Finite Group Databasebeta

← C127×#17523 #44581196

Subgroups — C127×#17523 #44581196

Order 32512.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3482C211yesmaximal, characteristic
2C127×#2990C211yesmaximal, characteristic, complemented
3C127×#2922C211yesmaximal, characteristic, complemented
4C127×#1497C212yesmaximal, complemented
5C127×#1493C211yesmaximal, characteristic, complemented
6C127×#1489C211yesmaximal, characteristic, complemented
7C127×#1488C212yesmaximal, complemented
8C127×#1485C211yesmaximal, characteristic, complemented
9C127×#1481C212yesmaximal, complemented
10C127×#1477C212yesmaximal, complemented
11C127×#1476C211yesmaximal, characteristic, complemented
12#557×C127C411yescharacteristic
13#556×C127C412yes—
14#556×C127C411yescharacteristic
15#552×C127—21no—
16#552×C127—22no—
17D4×Q8×C127C411yescharacteristic
18#548×C127—21no—
19#547×C127C411yescharacteristic
20#547×C127C412yes—
21#533×C127C2212yes—
22#533×C127C2211yescharacteristic
23#529×C127—21no—
24#529×C127—22no—
25C4×#41×C127C2211yescharacteristic
26C4×#41×C127C2212yes—
27C4×#41×C127C2211yescharacteristic
28C2×C4×Q8×C127—21no—
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—
46Q8⋊C8×C127C2212yes—
47Q8⋊C8×C127C2212yes—
48Q8⋊C8×C127C2212yes—
49Q8⋊C8×C127C2212yes—
50D4⋊C8×C127C2211yescharacteristic
51D4⋊C8×C127C2211yescharacteristic
52D4⋊C8×C127C2212yes—
53C2×#41×C127C2×C411yescharacteristic
54C2×#41×C127C2×C412yes—
55C2×#41×C127D411yescharacteristic
56C22×Q8×C127—21no—
57C2×M16×C127—21no—
58C2×M16×C127—21no—
59C22×C8×C127—21no—
60C22×C8×C127—22no—
61C4⋊Q8×C127C2×C412yes—
62C4⋊Q8×C127C2×C411yescharacteristic
63C4⋊Q8×C127C2×C412yes—
64C4⋊Q8×C127C2×C411yescharacteristic
65#126×C127—22no—
66#126×C127—22no—
67#126×C127—22no—
68#125×C127—22no—
69#125×C127—21no—
70#125×C127—21no—
71#125×C127—22no—
72#124×C127—21no—
73#124×C127—22no—
74#123×C127—22no—
75#123×C127—22no—
76#123×C127—22no—
77C22⋊Q8×C127—21no—
78C22⋊Q8×C127—22no—
79C22⋊Q8×C127—22no—
80C22⋊Q8×C127—21no—
81C22⋊Q8×C127C2×C412yes—
82C22⋊Q8×C127C2×C412yes—
83C22⋊Q8×C127C2×C412yes—
84#121×C127—21no—
85#121×C127—22no—
86C4×Q8×C127—21no—
87C4×Q8×C127—22no—
88C4×Q8×C127—21no—
89C4×Q8×C127—21no—
90C4×Q8×C127—22no—
91C4×Q8×C127C2×C411yescharacteristic
92C4×Q8×C127C2×C412yes—
93C4×Q8×C127C2×C411yescharacteristic
94C4×Q8×C127D411yescharacteristic
95C4×Q8×C127C2311yescharacteristic
96C4×Q8×C127C2312yes—
97C4×D4×C127—22no—
98C4×D4×C127—21no—
99C4×D4×C127—21no—
100C4×D4×C127—22no—
101C4×D4×C127C2×C411yescharacteristic
102C4×D4×C127C2×C412yes—
103C4×D4×C127C2×C411yescharacteristic
104C4×D4×C127C2×C412yes—
105C4×D4×C127D411yescharacteristic
106C4×D4×C127D412yes—
107C4×D4×C127C2311yescharacteristic
108#117×C127—22no—
109#117×C127—21no—
110#117×C127—21no—
111#117×C127—22no—
