Finite Group Databasebeta

← C127×#17526 #44581199

Subgroups — C127×#17526 #44581199

Order 32512.

990 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×#3483C211yesmaximal, characteristic
2C127×#2990C211yesmaximal, characteristic, complemented
3C127×#2922C211yesmaximal, characteristic, complemented
4C127×#1496C212yesmaximal, complemented
5C127×#1492C211yesmaximal, characteristic, complemented
6C127×#1489C211yesmaximal, characteristic, complemented
7C127×#1488C212yesmaximal, complemented
8C127×#1484C211yesmaximal, characteristic, complemented
9C127×#1480C212yesmaximal, complemented
10C127×#1477C212yesmaximal, complemented
11C127×#1476C211yesmaximal, characteristic, complemented
12Q82×C127C411yescharacteristic
13#555×C127—21no—
14#555×C127—22no—
15#555×C127C412yes—
16#555×C127C411yescharacteristic
17#550×C127—21no—
18#550×C127C411yescharacteristic
19#550×C127C412yes—
20#550×C127C411yescharacteristic
21#532×C127—21no—
22#532×C127—22no—
23#532×C127C2212yes—
24#532×C127C2211yescharacteristic
25C4×#41×C127—21no—
26C4×#41×C127C2211yescharacteristic
27C4×#41×C127C2212yes—
28C4×#41×C127C2211yescharacteristic
29#446×C127—21no—
30#435×C127—22no—
31C8×D4×C127—21no—
32#433×C127—22no—
33#432×C127C2212yes—
34#431×C127C2211yescharacteristic
35#431×C127C2212yes—
36#431×C127C2212yes—
37#424×C127—21no—
38#423×C127C2211yescharacteristic
39C2×C4⋊C8×C127C2212yes—
40#408×C127—21no—
41C4×M16×C127C2211yescharacteristic
42C2×C4×C8×C127C2211yescharacteristic
43Q8⋊C8×C127C2211yescharacteristic
44Q8⋊C8×C127C2211yescharacteristic
45Q8⋊C8×C127C2212yes—
46D4⋊C8×C127C2211yescharacteristic
47D4⋊C8×C127C2211yescharacteristic
48D4⋊C8×C127C2212yes—
49D4⋊C8×C127C2212yes—
50D4⋊C8×C127C2212yes—
51D4⋊C8×C127C2212yes—
52D4⋊C8×C127C2212yes—
53C2×#41×C127—21no—
54C2×#41×C127—22no—
55C2×#41×C127—21no—
56C2×#41×C127C2×C411yescharacteristic
57C2×#41×C127C2×C412yes—
58C2×#41×C127D411yescharacteristic
59C2×M16×C127—21no—
60C2×M16×C127—21no—
61C22×C8×C127—21no—
62C22×C8×C127—22no—
63C4⋊Q8×C127—22no—
64C4⋊Q8×C127—21no—
65C4⋊Q8×C127C2×C412yes—
66C4⋊Q8×C127C2×C411yescharacteristic
67#127×C127—21no—
68#127×C127—22no—
69#127×C127C2×C412yes—
70#127×C127C2×C411yescharacteristic
71#124×C127—22no—
72#124×C127—22no—
73#124×C127—22no—
74#124×C127—21no—
75#124×C127—22no—
76#121×C127—21no—
77#121×C127—22no—
78#121×C127—21no—
79#121×C127—22no—
80#121×C127—22no—
81#121×C127—21no—
82#121×C127—22no—
83#121×C127—22no—
84#121×C127—22no—
85#121×C127C2×C412yes—
86#121×C127C2×C412yes—
87#121×C127C2×C412yes—
88C4×Q8×C127—21no—
89C4×Q8×C127C2×C411yescharacteristic
90C4×Q8×C127C2×C412yes—
91C4×Q8×C127C2×C411yescharacteristic
92C4×Q8×C127D411yescharacteristic
93C4×Q8×C127C2311yescharacteristic
94C4×D4×C127—22no—
95C4×D4×C127—21no—
96C4×D4×C127—21no—
97C4×D4×C127—22no—
98C4×D4×C127—22no—
99C4×D4×C127—21no—
100C4×D4×C127—21no—
101C4×D4×C127—22no—
102C4×D4×C127C2×C411yescharacteristic
103C4×D4×C127C2×C412yes—
104C4×D4×C127C2×C411yescharacteristic
105C4×D4×C127C2×C412yes—
106C4×D4×C127D411yescharacteristic
107C4×D4×C127D412yes—
