Finite Group Databasebeta

← C127×#18312 #44581985

Subgroups — C127×#18312 #44581985

Order 32512.

930 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×#3455C211yesmaximal, characteristic
2C127×#3229C211yesmaximal, characteristic, complemented
3C127×#3226C211yesmaximal, characteristic, complemented
4C127×#2621C211yesmaximal, characteristic, complemented
5C127×#1612C214yesmaximal, complemented
6C127×#1609C214yesmaximal, complemented
7C127×#1518C211yesmaximal, characteristic, complemented
8C127×#1517C211yesmaximal, characteristic, complemented
9C127×#1458C211yesmaximal, characteristic, complemented
10#542×C127—21no—
11#542×C127—21no—
12#541×C127—21no—
13#540×C127—21no—
14#539×C127—21no—
15#539×C127C2212yes—
16#538×C127—21no—
17#538×C127C2212yes—
18#533×C127C2211yescharacteristic
19#532×C127C2211yescharacteristic
20C2×C4⋊Q8×C127C2211yescharacteristic
21#529×C127—22no—
22#529×C127—22no—
23#525×C127—21no—
24C2×C22⋊Q8×C127—21no—
25C4×#41×C127—21no—
26C2×#117×C127C2211yescharacteristic
27#500×C127—21no—
28#498×C127—21no—
29#487×C127—21no—
30#486×C127—21no—
31#483×C127C2214yes—
32#480×C127C2214yes—
33#475×C127—22no—
34#474×C127—22no—
35#431×C127C2211yescharacteristic
36#431×C127C2211yescharacteristic
37#398×C127—21no—
38#398×C127C2212yes—
39#397×C127—21no—
40#397×C127C2212yes—
41#391×C127—21no—
42#382×C127C2211yescharacteristic
43#328×C127C2214yes—
44#327×C127C2214yes—
45#324×C127C2212yes—
46#323×C127C2212yes—
47C2×#41×C127—21no—
48C22×Q8×C127—21no—
49C23×C4×C127D411yescharacteristic
50C4⋊Q8×C127—21no—
51C4⋊Q8×C127—21no—
52C4⋊Q8×C127—21no—
53C4⋊Q8×C127—21no—
54C4⋊Q8×C127D411yescharacteristic
55C4⋊Q8×C127D411yescharacteristic
56C4⋊Q8×C127D411yescharacteristic
57#127×C127D411yescharacteristic
58#126×C127—22no—
59#126×C127—22no—
60#126×C127—22no—
61#126×C127—22no—
62#125×C127—22no—
63#125×C127—22no—
64#125×C127—21no—
65#124×C127—22no—
66#124×C127—22no—
67#124×C127—21no—
68#123×C127—22no—
69#123×C127—22no—
70#123×C127—22no—
71#123×C127—22no—
72#123×C127—22no—
73C22⋊Q8×C127—21no—
74C22⋊Q8×C127—21no—
75C22⋊Q8×C127—22no—
76C22⋊Q8×C127—22no—
77C22⋊Q8×C127—21no—
78C22⋊Q8×C127—44no—
79C22⋊Q8×C127C2312yes—
80#121×C127—21no—
81#121×C127—21no—
82#121×C127—22no—
83#121×C127—22no—
84#121×C127—21no—
85#121×C127C2312yes—
86#120×C127—22no—
87C4×Q8×C127—21no—
88C4×Q8×C127—21no—
89C4×Q8×C127—21no—
90C4×Q8×C127D412yes—
91C4×D4×C127—21no—
92C4×D4×C127—21no—
93C4×D4×C127—22no—
94C4×D4×C127—22no—
95C4×D4×C127—22no—
96C4×D4×C127—22no—
97C4×D4×C127—21no—
98C4×D4×C127D412yes—
99#117×C127—22no—
100#117×C127—21no—
101#117×C127—21no—
102#117×C127—21no—
103#117×C127—21no—
104#117×C127—21no—
105C2×C4⋊C4×C127—21no—
106C2×C4⋊C4×C127—22no—
107C2×C4⋊C4×C127—21no—
108C2×C4⋊C4×C127D411yescharacteristic
109C2×C4⋊C4×C127C2312yes—
110C2×C4⋊C4×C127C2311yescharacteristic
111C2×C22⋊C4×C127—22no—
112C2×C22⋊C4×C127D411yescharacteristic
113C2×C22⋊C4×C127C2311yescharacteristic
114C2×C42×C127—22no—
115C2×C42×C127D411yescharacteristic
116C2×C42×C127C2311yescharacteristic
