—22no—
216C42×C127—22no—
217C42×C127—22no—
218C42×C127—21no—
219C42×C127C2×D411yescharacteristic
220C42×C127C2×D411yescharacteristic
221C42×C127C2×D411yescharacteristic
222C42×C127C2×D411yescharacteristic
223C23×C127—21no—
224C23×C127C22×D411yescharacteristic
225Q8×C127—41no—
226Q8×C127—42no—
227Q8×C127—42no—
228Q8×C127—82no—
229D4×C127—41no—
230D4×C127—41no—
231D4×C127—41no—
232C2×C4×C127—21no—
233C2×C4×C127—21no—
234C2×C4×C127—22no—
235C2×C4×C127—21no—
236C2×C4×C127—21no—
237C2×C4×C127—22no—
238C2×C4×C127—21no—
239C2×C4×C127—22no—
240C2×C4×C127—22no—
241C2×C4×C127—22no—
242C2×C4×C127—22no—
243C2×C4×C127—41no—
244C2×C4×C127—41no—
245C2×C4×C127—42no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—42no—
249C2×C4×C127—42no—
250C2×C4×C127—41no—
251C2×C4×C127—41no—
252C2×C4×C127—42no—
253C2×C4×C127—41no—
254C2×C4×C127—41no—
255C2×C4×C127—82no—
256C2×C4×C127#12011yescharacteristic
257C2×C4×C127#12011yescharacteristic
258C2×C4×C127#12011yescharacteristic
259C2×C4×C127#12011yescharacteristic
260C2×C4×C127C22×D411yescharacteristic
261C2×C4×C127C22×D411yescharacteristic
262C8×C127—41no—
263C8×C127—41no—
264C8×C127—42no—
265C22×C127—21no—
266C22×C127—21no—
267C22×C127—41no—
268C22×C127—41no—
269C22×C127—41no—
270C22×C127C2×#12011yescharacteristic
271C22×C127#57711yescharacteristic
272C22×C127#57811yescharacteristic
273C4×C127—21no—
274C4×C127—21no—
275C4×C127—21no—
276C4×C127—21no—
277C4×C127—21no—
278C4×C127—21no—
279C4×C127—42no—
280C4×C127—41no—
281C4×C127—41no—
282C4×C127—42no—
283C4×C127—42no—
284C4×C127—42no—
285C4×C127—41no—
286C4×C127—82no—
287C4×C127—81no—
288C4×C127—81no—
289#18469C12711yesmaximal, characteristic, direct product factor, Sylow
290C2×C127—21no—
291C2×C127—21no—
292C2×C127—81no—
293C2×C127#301311yescharacteristic
294C2×C127#302011yescharacteristic
295C2×C127#302811yescharacteristic
296#3563C2×C12711yescharacteristic
297#3236C2×C12711yescharacteristic, complemented
298#3232C2×C12711yescharacteristic
299#2822C2×C12711yescharacteristic, complemented
300#1617C2×C12712yescomplemented
301#1615C2×C12712yes—
302#1611C2×C12712yescomplemented
303#1609C2×C12712yes—
304#1518C2×C12711yescharacteristic, complemented
305#1512C2×C12711yescharacteristic
306#1470C2×C12711yescharacteristic, complemented
307C127#1846911yescharacteristic, direct product factor, Sylow
308#564—21no—
309#563—21no—
310#557—21no—
311#554—21no—
312#552—21no—
313#552—22no—
314#552—21no—
315#544—21no—
316#544C22×C12712yes—
317#541—21no—
318#541—22no—
319#539C22×C12712yes—
320#533C22×C12711yescharacteristic
321#529C22×C12711yescharacteristic
322C2×#125C22×C12711yescharacteristic
323C2×#125C22×C12711yescharacteristic
324#501—21no—
325#499—21no—
326#491—21no—
327#488—21no—
328#484C22×C12712yes—
329#484C22×C12711yescharacteristic
330#483C22×C12712yes—
331#483C22×C12711yescharacteristic
332#481C22×C12711yescharacteristic
333#480C22×C12711yescharacteristic
334#479—21no—
335#479—22no—
336#478—21no—
337#432C22×C12711yescharacteristic
338#431C22×C12711yescharacteristic
339#400—21no—
340#400C22×C12712yes—
341#398—21no—
342#398—22no—
343#398C22×C12712yes—
344#382C22×C12711yescharacteristic
345#328C22×C12712yes—
346#328C22×C12712yes—
347#328C22×C12712yes—
348#327C22×C12712yes—
349#324C22×C12712yes—
350#324C22×C12712yes—
351C4⋊Q8—21no—
352C4⋊Q8—22no—
353C4⋊Q8—22no—
354C4⋊Q8D4×C12711yescharacteristic
355C4⋊Q8D4×C12711yescharacteristic
356#126—22no—
357#126—22no—
358#126—22no—
359#126—22no—
360#126—21no—
361#125—22no—
362#125—22no—
363#125—22no—
364#125—22no—
365#125—21no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—21no—