108C4×D4×C127C2311yescharacteristic
109C4×D4×C127C2312yes—
110#117×C127—22no—
111#117×C127—21no—
112#117×C127—21no—
113#117×C127—22no—
114#117×C127C2×C411yescharacteristic
115#117×C127C2×C412yes—
116#117×C127D411yescharacteristic
117#117×C127D412yes—
118C2×C42×C127—21no—
119C2×C42×C127—22no—
120C2×C42×C127C2311yescharacteristic
121C2×C42×C127C2312yes—
122C4⋊C8×C127—22no—
123C4⋊C8×C127—21no—
124C4⋊C8×C127—22no—
125C4⋊C8×C127—22no—
126C4⋊C8×C127C2311yescharacteristic
127C4⋊C8×C127C2312yes—
128C4⋊C8×C127C2311yescharacteristic
129C22⋊C8×C127—22no—
130C22⋊C8×C127—22no—
131C22⋊C8×C127—21no—
132C22⋊C8×C127—22no—
133#97×C127—21no—
134#97×C127—22no—
135C4×C8×C127—21no—
136C4×C8×C127—21no—
137C4×C8×C127C2311yescharacteristic
138C4×C8×C127C2311yescharacteristic
139C4×C8×C127C2312yes—
140#41×C127—22no—
141#41×C127—24no—
142#41×C127—41no—
143#41×C127—42no—
144#41×C127—42no—
145#41×C127—44no—
146#41×C127—82no—
147#41×C127—81no—
148#41×C127—82no—
149#41×C127—81no—
150C2×Q8×C127—21no—
151C2×Q8×C127—22no—
152C2×Q8×C127C22⋊C411yescharacteristic
153C2×Q8×C127C22×C411yescharacteristic
154C2×D4×C127—21no—
155C2×D4×C127—22no—
156C2×D4×C127—22no—
157C2×D4×C127—22no—
158C2×D4×C127—22no—
159C2×D4×C127—22no—
160C2×D4×C127—21no—
161C2×D4×C127—21no—
162C2×D4×C127—22no—
163C2×D4×C127C22⋊C411yescharacteristic
164C2×D4×C127C22⋊C412yes—
165C2×D4×C127C22×C411yescharacteristic
166C2×D4×C127C22×C412yes—
167C22×C4×C127—21no—
168C22×C4×C127—22no—
169C22×C4×C127—22no—
170C22×C4×C127—21no—
171C22×C4×C127—21no—
172C22×C4×C127—22no—
173C22×C4×C127C22×C411yescharacteristic
174C22×C4×C127C22×C412yes—
175C22×C4×C127C2×D411yescharacteristic
176C22×C4×C127C2×D412yes—
177C22×C4×C127C2×D411yescharacteristic
178C22×C4×C127C2×D412yes—
179M16×C127—42no—
180M16×C127—81no—
181C2×C8×C127—21no—
182C2×C8×C127—21no—
183C2×C8×C127—22no—
184C2×C8×C127—21no—
185C2×C8×C127—21no—
186C2×C8×C127—22no—
187C2×C8×C127—42no—
188C2×C8×C127—82no—
189C4⋊C4×C127—22no—
190C4⋊C4×C127—21no—
191C4⋊C4×C127—21no—
192C4⋊C4×C127—22no—
193C4⋊C4×C127—21no—
194C4⋊C4×C127C22⋊C412yes—
195C4⋊C4×C127C22⋊C411yescharacteristic
196C4⋊C4×C127C22⋊C412yes—
197C4⋊C4×C127C22⋊C411yescharacteristic
198C4⋊C4×C127C22×C411yescharacteristic
199C4⋊C4×C127C22×C411yescharacteristic
200C4⋊C4×C127C22×C412yes—
201C22⋊C4×C127—21no—
202C22⋊C4×C127—22no—
203C22⋊C4×C127—22no—
204C22⋊C4×C127—22no—
205C22⋊C4×C127—22no—
206C22⋊C4×C127—21no—
207C22⋊C4×C127—22no—
208C22⋊C4×C127—21no—
209C22⋊C4×C127—22no—
210C22⋊C4×C127C22⋊C412yes—
211C22⋊C4×C127C22⋊C412yes—
212C22⋊C4×C127C22⋊C412yes—
213C42×C127—21no—
214C42×C127—22no—
215C42×C127—22no—
216C42×C127—21no—
217C42×C127—21no—
218C42×C127C22×C412yes—
219C42×C127C22×C411yescharacteristic
220C42×C127C2×D411yescharacteristic
221C42×C127C2×D412yes—
222C42×C127C2×D411yescharacteristic
223C42×C127C2×D412yes—
224C42×C127C2411yescharacteristic
225C23×C127—22no—
226C23×C127—21no—
227C23×C127C2×C22⋊C411yescharacteristic
228C23×C127C2×C22⋊C412yes—
229Q8×C127—21no—
230Q8×C127—22no—
231Q8×C127—82no—
232Q8×C127—81no—
233Q8×C127#13112yes—