117#107×C127—24no—
118#106×C127—24no—
119C4⋊C8×C127—21no—
120C4⋊C8×C127—21no—
121#103×C127—24no—
122#102×C127—24no—
123C22⋊C8×C127C2312yes—
124C22⋊C8×C127C2312yes—
125C4×C8×C127—21no—
126C4×C8×C127—21no—
127#95×C127—21no—
128#95×C127—22no—
129#95×C127—22no—
130#95×C127C2312yes—
131C24×C127C2×D411yescharacteristic
132#41×C127—42no—
133#41×C127—42no—
134C2×Q8×C127—21no—
135C2×Q8×C127—21no—
136C2×Q8×C127—41no—
137C2×Q8×C127—41no—
138C2×Q8×C127—42no—
139C2×Q8×C127—42no—
140C2×Q8×C127C2×D412yes—
141C2×D4×C127—22no—
142C2×D4×C127—22no—
143C2×D4×C127—22no—
144C2×D4×C127—21no—
145C2×D4×C127—21no—
146C2×D4×C127C2×D412yes—
147C22×C4×C127—22no—
148C22×C4×C127—22no—
149C22×C4×C127—22no—
150C22×C4×C127—21no—
151C22×C4×C127—21no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127C2×D411yescharacteristic
157C22×C4×C127C2×D411yescharacteristic
158C22×C4×C127C2×D411yescharacteristic
159C22×C4×C127C2×D411yescharacteristic
160C22×C4×C127C2×D411yescharacteristic
161C22×C4×C127C2411yescharacteristic
162C2×C8×C127—22no—
163C2×C8×C127—22no—
164C4⋊C4×C127—22no—
165C4⋊C4×C127—22no—
166C4⋊C4×C127—22no—
167C4⋊C4×C127—22no—
168C4⋊C4×C127—21no—
169C4⋊C4×C127—21no—
170C4⋊C4×C127—21no—
171C4⋊C4×C127—21no—
172C4⋊C4×C127—21no—
173C4⋊C4×C127—21no—
174C4⋊C4×C127—22no—
175C4⋊C4×C127—21no—
176C4⋊C4×C127—21no—
177C4⋊C4×C127—22no—
178C4⋊C4×C127—22no—
179C4⋊C4×C127—22no—
180C4⋊C4×C127—44no—
181C4⋊C4×C127—41no—
182C4⋊C4×C127—41no—
183C4⋊C4×C127—42no—
184C4⋊C4×C127C2×D412yes—
185C4⋊C4×C127C2×D412yes—
186C22⋊C4×C127—22no—
187C22⋊C4×C127—22no—
188C22⋊C4×C127—22no—
189C22⋊C4×C127—22no—
190C22⋊C4×C127—22no—
191C22⋊C4×C127—22no—
192C22⋊C4×C127—22no—
193C22⋊C4×C127—21no—
194C22⋊C4×C127—21no—
195C22⋊C4×C127—22no—
196C22⋊C4×C127—21no—
197C22⋊C4×C127—21no—
198C22⋊C4×C127—22no—
199C22⋊C4×C127—44no—
200C42×C127—22no—
201C42×C127—22no—
202C42×C127—21no—
203C42×C127—22no—
204C42×C127—21no—
205C42×C127C2×D411yescharacteristic
206C42×C127C2×D411yescharacteristic
207C42×C127C2×D411yescharacteristic
208C42×C127C2×D411yescharacteristic
209C23×C127—21no—
210C23×C127—22no—
211C23×C127—21no—
212C23×C127—21no—
213C23×C127—41no—
214C23×C127—41no—
215C23×C127C22×D411yescharacteristic
216Q8×C127—42no—
217Q8×C127—42no—
218Q8×C127—81no—
219Q8×C127—82no—
220Q8×C127—82no—
221D4×C127—42no—
222D4×C127—42no—
223D4×C127—42no—
224D4×C127—42no—
225D4×C127—81no—
226D4×C127—82no—
227D4×C127—82no—
228C2×C4×C127—21no—
229C2×C4×C127—21no—
230C2×C4×C127—22no—
231C2×C4×C127—22no—
232C2×C4×C127—22no—
233C2×C4×C127—21no—
234C2×C4×C127—21no—
235C2×C4×C127—21no—
236C2×C4×C127—41no—
237C2×C4×C127—41no—
238C2×C4×C127—42no—
239C2×C4×C127—42no—
240C2×C4×C127—44no—
241C2×C4×C127—41no—
242C2×C4×C127—41no—
243C2×C4×C127—42no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—42no—
249C2×C4×C127—82no—
250C2×C4×C127—82no—
251C2×C4×C127#12011yescharacteristic
252C2×C4×C127#12011yescharacteristic
253C2×C4×C127#12011yescharacteristic
254C2×C4×C127#12011yescharacteristic
255C2×C4×C127#13512yes—
256C2×C4×C127AGL(1,ℤ/8)12yes—
