Finite Group Databasebeta

← C127×#16978 #44580651

Subgroups — C127×#16978 #44580651

Order 32512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389×C127C211yesmaximal, characteristic, complemented
2C2×#388×C127C211yesmaximal, characteristic
3C127×#1444C211yesmaximal, characteristic, complemented
4C127×#1442C211yesmaximal, characteristic, complemented
5C127×#1442C212yesmaximal, complemented
6C127×#1441C212yesmaximal, complemented
7C127×#1439C211yesmaximal, characteristic
8C127×#1439C211yesmaximal, characteristic
9C127×#1439C212yesmaximal
10C127×#1439C212yesmaximal
11C127×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2212yes—
20C22×C22⋊C4×C127C2211yescharacteristic
21C22×C22⋊C4×C127C2211yescharacteristic
22C22×C42×C127C2211yescharacteristic
23#389×C127—24no—
24#388×C127—24no—
25C2×#95×C127C2212yes—
26C2×#95×C127C2211yescharacteristic
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2211yescharacteristic
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C2×#95×C127C2212yes—
36C2×#95×C127C2212yes—
37C25×C127C2311yescharacteristic
38C23×C4×C127C2311yescharacteristic
39C23×C4×C127C2312yes—
40C23×C4×C127C2311yescharacteristic
41C23×C4×C127C2312yes—
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2312yes—
47C23×C4×C127C2311yescharacteristic
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—22no—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—42no—
55C2×C4⋊C4×C127—42no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127—42no—
58C2×C4⋊C4×C127—42no—
59C2×C4⋊C4×C127—42no—
60C2×C22⋊C4×C127—22no—
61C2×C22⋊C4×C127—22no—
62C2×C22⋊C4×C127—22no—
63C2×C22⋊C4×C127—22no—
64C2×C22⋊C4×C127—24no—
65C2×C22⋊C4×C127—24no—
66C2×C22⋊C4×C127—42no—
67C2×C22⋊C4×C127—42no—
68C2×C22⋊C4×C127—42no—
69C2×C22⋊C4×C127—42no—
70C2×C22⋊C4×C127—42no—
71C2×C22⋊C4×C127—42no—
72C2×C22⋊C4×C127Q814yes—
73C2×C42×C127—42no—
74C2×C42×C127D414yes—
75#95×C127—22no—
76#95×C127—22no—
77#95×C127—24no—
78#95×C127—22no—
79#95×C127—24no—
80#95×C127—24no—
81#95×C127—22no—
82#95×C127—42no—
83#95×C127—42no—
84#95×C127—42no—
85#95×C127—42no—
86#95×C127—42no—
87C24×C127—21no—
88C24×C127—21no—
89C24×C127—22no—
90C24×C127—22no—
91C24×C127—21no—
92C24×C127—21no—
93C24×C127—22no—
94C24×C127—22no—
95C24×C127—21no—
96C24×C127—21no—
97C24×C127C2×Q812yes—
98C24×C127C2411yescharacteristic
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—21no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—21no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—22no—
119C22×C4×C127—21no—
120C22×C4×C127—21no—
121C22×C4×C127—22no—
122C22×C4×C127—21no—
123C22×C4×C127—21no—
124C22×C4×C127—22no—
125C22×C4×C127—21no—
126C22×C4×C127—21no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—22no—
133C22×C4×C127—22no—
134C22×C4×C127—21no—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—22no—
140C22×C4×C127—22no—
141C22×C4×C127—22no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—22no—
145C22×C4×C127—21no—
146C22×C4×C127—21no—
147C22×C4×C127—21no—
148C22×C4×C127—22no—
149C22×C4×C127C2×D412yes—
150C22×C4×C127C2×D412yes—
151C22×C4×C127C2×D412yes—
152C22×C4×C127C2×Q812yes—
153C22×C4×C127C2×Q812yes—
154C22×C4×C127#4114yes—
155C22×C4×C127#4114yes—
156C22×C4×C127#4112yes—
157C4⋊C4×C127—28no—
158C4⋊C4×C127—44no—
159C4⋊C4×C127—44no—
160C4⋊C4×C127—48no—
161C4⋊C4×C127—48no—
162C22⋊C4×C127—28no—
163C22⋊C4×C127—44no—
164C22⋊C4×C127—44no—
165C22⋊C4×C127—48no—
166C22⋊C4×C127—48no—
167C42×C127—44no—
168C23×C127—22no—
169C23×C127—22no—
170C23×C127—22no—
171C23×C127—24no—
172C23×C127—24no—
173C23×C127—41no—
174C23×C127—41no—
175C23×C127—41no—
176C23×C127—41no—
177C23×C127—41no—
178C23×C127—41no—
179C23×C127—42no—
180C23×C127—42no—
181C23×C127—42no—
182C23×C127—42no—
183C23×C127—41no—
184C23×C127—41no—
185C23×C127—42no—
