Finite Group Databasebeta

← C127×#17015 #44580688

Subgroups — C127×#17015 #44580688

Order 32512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398×C127C211yesmaximal, characteristic, complemented
2C2×#396×C127C212yesmaximal, complemented
3C2×#394×C127C211yesmaximal, characteristic
4C2×#388×C127C212yesmaximal, complemented
5C127×#1445C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1439C212yesmaximal, complemented
8C127×#1438C212yesmaximal
9C127×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2212yes—
20C22×C22⋊C4×C127C2212yes—
21C22×C22⋊C4×C127C2212yes—
22C22×C42×C127C2211yescharacteristic
23#398×C127—24no—
24#396×C127—28no—
25#394×C127—24no—
26#388×C127—28no—
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2212yes—
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C25×C127C2312yes—
36C22×D4×C127—24no—
37C22×D4×C127—22no—
38C22×D4×C127—42no—
39C22×D4×C127—41no—
40C22×D4×C127—41no—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2312yes—
48C23×C4×C127C2311yescharacteristic
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—24no—
55C2×C4⋊C4×C127—41no—
56C2×C4⋊C4×C127—41no—
57C2×C4⋊C4×C127—41no—
58C2×C4⋊C4×C127—41no—
59C2×C4⋊C4×C127—42no—
60C2×C4⋊C4×C127—41no—
61C2×C4⋊C4×C127—41no—
62C2×C4⋊C4×C127D414yes—
63C2×C4⋊C4×C127D414yescomplemented
64C2×C22⋊C4×C127—24no—
65C2×C22⋊C4×C127—24no—
66C2×C22⋊C4×C127—24no—
67C2×C22⋊C4×C127—24no—
68C2×C22⋊C4×C127—24no—
69C2×C22⋊C4×C127—24no—
70C2×C22⋊C4×C127—24no—
71C2×C22⋊C4×C127—24no—
72C2×C22⋊C4×C127—24no—
73C2×C22⋊C4×C127—24no—
74C2×C22⋊C4×C127—24no—
75C2×C22⋊C4×C127—24no—
76C2×C22⋊C4×C127—42no—
77C2×C22⋊C4×C127—42no—
78C2×C22⋊C4×C127—42no—
79C2×C22⋊C4×C127—42no—
80C2×C22⋊C4×C127—42no—
81C2×C22⋊C4×C127—42no—
82C2×C42×C127—41no—
83C2×C42×C127—41no—
84C2×C42×C127D414yescomplemented
85#95×C127—24no—
86#95×C127—24no—
87#95×C127—22no—
88#95×C127—24no—
89#95×C127—22no—
90#95×C127—24no—
91#95×C127—42no—
92#95×C127—42no—
93C24×C127—22no—
94C24×C127—22no—
95C24×C127—22no—
96C24×C127—22no—
97C24×C127—22no—
98C24×C127—22no—
99C24×C127—22no—
100C24×C127—22no—
101C24×C127—22no—
102C24×C127—22no—
103C24×C127—22no—
104C24×C127—22no—
105C24×C127—22no—
106C24×C127—22no—
107C24×C127#4114yes—
108C24×C127C2411yescharacteristic
109C2×D4×C127—48no—
110C2×D4×C127—44no—
111C2×D4×C127—44no—
112C2×D4×C127—48no—
113C2×D4×C127—44no—
114C22×C4×C127—21no—
115C22×C4×C127—21no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
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—22no—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—21no—
139C22×C4×C127—21no—
140C22×C4×C127—21no—
141C22×C4×C127—21no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—22no—
145C22×C4×C127—22no—
146C22×C4×C127—22no—
147C22×C4×C127—22no—
148C22×C4×C127—22no—
149C22×C4×C127—22no—
150C22×C4×C127—22no—
151C22×C4×C127—22no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127C2×D412yes—
159C22×C4×C127C2×D414yes—
160C22×C4×C127C2×D412yes—
161C22×C4×C127C2×D414yes—
162C22×C4×C127C2×D412yes—
163C22×C4×C127C2×D412yes—
164C22×C4×C127C2×D412yes—
165C22×C4×C127#4114yes—
166C4⋊C4×C127—44no—
167C4⋊C4×C127—44no—
168C4⋊C4×C127—48no—
169C4⋊C4×C127—44no—
170C4⋊C4×C127—48no—
171C22⋊C4×C127—216no—
172C22⋊C4×C127—48no—
173C22⋊C4×C127—48no—
174C22⋊C4×C127—48no—
175C22⋊C4×C127—48no—
176C22⋊C4×C127—48no—
177C42×C127—44no—
178C23×C127—24no—
179C23×C127—24no—
180C23×C127—24no—
181C23×C127—24no—
182C23×C127—24no—
183C23×C127—24no—
184C23×C127—24no—
185C23×C127—42no—
186C23×C127—42no—
187C23×C127—42no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127—42no—
192C23×C127—42no—
193C23×C127—42no—
194C23×C127—42no—
195C23×C127—42no—
196C23×C127—42no—
197C23×C127—42no—
198C23×C127—42no—
199C23×C127—42no—
200C23×C127—42no—
201C23×C127—42no—
202C23×C127—42no—
203C23×C127—42no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—42no—
