Finite Group Databasebeta

← C127×#17519 #44581192

Subgroups — C127×#17519 #44581192

Order 32512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3482C211yesmaximal, characteristic
2C127×#2990C211yesmaximal, characteristic, complemented
3C4×#131×C127C211yesmaximal, characteristic, complemented
4C127×#1494C212yesmaximal, complemented
5C127×#1493C211yesmaximal, characteristic, complemented
6C127×#1491C212yesmaximal, complemented
7C127×#1489C211yesmaximal, characteristic, complemented
8C127×#1485C211yesmaximal, characteristic, complemented
9C127×#1483C212yesmaximal, complemented
10C127×#1476C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8×C127C212yesmaximal, complemented
12C2×#131×C127—21no—
13#557×C127C411yescharacteristic
14#556×C127—21no—
15#556×C127C411yescharacteristic
16#552×C127—22no—
17#552×C127C412yes—
18D4×Q8×C127C411yescharacteristic
19#548×C127C412yes—
20#547×C127—21no—
21#547×C127C411yescharacteristic
22#533×C127—21no—
23#533×C127C2211yescharacteristic
24#529×C127—22no—
25#529×C127C2212yes—
26C4×#41×C127—21no—
27C4×#41×C127C2211yescharacteristic
28C4×#41×C127C2211yescharacteristic
29C2×C4×Q8×C127C2212yes—
30#446×C127—21no—
31#435×C127—22no—
32C8×D4×C127—21no—
33#432×C127C2212yes—
34#431×C127C2211yescharacteristic
35#423×C127C2211yescharacteristic
36C2×C4⋊C8×C127C2212yes—
37#405×C127—22no—
38#405×C127—21no—
39C4×M16×C127C2212yes—
40C4×M16×C127C2211yescharacteristic
41C2×C4×C8×C127C2212yes—
42C2×C4×C8×C127C2211yescharacteristic
43Q8⋊C8×C127C2211yescharacteristic
44Q8⋊C8×C127C2212yes—
45Q8⋊C8×C127C2212yes—
46Q8⋊C8×C127C2212yes—
47Q8⋊C8×C127C2211yescharacteristic
48Q8⋊C8×C127C2212yes—
49Q8⋊C8×C127C2212yes—
50D4⋊C8×C127C2211yescharacteristic
51D4⋊C8×C127C2211yescharacteristic
52D4⋊C8×C127C2212yes—
53C2×#41×C127—21no—
54C2×#41×C127C2×C411yescharacteristic
55C2×#41×C127D411yescharacteristic
56C22×Q8×C127C2×C412yes—
57#131×C127—28no—
58C2×M16×C127—22no—
59C2×M16×C127—21no—
60C2×M16×C127—21no—
61C22×C8×C127—22no—
62C22×C8×C127—21no—
63C22×C8×C127—22no—
64C4⋊Q8×C127—21no—
65C4⋊Q8×C127—21no—
66C4⋊Q8×C127C2×C411yescharacteristic
67C4⋊Q8×C127C2×C411yescharacteristic
68#126×C127—22no—
69#126×C127—22no—
70#126×C127—22no—
71#125×C127—22no—
72#125×C127—22no—
73#125×C127—21no—
74#125×C127C2×C412yes—
75#124×C127—22no—
76#124×C127C2×C412yes—
77#123×C127—22no—
78#123×C127—22no—
79#123×C127—22no—
80C22⋊Q8×C127—21no—
81C22⋊Q8×C127—21no—
82C22⋊Q8×C127—21no—
83C22⋊Q8×C127—22no—
84C22⋊Q8×C127—22no—
85C22⋊Q8×C127C2×C412yes—
86C22⋊Q8×C127C2×C412yes—
87#121×C127—22no—
88#121×C127C2×C412yes—
89C4×Q8×C127—21no—
90C4×Q8×C127—22no—
91C4×Q8×C127—22no—
92C4×Q8×C127—21no—
93C4×Q8×C127C2×C412yes—
94C4×Q8×C127C2×C412yes—
95C4×Q8×C127C2×C411yescharacteristic
96C4×Q8×C127C2×C411yescharacteristic
97C4×Q8×C127D411yescharacteristic
98C4×Q8×C127C2311yescharacteristic
99C4×Q8×C127C2312yes—
100C4×D4×C127—21no—
101C4×D4×C127—22no—
102C4×D4×C127—21no—
103C4×D4×C127—21no—
104C4×D4×C127—22no—
105C4×D4×C127C2×C411yescharacteristic