234D4×C127—21no—
235D4×C127—22no—
236D4×C127—21no—
237D4×C127—22no—
238D4×C127—22no—
239D4×C127—42no—
240D4×C127—42no—
241D4×C127—44no—
242D4×C127—44no—
243D4×C127—42no—
244D4×C127—82no—
245D4×C127—81no—
246D4×C127—82no—
247D4×C127—82no—
248D4×C127—82no—
249C2×C4×C127—21no—
250C2×C4×C127—21no—
251C2×C4×C127—22no—
252C2×C4×C127—21no—
253C2×C4×C127—22no—
254C2×C4×C127—22no—
255C2×C4×C127—21no—
256C2×C4×C127—22no—
257C2×C4×C127—21no—
258C2×C4×C127—22no—
259C2×C4×C127—21no—
260C2×C4×C127—22no—
261C2×C4×C127—21no—
262C2×C4×C127—42no—
263C2×C4×C127—42no—
264C2×C4×C127—44no—
265C2×C4×C127—44no—
266C2×C4×C127—41no—
267C2×C4×C127—42no—
268C2×C4×C127—41no—
269C2×C4×C127—42no—
270C2×C4×C127—42no—
271C2×C4×C127C2×C22⋊C412yes—
272C2×C4×C127C2×C22⋊C411yescharacteristic
273C2×C4×C127C2×C22⋊C412yes—
274C2×C4×C127C2×C22⋊C411yescharacteristic
275C2×C4×C127C2×C22⋊C412yes—
276C2×C4×C127C2×C22⋊C411yescharacteristic
277C2×C4×C127C23×C411yescharacteristic
278C2×C4×C127C22×D411yescharacteristic
279C2×C4×C127C22×D411yescharacteristic
280C8×C127—41no—
281C8×C127—41no—
282C8×C127—42no—
283C8×C127—41no—
284C8×C127—42no—
285C8×C127—41no—
286C22×C127—21no—
287C22×C127—22no—
288C22×C127—22no—
289C22×C127—21no—
290C22×C127—21no—
291C22×C127—22no—
292C22×C127—42no—
293C22×C127—41no—
294C22×C127—42no—
295C22×C127—41no—
296C22×C127—42no—
297C22×C127—41no—
298C22×C127C22×C22⋊C411yescharacteristic
299C4×C127—21no—
300C4×C127—22no—
301C4×C127—22no—
302C4×C127—22no—
303C4×C127—21no—
304C4×C127—21no—
305C4×C127—22no—
306C4×C127—42no—
307C4×C127—41no—
308C4×C127—41no—
309C4×C127#42112yes—
310C4×C127C2×#13111yescharacteristic
311C4×C127#56811yescharacteristic
312C4×C127#57611yescharacteristic
313C4×C127#57711yescharacteristic
314#17526C12711yesmaximal, characteristic, direct product factor, Sylow
315C2×C127—41no—
316C2×C127—42no—
317C2×C127—82no—
318C2×C127—81no—
319C2×C127#288011yescharacteristic
320C2×C127#289711yescharacteristic
321C2×C127C2×#42111yescharacteristic
322#3483C2×C12711yescharacteristic
323#2990C2×C12711yescharacteristic, complemented
324#2922C2×C12711yescharacteristic, complemented
325#1496C2×C12712yescomplemented
326#1492C2×C12711yescharacteristic, complemented
327#1489C2×C12711yescharacteristic, complemented
328#1488C2×C12712yescomplemented
329#1484C2×C12711yescharacteristic, complemented
330#1480C2×C12712yescomplemented
331#1477C2×C12712yescomplemented
332#1476C2×C12711yescharacteristic, complemented
333C127#1752611yescharacteristic, direct product factor, Sylow
334Q82C4×C12711yescharacteristic
335#555—21no—
336#555—22no—
337#555C4×C12712yes—
338#555C4×C12711yescharacteristic
339#550—21no—
340#550C4×C12711yescharacteristic
341#550C4×C12712yes—
342#550C4×C12711yescharacteristic
343#532—21no—
344#532—22no—
345#532C22×C12712yes—
346#532C22×C12711yescharacteristic
347C4×#41—21no—
348C4×#41C22×C12711yescharacteristic
349C4×#41C22×C12712yes—
350C4×#41C22×C12711yescharacteristic
351#446—21no—
352#435—22no—
353C8×D4—21no—
354#433—22no—
355#432C22×C12712yes—
356#431C22×C12711yescharacteristic