257C2×C4×C127C22×D411yescharacteristic
258C2×C4×C127C22×D411yescharacteristic
259C8×C127—42no—
260C8×C127—42no—
261C22×C127—21no—
262C22×C127—21no—
263C22×C127—42no—
264C22×C127—41no—
265C22×C127—42no—
266C22×C127—42no—
267C22×C127—41no—
268C22×C127—42no—
269C22×C127—41no—
270C22×C127—81no—
271C22×C127C2×#12011yescharacteristic
272C22×C127C2×#13511yescharacteristic
273C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
274C4×C127—21no—
275C4×C127—21no—
276C4×C127—21no—
277C4×C127—21no—
278C4×C127—41no—
279C4×C127—42no—
280C4×C127—41no—
281C4×C127—42no—
282C4×C127—41no—
283C4×C127—42no—
284C4×C127—82no—
285C4×C127—82no—
286C4×C127#44912yes—
287C4×C127#44912yes—
288#18312C12711yesmaximal, characteristic, direct product factor, Sylow
289C2×C127—21no—
290C2×C127—21no—
291C2×C127—82no—
292C2×C127—81no—
293C2×C127C2×#44911yescharacteristic
294C2×C127#302211yescharacteristic
295C2×C127#302811yescharacteristic
296#3455C2×C12711yescharacteristic
297#3229C2×C12711yescharacteristic, complemented
298#3226C2×C12711yescharacteristic, complemented
299#2621C2×C12711yescharacteristic, complemented
300#1612C2×C12714yescomplemented
301#1609C2×C12714yescomplemented
302#1518C2×C12711yescharacteristic, complemented
303#1517C2×C12711yescharacteristic, complemented
304#1458C2×C12711yescharacteristic, complemented
305C127#1831211yescharacteristic, direct product factor, Sylow
306#542—21no—
307#542—21no—
308#541—21no—
309#540—21no—
310#539—21no—
311#539C22×C12712yes—
312#538—21no—
313#538C22×C12712yes—
314#533C22×C12711yescharacteristic
315#532C22×C12711yescharacteristic
316C2×C4⋊Q8C22×C12711yescharacteristic
317#529—22no—
318#529—22no—
319#525—21no—
320C2×C22⋊Q8—21no—
321C4×#41—21no—
322C2×#117C22×C12711yescharacteristic
323#500—21no—
324#498—21no—
325#487—21no—
326#486—21no—
327#483C22×C12714yes—
328#480C22×C12714yes—
329#475—22no—
330#474—22no—
331#431C22×C12711yescharacteristic
332#431C22×C12711yescharacteristic
333#398—21no—
334#398C22×C12712yes—
335#397—21no—
336#397C22×C12712yes—
337#391—21no—
338#382C22×C12711yescharacteristic
339#328C22×C12714yes—
340#327C22×C12714yes—
341#324C22×C12712yes—
342#323C22×C12712yes—
343C2×#41—21no—
344C22×Q8—21no—
345C23×C4D4×C12711yescharacteristic
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8D4×C12711yescharacteristic
351C4⋊Q8D4×C12711yescharacteristic
352C4⋊Q8D4×C12711yescharacteristic
353#127D4×C12711yescharacteristic
354#126—22no—
355#126—22no—
356#126—22no—
357#126—22no—
358#125—22no—
359#125—22no—
360#125—21no—
361#124—22no—
362#124—22no—
363#124—21no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—21no—
374C22⋊Q8—44no—
375C22⋊Q8C23×C12712yes—
376#121—21no—
377#121—21no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121C23×C12712yes—
382#120—22no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8D4×C12712yes—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4D4×C12712yes—
395#117—22no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4D4×C12711yescharacteristic
405C2×C4⋊C4C23×C12712yes—