186C23×C127—42no—
187C23×C127—42no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127C22×D411yescharacteristic
192C23×C127C22×Q811yescharacteristic
193C23×C127C2×#4111yescharacteristic
194C23×C127C2×#4112yes—
195C23×C127C2×#4112yes—
196C23×C127H(5,𝔽2)12yes—
197C23×C127H(5,𝔽2)11yescharacteristic
198C23×C127H(5,𝔽2)11yescharacteristic
199C23×C127H(5,𝔽2)12yes—
200C23×C1272−1+411yescharacteristic
201C23×C1272−1+411yescharacteristic
202C2×C4×C127—22no—
203C2×C4×C127—22no—
204C2×C4×C127—24no—
205C2×C4×C127—24no—
206C2×C4×C127—22no—
207C2×C4×C127—24no—
208C2×C4×C127—22no—
209C2×C4×C127—22no—
210C2×C4×C127—22no—
211C2×C4×C127—22no—
212C2×C4×C127—22no—
213C2×C4×C127—22no—
214C2×C4×C127—24no—
215C2×C4×C127—24no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—24no—
219C2×C4×C127—24no—
220C2×C4×C127—41no—
221C2×C4×C127—41no—
222C2×C4×C127—42no—
223C2×C4×C127—42no—
224C2×C4×C127—42no—
225C2×C4×C127—42no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127—42no—
235C2×C4×C127—41no—
236C2×C4×C127—42no—
237C2×C4×C127—41no—
238C2×C4×C127—42no—
239C2×C4×C127—41no—
240C2×C4×C127—41no—
241C2×C4×C127—42no—
242C2×C4×C127—41no—
243C2×C4×C127—41no—
244C2×C4×C127—42no—
245C2×C4×C127—42no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—42no—
249C2×C4×C127—42no—
250C2×C4×C127—42no—
251C2×C4×C127—42no—
252C2×C4×C127—42no—
253C2×C4×C127—42no—
254C2×C4×C127—42no—
255C2×C4×C127—42no—
256C2×C4×C127—42no—
257C2×C4×C127—42no—
258C2×C4×C127—42no—
259C2×C4×C127C22⋊Q814yes—
260C22×C127—42no—
261C22×C127—42no—
262C22×C127—42no—
263C22×C127—44no—
264C22×C127—44no—
265C22×C127—81no—
266C22×C127—81no—
267C22×C127—81no—
268C22×C127—81no—
269C22×C127—82no—
270C22×C127—82no—
271C22×C127C2×C22⋊Q811yescharacteristic
272C22×C127#52511yescharacteristic
273C22×C127#53211yescharacteristic
274C22×C127#53411yescharacteristic
275C22×C127#53511yescharacteristic
276C22×C127#53511yescharacteristic
277C22×C127#53611yescharacteristic
278C22×C127#53812yes—
279C22×C127#53812yes—
280C22×C127#53812yes—
281C22×C127#54012yes—
282C22×C127#54211yescharacteristic
283C22×C127#54211yescharacteristic
284C22×C127#54411yescharacteristic
285C22×C127#54411yescharacteristic
286C22×C127#55212yes—
287C22×C127#55312yes—
288C22×C127#55312yes—
289C22×C127#55412yes—
290C22×C127#55412yes—
291C22×C127#55612yes—
292C22×C127#56212yes—
293C22×C127#56212yes—
294C4×C127—24no—
295C4×C127—42no—
296C4×C127—42no—
297C4×C127—44no—
298C4×C127—42no—
299C4×C127—44no—
300C4×C127—81no—
301C4×C127—81no—
302C4×C127—82no—
303C4×C127—82no—
304#16978C12711yesmaximal, characteristic, direct product factor, Sylow
305C2×C127—82no—
306C2×C127#256611yescharacteristic
307C2×C127#257911yescharacteristic
308C2×C127#260411yescharacteristic
309C2×C127#262811yescharacteristic
310C2×C127#263111yescharacteristic
311C2×C127#266111yescharacteristic
312C2×C127#276812yes—
313C2×C127#277912yes—
314C2×C127#278812yes—
315C2×C127#280012yes—
316C2×C127#281611yescharacteristic
317C2×#389C2×C12711yescharacteristic, complemented
318C2×#388C2×C12711yescharacteristic
319#1444C2×C12711yescharacteristic, complemented
320#1442C2×C12711yescharacteristic, complemented
321#1442C2×C12712yescomplemented
322#1441C2×C12712yescomplemented
323#1439C2×C12711yescharacteristic
324#1439C2×C12711yescharacteristic
325#1439C2×C12712yes—
326#1439C2×C12712yes—
327#1434C2×C12711yescharacteristic, complemented
328C127#1697811yescharacteristic, direct product factor, Sylow
329C22×C4⋊C4C22×C12711yescharacteristic
330C22×C4⋊C4C22×C12712yes—
331C22×C4⋊C4C22×C12712yes—
332C22×C4⋊C4C22×C12711yescharacteristic
333C22×C4⋊C4C22×C12711yescharacteristic
334C22×C22⋊C4C22×C12711yescharacteristic
335C22×C22⋊C4C22×C12712yes—
336C22×C22⋊C4C22×C12712yes—
337C22×C22⋊C4C22×C12711yescharacteristic
338C22×C22⋊C4C22×C12711yescharacteristic