106C4×D4×C127C2×C411yescharacteristic
107C4×D4×C127C2×C412yes—
108C4×D4×C127D411yescharacteristic
109C4×D4×C127D412yes—
110C4×D4×C127C2311yescharacteristic
111#117×C127—21no—
112#117×C127—21no—
113#117×C127—22no—
114#117×C127—21no—
115#117×C127—22no—
116#117×C127C2×C411yescharacteristic
117#117×C127D412yes—
118#117×C127D411yescharacteristic
119C2×C4⋊C4×C127—22no—
120C2×C4⋊C4×C127C2×C412yes—
121C2×C42×C127—22no—
122C2×C42×C127—21no—
123C2×C42×C127C2312yes—
124C2×C42×C127C2311yescharacteristic
125C4⋊C8×C127—22no—
126C4⋊C8×C127—21no—
127C4⋊C8×C127C2311yescharacteristic
128C4⋊C8×C127C2312yes—
129C4⋊C8×C127C2311yescharacteristic
130C22⋊C8×C127—21no—
131C22⋊C8×C127—22no—
132#97×C127—21no—
133#97×C127—22no—
134#97×C127—22no—
135C4×C8×C127—21no—
136C4×C8×C127—22no—
137C4×C8×C127—21no—
138C4×C8×C127C2311yescharacteristic
139C4×C8×C127C2311yescharacteristic
140C4×C8×C127C2312yes—
141#41×C127—24no—
142#41×C127—22no—
143#41×C127—41no—
144#41×C127—44no—
145C2×Q8×C127—22no—
146C2×Q8×C127—21no—
147C2×Q8×C127—24no—
148C2×Q8×C127—42no—
149C2×Q8×C127C22⋊C411yescharacteristic
150C2×Q8×C127C22×C412yes—
151C2×Q8×C127C22×C411yescharacteristic
152C2×D4×C127—21no—
153C2×D4×C127—22no—
154C2×D4×C127C22⋊C412yes—
155C2×D4×C127C22⋊C411yescharacteristic
156C2×D4×C127C22×C411yescharacteristic
157C22×C4×C127—22no—
158C22×C4×C127—22no—
159C22×C4×C127—21no—
160C22×C4×C127—21no—
161C22×C4×C127—21no—
162C22×C4×C127C22×C412yes—
163C22×C4×C127C22×C411yescharacteristic
164C22×C4×C127C2×D412yes—
165C22×C4×C127C2×D411yescharacteristic
166C22×C4×C127C2×D412yes—
167C22×C4×C127C2×D411yescharacteristic
168M16×C127—28no—
169M16×C127—24no—
170M16×C127—81no—
171C2×C8×C127—28no—
172C2×C8×C127—24no—
173C2×C8×C127—21no—
174C2×C8×C127—21no—
175C2×C8×C127—22no—
176C2×C8×C127—21no—
177C2×C8×C127—21no—
178C2×C8×C127—22no—
179C2×C8×C127—82no—
180C4⋊C4×C127—22no—
181C4⋊C4×C127—22no—
182C4⋊C4×C127—22no—
183C4⋊C4×C127—21no—
184C4⋊C4×C127—22no—
185C4⋊C4×C127—22no—
186C4⋊C4×C127—21no—
187C4⋊C4×C127—21no—
188C4⋊C4×C127—22no—
189C4⋊C4×C127—21no—
190C4⋊C4×C127—22no—
191C4⋊C4×C127—21no—
192C4⋊C4×C127—22no—
193C4⋊C4×C127C22⋊C411yescharacteristic
194C4⋊C4×C127C22⋊C412yes—
195C4⋊C4×C127C22⋊C411yescharacteristic
196C4⋊C4×C127C22⋊C412yes—
197C4⋊C4×C127C22×C411yescharacteristic
198C4⋊C4×C127C22×C412yes—
199C4⋊C4×C127C22×C411yescharacteristic
200C22⋊C4×C127—22no—
201C22⋊C4×C127—22no—
202C22⋊C4×C127—21no—
203C22⋊C4×C127—22no—
204C22⋊C4×C127—21no—
205C22⋊C4×C127—22no—
206C22⋊C4×C127—22no—
207C22⋊C4×C127—21no—
208C22⋊C4×C127C22⋊C412yes—
209C22⋊C4×C127C22⋊C412yes—
210C22⋊C4×C127C22⋊C412yes—
211C42×C127—21no—
212C42×C127—22no—
213C42×C127—21no—
214C42×C127—22no—
215C42×C127—21no—
216C42×C127—22no—
217C42×C127C22×C411yescharacteristic
218C42×C127C22×C412yes—
219C42×C127C2×D411yescharacteristic
220C42×C127C2×D412yes—
221C42×C127C2×D411yescharacteristic
222C42×C127C2×D412yes—
223C42×C127C2411yescharacteristic
224C23×C127—21no—
225C23×C127C2×C22⋊C411yescharacteristic
226C23×C127C2×C22⋊C412yes—
227Q8×C127—21no—