112#117×C127C2×C411yescharacteristic
113#117×C127C2×C412yes—
114#117×C127D411yescharacteristic
115#117×C127D412yes—
116C2×C4⋊C4×C127—22no—
117C2×C4⋊C4×C127—21no—
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—
145C2×Q8×C127—22no—
146C2×Q8×C127—21no—
147C2×Q8×C127—44no—
148C2×Q8×C127C22⋊C411yescharacteristic
149C2×Q8×C127C22×C411yescharacteristic
150C2×Q8×C127C22×C412yes—
151C2×D4×C127—21no—
152C2×D4×C127—22no—
153C2×D4×C127C22⋊C411yescharacteristic
154C2×D4×C127C22⋊C412yes—
155C2×D4×C127C22×C411yescharacteristic
156C22×C4×C127—21no—
157C22×C4×C127—22no—
158C22×C4×C127—22no—
159C22×C4×C127—21no—
160C22×C4×C127—21no—
161C22×C4×C127C22×C411yescharacteristic
162C22×C4×C127C22×C412yes—
163C22×C4×C127C2×D411yescharacteristic
164C22×C4×C127C2×D412yes—
165C22×C4×C127C2×D411yescharacteristic
166C22×C4×C127C2×D412yes—
167M16×C127—42no—
168M16×C127—81no—
169C2×C8×C127—21no—
170C2×C8×C127—21no—
171C2×C8×C127—22no—
172C2×C8×C127—21no—
173C2×C8×C127—21no—
174C2×C8×C127—22no—
175C2×C8×C127—42no—
176C2×C8×C127—82no—
177C4⋊C4×C127—22no—
178C4⋊C4×C127—21no—
179C4⋊C4×C127—22no—
180C4⋊C4×C127—22no—
181C4⋊C4×C127—22no—
182C4⋊C4×C127—21no—
183C4⋊C4×C127—22no—
184C4⋊C4×C127—21no—
185C4⋊C4×C127—22no—
186C4⋊C4×C127—21no—
187C4⋊C4×C127—22no—
188C4⋊C4×C127—21no—
189C4⋊C4×C127—22no—
190C4⋊C4×C127C22⋊C412yes—
191C4⋊C4×C127C22⋊C411yescharacteristic
192C4⋊C4×C127C22⋊C412yes—
193C4⋊C4×C127C22⋊C411yescharacteristic
194C4⋊C4×C127C22×C411yescharacteristic
195C4⋊C4×C127C22×C411yescharacteristic
196C4⋊C4×C127C22×C412yes—
197C22⋊C4×C127—22no—
198C22⋊C4×C127—22no—
199C22⋊C4×C127—22no—
200C22⋊C4×C127—21no—
201C22⋊C4×C127—22no—
202C22⋊C4×C127—21no—
203C22⋊C4×C127—22no—
204C22⋊C4×C127—21no—
205C22⋊C4×C127C22⋊C412yes—
206C22⋊C4×C127C22⋊C412yes—
207C22⋊C4×C127C22⋊C412yes—
208C42×C127—21no—
209C42×C127—22no—
210C42×C127—22no—
211C42×C127—21no—
212C42×C127—21no—
213C42×C127—22no—
214C42×C127C22×C412yes—
215C42×C127C22×C411yescharacteristic
216C42×C127C2×D411yescharacteristic
217C42×C127C2×D412yes—
218C42×C127C2×D411yescharacteristic
219C42×C127C2×D412yes—
220C42×C127C2411yescharacteristic
221C23×C127—21no—
222C23×C127C2×C22⋊C411yescharacteristic
223C23×C127C2×C22⋊C412yes—
224Q8×C127—22no—
225Q8×C127—21no—
226Q8×C127—22no—
227Q8×C127—21no—
228Q8×C127—22no—
229Q8×C127—44no—
230Q8×C127—42no—
231D4×C127—21no—
232D4×C127—22no—
233D4×C127—42no—
234D4×C127—42no—
235D4×C127—44no—
236D4×C127#13112yes—
237C2×C4×C127—21no—
238C2×C4×C127—22no—
239C2×C4×C127—21no—
240C2×C4×C127—22no—
241C2×C4×C127—22no—
242C2×C4×C127—21no—
243C2×C4×C127—22no—
244C2×C4×C127—21no—
245C2×C4×C127—22no—
246C2×C4×C127—21no—
247C2×C4×C127—22no—
248C2×C4×C127—21no—
249C2×C4×C127—22no—
250C2×C4×C127—21no—
251C2×C4×C127—42no—
252C2×C4×C127—42no—
253C2×C4×C127—44no—