357#431C22×C12712yes—
358#431C22×C12712yes—
359#424—21no—
360#423C22×C12711yescharacteristic
361C2×C4⋊C8C22×C12712yes—
362#408—21no—
363C4×M16C22×C12711yescharacteristic
364C2×C4×C8C22×C12711yescharacteristic
365Q8⋊C8C22×C12711yescharacteristic
366Q8⋊C8C22×C12711yescharacteristic
367Q8⋊C8C22×C12712yes—
368D4⋊C8C22×C12711yescharacteristic
369D4⋊C8C22×C12711yescharacteristic
370D4⋊C8C22×C12712yes—
371D4⋊C8C22×C12712yes—
372D4⋊C8C22×C12712yes—
373D4⋊C8C22×C12712yes—
374D4⋊C8C22×C12712yes—
375C2×#41—21no—
376C2×#41—22no—
377C2×#41—21no—
378C2×#41C2×C4×C12711yescharacteristic
379C2×#41C2×C4×C12712yes—
380C2×#41D4×C12711yescharacteristic
381C2×M16—21no—
382C2×M16—21no—
383C22×C8—21no—
384C22×C8—22no—
385C4⋊Q8—22no—
386C4⋊Q8—21no—
387C4⋊Q8C2×C4×C12712yes—
388C4⋊Q8C2×C4×C12711yescharacteristic
389#127—21no—
390#127—22no—
391#127C2×C4×C12712yes—
392#127C2×C4×C12711yescharacteristic
393#124—22no—
394#124—22no—
395#124—22no—
396#124—21no—
397#124—22no—
398#121—21no—
399#121—22no—
400#121—21no—
401#121—22no—
402#121—22no—
403#121—21no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121C2×C4×C12712yes—
408#121C2×C4×C12712yes—
409#121C2×C4×C12712yes—
410C4×Q8—21no—
411C4×Q8C2×C4×C12711yescharacteristic
412C4×Q8C2×C4×C12712yes—
413C4×Q8C2×C4×C12711yescharacteristic
414C4×Q8D4×C12711yescharacteristic
415C4×Q8C23×C12711yescharacteristic
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—22no—
424C4×D4C2×C4×C12711yescharacteristic
425C4×D4C2×C4×C12712yes—
426C4×D4C2×C4×C12711yescharacteristic
427C4×D4C2×C4×C12712yes—
428C4×D4D4×C12711yescharacteristic
429C4×D4D4×C12712yes—
430C4×D4C23×C12711yescharacteristic
431C4×D4C23×C12712yes—
432#117—22no—
433#117—21no—
434#117—21no—
435#117—22no—
436#117C2×C4×C12711yescharacteristic
437#117C2×C4×C12712yes—
438#117D4×C12711yescharacteristic
439#117D4×C12712yes—
440C2×C42—21no—
441C2×C42—22no—
442C2×C42C23×C12711yescharacteristic
443C2×C42C23×C12712yes—
444C4⋊C8—22no—
445C4⋊C8—21no—
446C4⋊C8—22no—
447C4⋊C8—22no—
448C4⋊C8C23×C12711yescharacteristic
449C4⋊C8C23×C12712yes—
450C4⋊C8C23×C12711yescharacteristic
451C22⋊C8—22no—
452C22⋊C8—22no—
453C22⋊C8—21no—
454C22⋊C8—22no—
455#97—21no—
456#97—22no—
457C4×C8—21no—
458C4×C8—21no—
459C4×C8C23×C12711yescharacteristic
460C4×C8C23×C12711yescharacteristic
461C4×C8C23×C12712yes—
462#41—22no—
463#41—24no—
464#41—41no—
465#41—42no—
466#41—42no—
467#41—44no—
468#41—82no—
469#41—81no—
470#41—82no—
471#41—81no—
472C2×Q8—21no—
473C2×Q8—22no—
474C2×Q8C22⋊C4×C12711yescharacteristic
475C2×Q8C22×C4×C12711yescharacteristic
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4C22⋊C4×C12711yescharacteristic
486C2×D4C22⋊C4×C12712yes—
487C2×D4C22×C4×C12711yescharacteristic
488C2×D4C22×C4×C12712yes—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4C22×C4×C12711yescharacteristic
496C22×C4C22×C4×C12712yes—
497C22×C4C2×D4×C12712yes—
498C22×C4C2×D4×C12711yescharacteristic
499C22×C4C2×D4×C12711yescharacteristic
500C22×C4C2×D4×C12712yes—
501M16—42no—
502M16—81no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—42no—