406C2×C4⋊C4C23×C12711yescharacteristic
407C2×C22⋊C4—22no—
408C2×C22⋊C4D4×C12711yescharacteristic
409C2×C22⋊C4C23×C12711yescharacteristic
410C2×C42—22no—
411C2×C42D4×C12711yescharacteristic
412C2×C42C23×C12711yescharacteristic
413#107—24no—
414#106—24no—
415C4⋊C8—21no—
416C4⋊C8—21no—
417#103—24no—
418#102—24no—
419C22⋊C8C23×C12712yes—
420C22⋊C8C23×C12712yes—
421C4×C8—21no—
422C4×C8—21no—
423#95—21no—
424#95—22no—
425#95—22no—
426#95C23×C12712yes—
427C24C2×D4×C12711yescharacteristic
428#41—42no—
429#41—42no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×D4×C12712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×D4×C12712yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4C2×D4×C12711yescharacteristic
453C22×C4C2×D4×C12711yescharacteristic
454C22×C4C2×D4×C12711yescharacteristic
455C22×C4C2×D4×C12711yescharacteristic
456C22×C4C2×D4×C12711yescharacteristic
457C22×C4C24×C12711yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4C2×D4×C12712yes—
481C4⋊C4C2×D4×C12712yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—44no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—22no—
500C42—21no—
501C42C2×D4×C12711yescharacteristic
502C42C2×D4×C12711yescharacteristic
503C42C2×D4×C12711yescharacteristic
504C42C2×D4×C12711yescharacteristic
505C23—21no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—41no—
511C23C22×D4×C12711yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—81no—
515Q8—82no—
516Q8—82no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—81no—
522D4—82no—
523D4—82no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#120×C12711yescharacteristic
548C2×C4#120×C12711yescharacteristic
549C2×C4#120×C12711yescharacteristic
550C2×C4#120×C12711yescharacteristic
551C2×C4#135×C12712yes—
552C2×C4AGL(1,ℤ/8)×C12712yes—
553C2×C4C22×D4×C12711yescharacteristic
554C2×C4C22×D4×C12711yescharacteristic
555C8—42no—
556C8—42no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—81no—
567C22C2×#120×C12711yescharacteristic
568C22C2×#135×C12711yescharacteristic
569C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4#449×C12712yes—
583C4#449×C12712yes—
584C2—21no—
585C2—21no—
586C2—81no—
587C2—82no—
588C2C2×#449×C12711yescharacteristic
589C2C127×#302211yescharacteristic
590C2C127×#302811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3455C2
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×#3229C2semidirect
3. C127×#3226C2semidirect
4. C127×#2621C2semidirect
5. C127×#1612C2semidirect
6. C127×#1609C2semidirect
7. C127×#1518C2semidirect
8. C127×#1517C2semidirect
9. C127×#1458C2semidirect
288. #18312C127direct (N × H)
297. #3229C2×C127semidirect
298. #3226C2×C127semidirect
299. #2621C2×C127semidirect
300. #1612C2×C127semidirect
301. #1609C2×C127semidirect
302. #1518C2×C127semidirect
303. #1517C2×C127semidirect
304. #1458C2×C127semidirect
305. C127#18312direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34552
C127×#32292
C127×#32262
C127×#26212
C127×#16122
C127×#16092
C127×#15182
C127×#15172
C127×#14582
#18312127