228Q8×C127—22no—
229Q8×C127—22no—
230Q8×C127—22no—
231Q8×C127—44no—
232Q8×C127—42no—
233Q8×C127#13112yescomplemented
234D4×C127—24no—
235D4×C127—21no—
236D4×C127—21no—
237D4×C127—42no—
238D4×C127—44no—
239D4×C127#10414yes—
240C2×C4×C127—21no—
241C2×C4×C127—24no—
242C2×C4×C127—22no—
243C2×C4×C127—21no—
244C2×C4×C127—22no—
245C2×C4×C127—21no—
246C2×C4×C127—22no—
247C2×C4×C127—21no—
248C2×C4×C127—22no—
249C2×C4×C127—21no—
250C2×C4×C127—22no—
251C2×C4×C127—21no—
252C2×C4×C127—21no—
253C2×C4×C127—22no—
254C2×C4×C127—22no—
255C2×C4×C127—42no—
256C2×C4×C127—44no—
257C2×C4×C127—44no—
258C2×C4×C127—41no—
259C2×C4×C127—41no—
260C2×C4×C127—42no—
261C2×C4×C127C2×C22⋊C412yes—
262C2×C4×C127C2×C22⋊C411yescharacteristic
263C2×C4×C127C2×C22⋊C411yescharacteristic
264C2×C4×C127C2×C22⋊C412yes—
265C2×C4×C127C2×C22⋊C412yes—
266C2×C4×C127C2×C22⋊C411yescharacteristic
267C2×C4×C127C23×C411yescharacteristic
268C2×C4×C127C22×D411yescharacteristic
269C2×C4×C127C22×D411yescharacteristic
270C8×C127—22no—
271C8×C127—24no—
272C8×C127—22no—
273C8×C127—41no—
274C8×C127—42no—
275C8×C127—41no—
276C22×C127—22no—
277C22×C127—22no—
278C22×C127—21no—
279C22×C127—21no—
280C22×C127—21no—
281C22×C127—41no—
282C22×C127—41no—
283C22×C127—41no—
284C22×C127C2×#10414yes—
285C22×C127C22×C22⋊C411yescharacteristic
286C4×C127—21no—
287C4×C127—22no—
288C4×C127—22no—
289C4×C127—22no—
290C4×C127—21no—
291C4×C127—21no—
292C4×C127—22no—
293C4×C127—42no—
294C4×C127—41no—
295C4×C127—41no—
296C4×C127—42no—
297C4×C127C2×#10412yes—
298C4×C127C2×#13111yescharacteristic
299C4×C127#56811yescharacteristic
300C4×C127#57711yescharacteristic
301C4×C127#57811yescharacteristic
302#17519C12711yesmaximal, characteristic, direct product factor, Sylow
303C2×C127—24no—
304C2×C127—41no—
305C2×C127—81no—
306C2×C127#288011yescharacteristic
307C2×C127#289811yescharacteristic
308C2×C127C22×#10411yescharacteristic
309#3482C2×C12711yescharacteristic
310#2990C2×C12711yescharacteristic, complemented
311C4×#131C2×C12711yescharacteristic, complemented
312#1494C2×C12712yescomplemented
313#1493C2×C12711yescharacteristic, complemented
314#1491C2×C12712yescomplemented
315#1489C2×C12711yescharacteristic, complemented
316#1485C2×C12711yescharacteristic, complemented
317#1483C2×C12712yescomplemented
318#1476C2×C12711yescharacteristic, complemented
319C2×Q8⋊C8C2×C12712yescomplemented
320C127#1751911yescharacteristic, direct product factor, Sylow
321C2×#131—21no—
322#557C4×C12711yescharacteristic
323#556—21no—
324#556C4×C12711yescharacteristic
325#552—22no—
326#552C4×C12712yes—
327D4×Q8C4×C12711yescharacteristic
328#548C4×C12712yes—
329#547—21no—
330#547C4×C12711yescharacteristic
331#533—21no—
332#533C22×C12711yescharacteristic
333#529—22no—
334#529C22×C12712yes—
335C4×#41—21no—
336C4×#41C22×C12711yescharacteristic
337C4×#41C22×C12711yescharacteristic
338C2×C4×Q8C22×C12712yes—
339#446—21no—
340#435—22no—
341C8×D4—21no—
342#432C22×C12712yes—
343#431C22×C12711yescharacteristic
344#423C22×C12711yescharacteristic