510C2×C8—82no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4C22⋊C4×C12712yes—
517C4⋊C4C22⋊C4×C12711yescharacteristic
518C4⋊C4C22⋊C4×C12712yes—
519C4⋊C4C22⋊C4×C12711yescharacteristic
520C4⋊C4C22×C4×C12711yescharacteristic
521C4⋊C4C22×C4×C12711yescharacteristic
522C4⋊C4C22×C4×C12712yes—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4C22⋊C4×C12712yes—
533C22⋊C4C22⋊C4×C12712yes—
534C22⋊C4C22⋊C4×C12712yes—
535C42—21no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42C22×C4×C12712yes—
541C42C22×C4×C12711yescharacteristic
542C42C2×D4×C12711yescharacteristic
543C42C2×D4×C12711yescharacteristic
544C42C2×D4×C12712yes—
545C42C2×D4×C12712yes—
546C42C24×C12711yescharacteristic
547C23—22no—
548C23—21no—
549C23C2×C22⋊C4×C12711yescharacteristic
550C23C2×C22⋊C4×C12712yes—
551Q8—21no—
552Q8—22no—
553Q8—82no—
554Q8—81no—
555Q8#131×C12712yes—
556D4—21no—
557D4—22no—
558D4—21no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—42no—
563D4—44no—
564D4—44no—
565D4—42no—
566D4—82no—
567D4—81no—
568D4—82no—
569D4—82no—
570D4—82no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4C2×C22⋊C4×C12712yes—
594C2×C4C2×C22⋊C4×C12711yescharacteristic
595C2×C4C2×C22⋊C4×C12712yes—
596C2×C4C2×C22⋊C4×C12711yescharacteristic
597C2×C4C2×C22⋊C4×C12712yes—
598C2×C4C2×C22⋊C4×C12711yescharacteristic
599C2×C4C23×C4×C12711yescharacteristic
600C2×C4C22×D4×C12711yescharacteristic
601C2×C4C22×D4×C12711yescharacteristic
602C8—41no—
603C8—41no—
604C8—42no—
605C8—41no—
606C8—42no—
607C8—41no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22C22×C22⋊C4×C12711yescharacteristic
621C4—21no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4#421×C12712yes—
632C4C2×#131×C12711yescharacteristic
633C4#568×C12711yescharacteristic
634C4#576×C12711yescharacteristic
635C4#577×C12711yescharacteristic
636C2—41no—
637C2—42no—
638C2—82no—
639C2—81no—
640C2C127×#288011yescharacteristic
641C2C127×#289711yescharacteristic
642C2C2×#421×C12711yescharacteristic
subgroupG/H
centerC2×C4×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC42C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC127×#3483C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C4×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2990C2semidirect
3. C127×#2922C2semidirect
4. C127×#1496C2semidirect
5. C127×#1492C2semidirect
6. C127×#1489C2semidirect
7. C127×#1488C2semidirect
8. C127×#1484C2semidirect
9. C127×#1480C2semidirect
10. C127×#1477C2semidirect
11. C127×#1476C2semidirect
314. #17526C127direct (N × H)
323. #2990C2×C127semidirect
324. #2922C2×C127semidirect
325. #1496C2×C127semidirect
326. #1492C2×C127semidirect
327. #1489C2×C127semidirect
328. #1488C2×C127semidirect
329. #1484C2×C127semidirect
330. #1480C2×C127semidirect
331. #1477C2×C127semidirect
332. #1476C2×C127semidirect
333. C127#17526direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34832
C127×#29902
C127×#29222
C127×#14962
C127×#14922
C127×#14892
C127×#14882
C127×#14842
C127×#14802
C127×#14772
C127×#14762
#17526127