371#125—21no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125—21no—
376#125—21no—
377#125—21no—
378#125—21no—
379#125D4×C12711yescharacteristic
380#124D4×C12711yescharacteristic
381#123—22no—
382#123—21no—
383#123—22no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—21no—
390C22⋊Q8—22no—
391C22⋊Q8C23×C12711yescharacteristic
392C22⋊Q8C23×C12712yes—
393#121C23×C12711yescharacteristic
394C4×Q8—21no—
395C4×Q8—22no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8D4×C12711yescharacteristic
399C4×Q8D4×C12712yes—
400C4×D4—21no—
401C4×D4—22no—
402C4×D4D4×C12711yescharacteristic
403#117—22no—
404#117—21no—
405#117—21no—
406#117—22no—
407#117—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4D4×C12711yescharacteristic
414C2×C4⋊C4D4×C12712yes—
415C2×C4⋊C4D4×C12711yescharacteristic
416C2×C4⋊C4C23×C12711yescharacteristic
417C2×C4⋊C4C23×C12711yescharacteristic
418C2×C4⋊C4C23×C12712yes—
419C2×C42C23×C12711yescharacteristic
420#107—21no—
421#107—21no—
422#107—22no—
423#106—21no—
424#106—21no—
425#106—22no—
426C4⋊C8—21no—
427C4⋊C8—21no—
428#103—21no—
429#103—22no—
430#103—21no—
431#103—22no—
432#102—21no—
433#102—21no—
434C22⋊C8C23×C12712yes—
435C22⋊C8C23×C12711yescharacteristic
436C22⋊C8C23×C12711yescharacteristic
437#97—21no—
438C4×C8—21no—
439#95—21no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95C23×C12712yes—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8C2×D4×C12712yes—
447C2×Q8C2×D4×C12711yescharacteristic
448C2×D4—21no—
449C2×D4C2×D4×C12711yescharacteristic
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4C2×D4×C12712yes—
456C22×C4C2×D4×C12711yescharacteristic
457C22×C4C2×D4×C12712yes—
458C22×C4C2×D4×C12711yescharacteristic
459C22×C4C24×C12711yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4C2×D4×C12711yescharacteristic
499C4⋊C4C2×D4×C12711yescharacteristic
500C4⋊C4C2×D4×C12712yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42C2×D4×C12711yescharacteristic
516C42C2×D4×C12711yescharacteristic
517C42C2×D4×C12711yescharacteristic
518C42C2×D4×C12711yescharacteristic
519C23—21no—
520C23C22×D4×C12711yescharacteristic
521Q8—41no—
522Q8—42no—
523Q8—42no—
524Q8—82no—
525D4—41no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4#120×C12711yescharacteristic
553C2×C4#120×C12711yescharacteristic
554C2×C4#120×C12711yescharacteristic
555C2×C4#120×C12711yescharacteristic
556C2×C4C22×D4×C12711yescharacteristic
557C2×C4C22×D4×C12711yescharacteristic
558C8—41no—
559C8—41no—
560C8—42no—
561C22—21no—
562C22—21no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×#120×C12711yescharacteristic
567C22#577×C12711yescharacteristic
568C22#578×C12711yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—82no—
583C4—81no—
584C4—81no—
585C2—21no—
586C2—21no—
587C2—81no—
588C2C127×#301311yescharacteristic
589C2C127×#302011yescharacteristic
590C2C127×#302811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radical#121×C127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3236C2semidirect
4. C127×#2822C2semidirect
5. C127×#1617C2semidirect
7. C127×#1611C2semidirect
9. C127×#1518C2semidirect
11. C127×#1470C2semidirect
289. #18469C127direct (N × H)
297. #3236C2×C127semidirect
299. #2822C2×C127semidirect
300. #1617C2×C127semidirect
302. #1611C2×C127semidirect
304. #1518C2×C127semidirect
306. #1470C2×C127semidirect
307. C127#18469direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35632
C127×#32362
C127×#32322
C127×#28222
C127×#16172
C127×#16152
C127×#16112
C127×#16092
C127×#15182
C127×#15122
C127×#14702
#18469127