254C2×C4×C127—44no—
255C2×C4×C127—41no—
256C2×C4×C127—41no—
257C2×C4×C127—42no—
258C2×C4×C127C2×C22⋊C412yes—
259C2×C4×C127C2×C22⋊C411yescharacteristic
260C2×C4×C127C2×C22⋊C412yes—
261C2×C4×C127C2×C22⋊C411yescharacteristic
262C2×C4×C127C2×C22⋊C412yes—
263C2×C4×C127C2×C22⋊C411yescharacteristic
264C2×C4×C127C23×C411yescharacteristic
265C2×C4×C127C22×D411yescharacteristic
266C2×C4×C127C22×D411yescharacteristic
267C8×C127—41no—
268C8×C127—41no—
269C8×C127—42no—
270C8×C127—41no—
271C8×C127—42no—
272C8×C127—41no—
273C22×C127—21no—
274C22×C127—22no—
275C22×C127—22no—
276C22×C127—22no—
277C22×C127—21no—
278C22×C127—21no—
279C22×C127—22no—
280C22×C127—41no—
281C22×C127—41no—
282C22×C127C22×C22⋊C411yescharacteristic
283C4×C127—21no—
284C4×C127—22no—
285C4×C127—22no—
286C4×C127—21no—
287C4×C127—21no—
288C4×C127—22no—
289C4×C127—42no—
290C4×C127—41no—
291C4×C127—41no—
292C4×C127—42no—
293C4×C127—41no—
294C4×C127#42112yes—
295C4×C127C2×#13111yescharacteristic
296C4×C127#56811yescharacteristic
297C4×C127#57711yescharacteristic
298C4×C127#57811yescharacteristic
299#17523C12711yesmaximal, characteristic, direct product factor, Sylow
300C2×C127—41no—
301C2×C127—42no—
302C2×C127—42no—
303C2×C127#288011yescharacteristic
304C2×C127#289811yescharacteristic
305C2×C127C2×#42111yescharacteristic
306#3482C2×C12711yescharacteristic
307#2990C2×C12711yescharacteristic, complemented
308#2922C2×C12711yescharacteristic, complemented
309#1497C2×C12712yescomplemented
310#1493C2×C12711yescharacteristic, complemented
311#1489C2×C12711yescharacteristic, complemented
312#1488C2×C12712yescomplemented
313#1485C2×C12711yescharacteristic, complemented
314#1481C2×C12712yescomplemented
315#1477C2×C12712yescomplemented
316#1476C2×C12711yescharacteristic, complemented
317C127#1752311yescharacteristic, direct product factor, Sylow
318#557C4×C12711yescharacteristic
319#556C4×C12712yes—
320#556C4×C12711yescharacteristic
321#552—21no—
322#552—22no—
323D4×Q8C4×C12711yescharacteristic
324#548—21no—
325#547C4×C12711yescharacteristic
326#547C4×C12712yes—
327#533C22×C12712yes—
328#533C22×C12711yescharacteristic
329#529—21no—
330#529—22no—
331C4×#41C22×C12711yescharacteristic
332C4×#41C22×C12712yes—
333C4×#41C22×C12711yescharacteristic
334C2×C4×Q8—21no—
335#446—21no—
336#435—22no—
337C8×D4—21no—
338#433—22no—
339#432C22×C12712yes—
340#431C22×C12711yescharacteristic
341#431C22×C12712yes—
342#431C22×C12712yes—
343#424—21no—
344#423C22×C12711yescharacteristic
345C2×C4⋊C8C22×C12712yes—
346#408—21no—
347C4×M16C22×C12711yescharacteristic
348C2×C4×C8C22×C12711yescharacteristic
349Q8⋊C8C22×C12711yescharacteristic
350Q8⋊C8C22×C12711yescharacteristic
351Q8⋊C8C22×C12712yes—
352Q8⋊C8C22×C12712yes—
353Q8⋊C8C22×C12712yes—
354Q8⋊C8C22×C12712yes—
355Q8⋊C8C22×C12712yes—