339C22×C42C22×C12711yescharacteristic
340#389—24no—
341#388—24no—
342C2×#95C22×C12712yes—
343C2×#95C22×C12711yescharacteristic
344C2×#95C22×C12711yescharacteristic
345C2×#95C22×C12712yes—
346C2×#95C22×C12712yes—
347C2×#95C22×C12711yescharacteristic
348C2×#95C22×C12711yescharacteristic
349C2×#95C22×C12712yes—
350C2×#95C22×C12712yes—
351C2×#95C22×C12712yes—
352C2×#95C22×C12712yes—
353C2×#95C22×C12712yes—
354C25C23×C12711yescharacteristic
355C23×C4C23×C12711yescharacteristic
356C23×C4C23×C12712yes—
357C23×C4C23×C12711yescharacteristic
358C23×C4C23×C12712yes—
359C23×C4C23×C12711yescharacteristic
360C23×C4C23×C12711yescharacteristic
361C23×C4C23×C12711yescharacteristic
362C23×C4C23×C12712yes—
363C23×C4C23×C12712yes—
364C23×C4C23×C12711yescharacteristic
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—42no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4Q8×C12714yes—
390C2×C42—42no—
391C2×C42D4×C12714yes—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—42no—
400#95—42no—
401#95—42no—
402#95—42no—
403#95—42no—
404C24—21no—
405C24—21no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24C2×Q8×C12712yes—
415C24C24×C12711yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4C2×D4×C12712yes—
467C22×C4C2×D4×C12712yes—
468C22×C4C2×D4×C12712yes—
469C22×C4C2×Q8×C12712yes—
470C22×C4C2×Q8×C12712yes—
471C22×C4#41×C12714yes—
472C22×C4#41×C12714yes—
473C22×C4#41×C12712yes—
474C4⋊C4—28no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C22⋊C4—28no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C42—44no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C22×D4×C12711yescharacteristic
509C23C22×Q8×C12711yescharacteristic
510C23C2×#41×C12711yescharacteristic
511C23C2×#41×C12712yes—
512C23C2×#41×C12712yes—
513C23H(5,𝔽2)×C12712yes—
514C23H(5,𝔽2)×C12711yescharacteristic
515C23H(5,𝔽2)×C12711yescharacteristic
516C23H(5,𝔽2)×C12712yes—
517C232−1+4×C12711yescharacteristic
518C232−1+4×C12711yescharacteristic
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4C22⋊Q8×C12714yes—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22—81no—
588C22C2×C22⋊Q8×C12711yescharacteristic
589C22#525×C12711yescharacteristic
590C22#532×C12711yescharacteristic
591C22#534×C12711yescharacteristic
592C22#535×C12711yescharacteristic
593C22#535×C12711yescharacteristic
594C22#536×C12711yescharacteristic
595C22#538×C12712yes—
596C22#538×C12712yes—
597C22#538×C12712yes—
598C22#540×C12712yes—
599C22#542×C12711yescharacteristic
600C22#542×C12711yescharacteristic
601C22#544×C12711yescharacteristic
602C22#544×C12711yescharacteristic
603C22#552×C12712yes—
604C22#553×C12712yes—
605C22#553×C12712yes—
606C22#554×C12712yes—
607C22#554×C12712yes—
608C22#556×C12712yes—
609C22#562×C12712yes—
610C22#562×C12712yes—
611C4—24no—
612C4—44no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—82no—
619C4—82no—
620C4—81no—
621C2—82no—
622C2C127×#256611yescharacteristic
623C2C127×#257911yescharacteristic
624C2C127×#260411yescharacteristic
625C2C127×#262811yescharacteristic
626C2C127×#263111yescharacteristic
627C2C127×#266111yescharacteristic
628C2C127×#276812yes—
629C2C127×#277912yes—
630C2C127×#278812yes—
631C2C127×#280012yes—
632C2C127×#281611yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#389×C127C2semidirect
3. C127×#1444C2semidirect
4. C127×#1442C2semidirect
5. C127×#1442C2semidirect
6. C127×#1441C2semidirect
11. C127×#1434C2semidirect
304. #16978C127direct (N × H)
317. C2×#389C2×C127semidirect
319. #1444C2×C127semidirect
320. #1442C2×C127semidirect
321. #1442C2×C127semidirect
322. #1441C2×C127semidirect
327. #1434C2×C127semidirect
328. C127#16978direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#389×C1272
C2×#388×C1272
C127×#14442
C127×#14422
C127×#14412
C127×#14392
C127×#14342
#16978127