207C23×C127—42no—
208C23×C127—42no—
209C23×C127—42no—
210C23×C127—42no—
211C23×C127—42no—
212C23×C127—42no—
213C23×C127C22×D411yescharacteristic
214C23×C127C22×D411yescharacteristic
215C23×C127C22×D411yescharacteristic
216C23×C127C2×#4112yes—
217C23×C127C2×#4112yes—
218C23×C127H(5,𝔽2)11yescharacteristic
219C23×C127H(5,𝔽2)11yescharacteristic
220C23×C127H(5,𝔽2)11yescharacteristic
221C23×C127H(5,𝔽2)12yes—
222C23×C1272−1+411yescharacteristic
223C23×C1272−1+412yes—
224D4×C127—416no—
225C2×C4×C127—22no—
226C2×C4×C127—22no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—24no—
232C2×C4×C127—24no—
233C2×C4×C127—24no—
234C2×C4×C127—24no—
235C2×C4×C127—22no—
236C2×C4×C127—22no—
237C2×C4×C127—24no—
238C2×C4×C127—24no—
239C2×C4×C127—24no—
240C2×C4×C127—24no—
241C2×C4×C127—24no—
242C2×C4×C127—24no—
243C2×C4×C127—24no—
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—41no—
257C2×C4×C127—41no—
258C2×C4×C127—42no—
259C2×C4×C127—41no—
260C2×C4×C127—42no—
261C2×C4×C127—41no—
262C2×C4×C127—42no—
263C2×C4×C127—42no—
264C2×C4×C127—42no—
265C2×C4×C127—42no—
266C2×C4×C127—42no—
267C2×C4×C127—42no—
268C2×C4×C127—42no—
269C2×C4×C127—42no—
270C2×C4×C127—42no—
271C2×C4×C127—42no—
272C2×C4×C127—42no—
273C2×C4×C127—42no—
274C2×C4×C127—42no—
275C2×C4×C127—42no—
276C2×C4×C127#12714yes—
277C22×C127—44no—
278C22×C127—44no—
279C22×C127—44no—
280C22×C127—44no—
281C22×C127—44no—
282C22×C127—44no—
283C22×C127—44no—
284C22×C127—82no—
285C22×C127—82no—
286C22×C127—82no—
287C22×C127—82no—
288C22×C127—82no—
289C22×C127—82no—
290C22×C127—82no—
291C22×C127—82no—
292C22×C127#52512yes—
293C22×C127C2×#12711yescharacteristic
294C22×C127#53411yescharacteristic
295C22×C127#53411yescharacteristic
296C22×C127#53411yescharacteristic
297C22×C127#53711yescharacteristic
298C22×C127#53712yes—
299C22×C127#53711yescharacteristic
300C22×C127#53711yescharacteristic
301C22×C127#53812yes—
302C22×C127#53912yes—
303C22×C127#54012yes—
304C22×C127#54212yes—
305C22×C127#54612yes—
306C22×C127#54612yes—
307C22×C127#54812yes—
308C22×C127#54812yes—
309C22×C127#55111yescharacteristic
310C22×C127#55212yes—
311C22×C127#55511yescharacteristic
312C22×C127#56011yescharacteristic
313C22×C127#56312yes—
314C22×C127#56311yescharacteristic
315C4×C127—24no—
316C4×C127—44no—
317C4×C127—42no—
318C4×C127—42no—
319C4×C127—44no—
320C4×C127—44no—
321C4×C127—82no—
322C4×C127—82no—
323#17015C12711yesmaximal, characteristic, direct product factor, Sylow
324C2×C127—84no—
325C2×C127#254012yes—
326C2×C127#260911yescharacteristic
327C2×C127#263012yes—
328C2×C127#265611yescharacteristic
329C2×C127#269711yescharacteristic
330C2×C127#269911yescharacteristic
331C2×C127#270212yes—
332C2×C127#270712yes—
333C2×C127#272811yescharacteristic
334C2×C127#273111yescharacteristic
335C2×C127#282311yescharacteristic
336C2×#398C2×C12711yescharacteristic, complemented
337C2×#396C2×C12712yescomplemented
338C2×#394C2×C12711yescharacteristic
339C2×#388C2×C12712yescomplemented
340#1445C2×C12712yescomplemented
341#1439C2×C12712yescomplemented
342#1439C2×C12712yescomplemented
343#1438C2×C12712yes—
344C4⋊#95C2×C12711yescharacteristic, complemented
345C127#1701511yescharacteristic, direct product factor, Sylow
346C23×D4C22×C12711yescharacteristic
347C22×C4⋊C4C22×C12712yes—
348C22×C4⋊C4C22×C12712yes—
349C22×C4⋊C4C22×C12711yescharacteristic
350C22×C4⋊C4C22×C12711yescharacteristic
351C22×C4⋊C4C22×C12711yescharacteristic
352C22×C22⋊C4C22×C12712yes—
353C22×C22⋊C4C22×C12712yes—
354C22×C22⋊C4C22×C12712yes—
355C22×C22⋊C4C22×C12712yes—
356C22×C22⋊C4C22×C12712yes—
357C22×C22⋊C4C22×C12712yes—
358C22×C42C22×C12711yescharacteristic
359#398—24no—
360#396—28no—
361#394—24no—
362#388—28no—
363C2×#95C22×C12711yescharacteristic
364C2×#95C22×C12712yes—
365C2×#95C22×C12712yes—
366C2×#95C22×C12711yescharacteristic
367C2×#95C22×C12712yes—