356D4⋊C8C22×C12711yescharacteristic
357D4⋊C8C22×C12711yescharacteristic
358D4⋊C8C22×C12712yes—
359C2×#41C2×C4×C12711yescharacteristic
360C2×#41C2×C4×C12712yes—
361C2×#41D4×C12711yescharacteristic
362C22×Q8—21no—
363C2×M16—21no—
364C2×M16—21no—
365C22×C8—21no—
366C22×C8—22no—
367C4⋊Q8C2×C4×C12712yes—
368C4⋊Q8C2×C4×C12711yescharacteristic
369C4⋊Q8C2×C4×C12712yes—
370C4⋊Q8C2×C4×C12711yescharacteristic
371#126—22no—
372#126—22no—
373#126—22no—
374#125—22no—
375#125—21no—
376#125—21no—
377#125—22no—
378#124—21no—
379#124—22no—
380#123—22no—
381#123—22no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8C2×C4×C12712yes—
388C22⋊Q8C2×C4×C12712yes—
389C22⋊Q8C2×C4×C12712yes—
390#121—21no—
391#121—22no—
392C4×Q8—21no—
393C4×Q8—22no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—22no—
397C4×Q8C2×C4×C12711yescharacteristic
398C4×Q8C2×C4×C12712yes—
399C4×Q8C2×C4×C12711yescharacteristic
400C4×Q8D4×C12711yescharacteristic
401C4×Q8C23×C12711yescharacteristic
402C4×Q8C23×C12712yes—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—22no—
407C4×D4C2×C4×C12711yescharacteristic
408C4×D4C2×C4×C12712yes—
409C4×D4C2×C4×C12711yescharacteristic
410C4×D4C2×C4×C12712yes—
411C4×D4D4×C12711yescharacteristic
412C4×D4D4×C12712yes—
413C4×D4C23×C12711yescharacteristic
414#117—22no—
415#117—21no—
416#117—21no—
417#117—22no—
418#117C2×C4×C12711yescharacteristic
419#117C2×C4×C12712yes—
420#117D4×C12711yescharacteristic
421#117D4×C12712yes—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C42—21no—
425C2×C42—22no—
426C2×C42C23×C12711yescharacteristic
427C2×C42C23×C12712yes—
428C4⋊C8—22no—
429C4⋊C8—21no—
430C4⋊C8—22no—
431C4⋊C8—22no—
432C4⋊C8C23×C12711yescharacteristic
433C4⋊C8C23×C12712yes—
434C4⋊C8C23×C12711yescharacteristic
435C22⋊C8—22no—
436C22⋊C8—22no—
437C22⋊C8—21no—
438C22⋊C8—22no—
439#97—21no—
440#97—22no—
441C4×C8—21no—
442C4×C8—21no—
443C4×C8C23×C12711yescharacteristic
444C4×C8C23×C12711yescharacteristic
445C4×C8C23×C12712yes—
446#41—22no—
447#41—24no—
448#41—41no—
449#41—42no—
450#41—42no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—44no—
454C2×Q8C22⋊C4×C12711yescharacteristic
455C2×Q8C22×C4×C12711yescharacteristic
456C2×Q8C22×C4×C12712yes—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4C22⋊C4×C12711yescharacteristic
460C2×D4C22⋊C4×C12712yes—
461C2×D4C22×C4×C12711yescharacteristic
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4C22×C4×C12711yescharacteristic
468C22×C4C22×C4×C12712yes—
469C22×C4C2×D4×C12712yes—
470C22×C4C2×D4×C12711yescharacteristic
471C22×C4C2×D4×C12711yescharacteristic
472C22×C4C2×D4×C12712yes—
473M16—42no—
474M16—81no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—42no—
482C2×C8—82no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4C22⋊C4×C12712yes—