345C2×C4⋊C8C22×C12712yes—
346#405—22no—
347#405—21no—
348C4×M16C22×C12712yes—
349C4×M16C22×C12711yescharacteristic
350C2×C4×C8C22×C12712yes—
351C2×C4×C8C22×C12711yescharacteristic
352Q8⋊C8C22×C12711yescharacteristic
353Q8⋊C8C22×C12712yes—
354Q8⋊C8C22×C12712yes—
355Q8⋊C8C22×C12712yes—
356Q8⋊C8C22×C12711yescharacteristic
357Q8⋊C8C22×C12712yes—
358Q8⋊C8C22×C12712yes—
359D4⋊C8C22×C12711yescharacteristic
360D4⋊C8C22×C12711yescharacteristic
361D4⋊C8C22×C12712yes—
362C2×#41—21no—
363C2×#41C2×C4×C12711yescharacteristic
364C2×#41D4×C12711yescharacteristic
365C22×Q8C2×C4×C12712yes—
366#131—28no—
367C2×M16—22no—
368C2×M16—21no—
369C2×M16—21no—
370C22×C8—22no—
371C22×C8—21no—
372C22×C8—22no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8C2×C4×C12711yescharacteristic
376C4⋊Q8C2×C4×C12711yescharacteristic
377#126—22no—
378#126—22no—
379#126—22no—
380#125—22no—
381#125—22no—
382#125—21no—
383#125C2×C4×C12712yes—
384#124—22no—
385#124C2×C4×C12712yes—
386#123—22no—
387#123—22no—
388#123—22no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8C2×C4×C12712yes—
395C22⋊Q8C2×C4×C12712yes—
396#121—22no—
397#121C2×C4×C12712yes—
398C4×Q8—21no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—21no—
402C4×Q8C2×C4×C12712yes—
403C4×Q8C2×C4×C12712yes—
404C4×Q8C2×C4×C12711yescharacteristic
405C4×Q8C2×C4×C12711yescharacteristic
406C4×Q8D4×C12711yescharacteristic
407C4×Q8C23×C12711yescharacteristic
408C4×Q8C23×C12712yes—
409C4×D4—21no—
410C4×D4—22no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—22no—
414C4×D4C2×C4×C12711yescharacteristic
415C4×D4C2×C4×C12711yescharacteristic
416C4×D4C2×C4×C12712yes—
417C4×D4D4×C12711yescharacteristic
418C4×D4D4×C12712yes—
419C4×D4C23×C12711yescharacteristic
420#117—21no—
421#117—21no—
422#117—22no—
423#117—21no—
424#117—22no—
425#117C2×C4×C12711yescharacteristic
426#117D4×C12712yes—
427#117D4×C12711yescharacteristic
428C2×C4⋊C4—22no—
429C2×C4⋊C4C2×C4×C12712yes—
430C2×C42—22no—
431C2×C42—21no—
432C2×C42C23×C12712yes—
433C2×C42C23×C12711yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—21no—
436C4⋊C8C23×C12711yescharacteristic
437C4⋊C8C23×C12712yes—
438C4⋊C8C23×C12711yescharacteristic
439C22⋊C8—21no—
440C22⋊C8—22no—
441#97—21no—
442#97—22no—
443#97—22no—
444C4×C8—21no—
445C4×C8—22no—
446C4×C8—21no—
447C4×C8C23×C12711yescharacteristic
448C4×C8C23×C12711yescharacteristic
449C4×C8C23×C12712yes—
450#41—24no—
451#41—22no—
452#41—41no—
453#41—44no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—24no—
457C2×Q8—42no—
458C2×Q8C22⋊C4×C12711yescharacteristic
459C2×Q8C22×C4×C12712yes—
460C2×Q8C22×C4×C12711yescharacteristic
461C2×D4—21no—
462C2×D4—22no—
463C2×D4C22⋊C4×C12712yes—
464C2×D4C22⋊C4×C12711yescharacteristic
465C2×D4C22×C4×C12711yescharacteristic
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4C22×C4×C12712yes—
472C22×C4C22×C4×C12711yescharacteristic
473C22×C4C2×D4×C12712yes—
474C22×C4C2×D4×C12711yescharacteristic
475C22×C4C2×D4×C12712yes—
476C22×C4C2×D4×C12711yescharacteristic