368C2×#95C22×C12712yes—
369C2×#95C22×C12712yes—
370C2×#95C22×C12712yes—
371C25C23×C12712yes—
372C22×D4—24no—
373C22×D4—22no—
374C22×D4—42no—
375C22×D4—41no—
376C22×D4—41no—
377C23×C4C23×C12712yes—
378C23×C4C23×C12712yes—
379C23×C4C23×C12711yescharacteristic
380C23×C4C23×C12712yes—
381C23×C4C23×C12712yes—
382C23×C4C23×C12711yescharacteristic
383C23×C4C23×C12712yes—
384C23×C4C23×C12711yescharacteristic
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4—41no—
393C2×C4⋊C4—41no—
394C2×C4⋊C4—41no—
395C2×C4⋊C4—42no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4D4×C12714yes—
399C2×C4⋊C4D4×C12714yescomplemented
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C42—41no—
419C2×C42—41no—
420C2×C42D4×C12714yescomplemented
421#95—24no—
422#95—24no—
423#95—22no—
424#95—24no—
425#95—22no—
426#95—24no—
427#95—42no—
428#95—42no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24#41×C12714yes—
444C24C24×C12711yescharacteristic
445C2×D4—48no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—48no—
449C2×D4—44no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
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—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4C2×D4×C12712yes—
495C22×C4C2×D4×C12714yes—
496C22×C4C2×D4×C12712yes—
497C22×C4C2×D4×C12714yes—
498C22×C4C2×D4×C12712yes—
499C22×C4C2×D4×C12712yes—
500C22×C4C2×D4×C12712yes—
501C22×C4#41×C12714yes—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C42—44no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C22×D4×C12711yescharacteristic
550C23C22×D4×C12711yescharacteristic
551C23C22×D4×C12711yescharacteristic
552C23C2×#41×C12712yes—
553C23C2×#41×C12712yes—
554C23H(5,𝔽2)×C12711yescharacteristic
555C23H(5,𝔽2)×C12711yescharacteristic
556C23H(5,𝔽2)×C12711yescharacteristic
557C23H(5,𝔽2)×C12712yes—
558C232−1+4×C12711yescharacteristic
559C232−1+4×C12712yes—
560D4—416no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4#127×C12714yes—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#525×C12712yes—
629C22C2×#127×C12711yescharacteristic
630C22#534×C12711yescharacteristic
631C22#534×C12711yescharacteristic
632C22#534×C12711yescharacteristic
633C22#537×C12711yescharacteristic
634C22#537×C12712yes—
635C22#537×C12711yescharacteristic
636C22#537×C12711yescharacteristic
637C22#538×C12712yes—
638C22#539×C12712yes—
639C22#540×C12712yes—
640C22#542×C12712yes—
641C22#546×C12712yes—
642C22#546×C12712yes—
643C22#548×C12712yes—
644C22#548×C12712yes—
645C22#551×C12711yescharacteristic
646C22#552×C12712yes—
647C22#555×C12711yescharacteristic
648C22#560×C12711yescharacteristic
649C22#563×C12712yes—
650C22#563×C12711yescharacteristic
651C4—24no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—82no—
659C2—84no—
660C2C127×#254012yes—
661C2C127×#260911yescharacteristic
662C2C127×#263012yes—
663C2C127×#265611yescharacteristic
664C2C127×#269711yescharacteristic
665C2C127×#269911yescharacteristic
666C2C127×#270212yes—
667C2C127×#270712yes—
668C2C127×#272811yescharacteristic
669C2C127×#273111yescharacteristic
670C2C127×#282311yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#398×C127C2semidirect
2. C2×#396×C127C2semidirect
4. C2×#388×C127C2semidirect
5. C127×#1445C2semidirect
6. C127×#1439C2semidirect
7. C127×#1439C2semidirect
9. C127×C4⋊#95C2semidirect
63. C2×C4⋊C4×C127D4semidirect
84. C2×C42×C127D4semidirect
323. #17015C127direct (N × H)
336. C2×#398C2×C127semidirect
337. C2×#396C2×C127semidirect
339. C2×#388C2×C127semidirect
340. #1445C2×C127semidirect
341. #1439C2×C127semidirect
342. #1439C2×C127semidirect
344. C4⋊#95C2×C127semidirect
345. C127#17015direct (N × H)
399. C2×C4⋊C4D4×C127semidirect
420. C2×C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#398×C1272
C2×#396×C1272
C2×#394×C1272
C2×#388×C1272
C127×#14452
C127×#14392
C127×#14382
C127×C4⋊#952
#17015127