497C4⋊C4C22⋊C4×C12711yescharacteristic
498C4⋊C4C22⋊C4×C12712yes—
499C4⋊C4C22⋊C4×C12711yescharacteristic
500C4⋊C4C22×C4×C12711yescharacteristic
501C4⋊C4C22×C4×C12711yescharacteristic
502C4⋊C4C22×C4×C12712yes—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4C22⋊C4×C12712yes—
512C22⋊C4C22⋊C4×C12712yes—
513C22⋊C4C22⋊C4×C12712yes—
514C42—21no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—22no—
520C42C22×C4×C12712yes—
521C42C22×C4×C12711yescharacteristic
522C42C2×D4×C12711yescharacteristic
523C42C2×D4×C12711yescharacteristic
524C42C2×D4×C12712yes—
525C42C2×D4×C12712yes—
526C42C24×C12711yescharacteristic
527C23—21no—
528C23C2×C22⋊C4×C12711yescharacteristic
529C23C2×C22⋊C4×C12712yes—
530Q8—22no—
531Q8—21no—
532Q8—22no—
533Q8—21no—
534Q8—22no—
535Q8—44no—
536Q8—42no—
537D4—21no—
538D4—22no—
539D4—42no—
540D4—42no—
541D4—44no—
542D4#131×C12712yes—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4C2×C22⋊C4×C12712yes—
565C2×C4C2×C22⋊C4×C12711yescharacteristic
566C2×C4C2×C22⋊C4×C12712yes—
567C2×C4C2×C22⋊C4×C12711yescharacteristic
568C2×C4C2×C22⋊C4×C12712yes—
569C2×C4C2×C22⋊C4×C12711yescharacteristic
570C2×C4C23×C4×C12711yescharacteristic
571C2×C4C22×D4×C12711yescharacteristic
572C2×C4C22×D4×C12711yescharacteristic
573C8—41no—
574C8—41no—
575C8—42no—
576C8—41no—
577C8—42no—
578C8—41no—
579C22—21no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22C22×C22⋊C4×C12711yescharacteristic
589C4—21no—
590C4—22no—
591C4—22no—
592C4—21no—
593C4—21no—
594C4—22no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4#421×C12712yes—
601C4C2×#131×C12711yescharacteristic
602C4#568×C12711yescharacteristic
603C4#577×C12711yescharacteristic
604C4#578×C12711yescharacteristic
605C2—41no—
606C2—42no—
607C2—42no—
608C2C127×#288011yescharacteristic
609C2C127×#289811yescharacteristic
610C2C2×#421×C12711yescharacteristic
subgroupG/H
centerC2×C4×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC42C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC127×#3482C2
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×#1497C2semidirect
5. C127×#1493C2semidirect
6. C127×#1489C2semidirect
7. C127×#1488C2semidirect
8. C127×#1485C2semidirect
9. C127×#1481C2semidirect
10. C127×#1477C2semidirect
11. C127×#1476C2semidirect
299. #17523C127direct (N × H)
307. #2990C2×C127semidirect
308. #2922C2×C127semidirect
309. #1497C2×C127semidirect
310. #1493C2×C127semidirect
311. #1489C2×C127semidirect
312. #1488C2×C127semidirect
313. #1485C2×C127semidirect
314. #1481C2×C127semidirect
315. #1477C2×C127semidirect
316. #1476C2×C127semidirect
317. C127#17523direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34822
C127×#29902
C127×#29222
C127×#14972
C127×#14932
C127×#14892
C127×#14882
C127×#14852
C127×#14812
C127×#14772
C127×#14762
#17523127