477M16—28no—
478M16—24no—
479M16—81no—
480C2×C8—28no—
481C2×C8—24no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—82no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4C22⋊C4×C12711yescharacteristic
503C4⋊C4C22⋊C4×C12712yes—
504C4⋊C4C22⋊C4×C12711yescharacteristic
505C4⋊C4C22⋊C4×C12712yes—
506C4⋊C4C22×C4×C12711yescharacteristic
507C4⋊C4C22×C4×C12712yes—
508C4⋊C4C22×C4×C12711yescharacteristic
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C22⋊C4C22⋊C4×C12712yes—
518C22⋊C4C22⋊C4×C12712yes—
519C22⋊C4C22⋊C4×C12712yes—
520C42—21no—
521C42—22no—
522C42—21no—
523C42—22no—
524C42—21no—
525C42—22no—
526C42C22×C4×C12711yescharacteristic
527C42C22×C4×C12712yes—
528C42C2×D4×C12711yescharacteristic
529C42C2×D4×C12712yes—
530C42C2×D4×C12711yescharacteristic
531C42C2×D4×C12712yes—
532C42C24×C12711yescharacteristic
533C23—21no—
534C23C2×C22⋊C4×C12711yescharacteristic
535C23C2×C22⋊C4×C12712yes—
536Q8—21no—
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—44no—
541Q8—42no—
542Q8#131×C12712yescomplemented
543D4—24no—
544D4—21no—
545D4—21no—
546D4—42no—
547D4—44no—
548D4#104×C12714yes—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4C2×C22⋊C4×C12712yes—
571C2×C4C2×C22⋊C4×C12711yescharacteristic
572C2×C4C2×C22⋊C4×C12711yescharacteristic
573C2×C4C2×C22⋊C4×C12712yes—
574C2×C4C2×C22⋊C4×C12712yes—
575C2×C4C2×C22⋊C4×C12711yescharacteristic
576C2×C4C23×C4×C12711yescharacteristic
577C2×C4C22×D4×C12711yescharacteristic
578C2×C4C22×D4×C12711yescharacteristic
579C8—22no—
580C8—24no—
581C8—22no—
582C8—41no—
583C8—42no—
584C8—41no—
585C22—22no—
586C22—22no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#104×C12714yes—
594C22C22×C22⋊C4×C12711yescharacteristic
595C4—21no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—21no—
600C4—21no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4C2×#104×C12712yes—
607C4C2×#131×C12711yescharacteristic
608C4#568×C12711yescharacteristic
609C4#577×C12711yescharacteristic
610C4#578×C12711yescharacteristic
611C2—24no—
612C2—41no—
613C2—81no—
614C2C127×#288011yescharacteristic
615C2C127×#289811yescharacteristic
616C2C22×#104×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. C4×#131×C127C2semidirect
4. C127×#1494C2semidirect
5. C127×#1493C2semidirect
6. C127×#1491C2semidirect
7. C127×#1489C2semidirect
8. C127×#1485C2semidirect
9. C127×#1483C2semidirect
10. C127×#1476C2semidirect
11. C2×Q8⋊C8×C127C2semidirect
233. Q8×C127#131semidirect
302. #17519C127direct (N × H)
310. #2990C2×C127semidirect
311. C4×#131C2×C127semidirect
312. #1494C2×C127semidirect
313. #1493C2×C127semidirect
314. #1491C2×C127semidirect
315. #1489C2×C127semidirect
316. #1485C2×C127semidirect
317. #1483C2×C127semidirect
318. #1476C2×C127semidirect
319. C2×Q8⋊C8C2×C127semidirect
320. C127#17519direct (N × H)
542. Q8#131×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34822
C127×#29902
C4×#131×C1272
C127×#14942
C127×#14932
C127×#14912
C127×#14892
C127×#14852
C127×#14832
C127×#14762
C2×Q8⋊C8×C1272
#17519127