Finite Group Databasebeta

← C127×#16810 #44580483

Subgroups — C127×#16810 #44580483

Order 32512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#397×C127C212yesmaximal, complemented
2C2×#392×C127C212yesmaximal, complemented
3C2×#392×C127C212yesmaximal, complemented
4C2×#390×C127C211yesmaximal, characteristic, complemented
5C2×#390×C127C211yesmaximal, characteristic, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1438C214yesmaximal, complemented
8C127×#1434C211yesmaximal, characteristic, complemented
9C23×D4×C127C2211yescharacteristic
10C23×D4×C127C2214yes—
11C23×D4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2214yes—
17C22×C22⋊C4×C127C2214yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2212yes—
20C22×C42×C127C2211yescharacteristic
21#397×C127—28no—
22#392×C127—28no—
23#392×C127—28no—
24#390×C127—24no—
25#390×C127—24no—
26C2×#95×C127C2212yes—
27C2×#95×C127C2214yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C25×C127C2312yes—
31C25×C127C2312yes—
32C22×D4×C127—24no—
33C22×D4×C127—28no—
34C22×D4×C127—28no—
35C22×D4×C127—22no—
36C22×D4×C127—22no—
37C22×D4×C127—28no—
38C22×D4×C127—24no—
39C22×D4×C127—22no—
40C22×D4×C127—42no—
41C22×D4×C127—42no—
42C22×D4×C127—42no—
43C22×D4×C127—44no—
44C22×D4×C127—44no—
45C22×D4×C127—44no—
46C22×D4×C127—44no—
47C22×D4×C127D414yescomplemented
48C23×C4×C127C2312yes—
49C23×C4×C127C2312yes—
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2314yes—
52C23×C4×C127C2311yescharacteristic
53C23×C4×C127C2311yescharacteristic
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127—24no—
57C2×C4⋊C4×C127—42no—
58C2×C22⋊C4×C127—28no—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127—28no—
64C2×C22⋊C4×C127—24no—
65C2×C22⋊C4×C127—24no—
66C2×C22⋊C4×C127—28no—
67C2×C22⋊C4×C127—24no—
68C2×C22⋊C4×C127—24no—
69C2×C22⋊C4×C127—44no—
70C2×C22⋊C4×C127—44no—
71C2×C22⋊C4×C127—44no—
72C2×C22⋊C4×C127—44no—
73C2×C22⋊C4×C127—44no—
74C2×C42×C127—42no—
75C2×C42×C127D414yescomplemented
76#95×C127—24no—
77#95×C127—28no—
78#95×C127—42no—
79#95×C127D418yescomplemented
80C24×C127—22no—
81C24×C127—22no—
82C24×C127—22no—
83C24×C127—22no—
84C24×C127—22no—
85C24×C127—24no—
86C24×C127—22no—
87C24×C127—22no—
88C24×C127—22no—
89C24×C127—24no—
90C24×C127—22no—
91C24×C127—22no—
92C24×C127—24no—
93C24×C127—24no—
94C24×C127—24no—
95C24×C127—24no—
96C24×C127—24no—
97C24×C127—22no—
98C24×C127—24no—
99C24×C127—22no—
100C24×C127C2×D414yes—
101C24×C127#4114yes—
102C24×C127C2411yescharacteristic
103C2×D4×C127—28no—
104C2×D4×C127—28no—
105C2×D4×C127—48no—
106C2×D4×C127—416no—
107C2×D4×C127—416no—
108C2×D4×C127—416no—
109C2×D4×C127—44no—
110C2×D4×C127—44no—
111C2×D4×C127—44no—
112C2×D4×C127—48no—
113C2×D4×C127—416no—
114C2×D4×C127—416no—
115C2×D4×C127—416no—
116C2×D4×C127—416no—
117C2×D4×C127—416no—
118C2×D4×C127—44no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—24no—
124C22×C4×C127—24no—
125C22×C4×C127—22no—
126C22×C4×C127—24no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—24no—
130C22×C4×C127—24no—
131C22×C4×C127—24no—
132C22×C4×C127—22no—
133C22×C4×C127—24no—
134C22×C4×C127—22no—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—24no—
140C22×C4×C127—24no—
141C22×C4×C127—24no—
142C22×C4×C127C2×D418yes—
143C22×C4×C127C2×D412yes—
144C22×C4×C127C2×D412yes—
145C22×C4×C127C2×D412yes—
146C22×C4×C127C2×D414yes—
147C22×C4×C127C2×D412yes—
148C22×C4×C127#4112yes—
149C4⋊C4×C127—44no—
150C4⋊C4×C127—44no—
151C22⋊C4×C127—416no—
152C22⋊C4×C127—48no—
153C22⋊C4×C127—48no—
154C22⋊C4×C127—416no—
155C22⋊C4×C127—48no—
156C22⋊C4×C127—48no—
157C42×C127—44no—
158C23×C127—24no—
159C23×C127—24no—
160C23×C127—28no—
161C23×C127—28no—
162C23×C127—24no—
163C23×C127—24no—
164C23×C127—24no—
165C23×C127—28no—
166C23×C127—28no—
167C23×C127—24no—
168C23×C127—42no—
169C23×C127—42no—
170C23×C127—42no—
171C23×C127—42no—
172C23×C127—42no—
173C23×C127—44no—
174C23×C127—42no—
175C23×C127—42no—
176C23×C127—44no—
177C23×C127—42no—
178C23×C127—42no—
179C23×C127—44no—
180C23×C127—44no—
181C23×C127—42no—
182C23×C127—44no—
183C23×C127—42no—
184C23×C127—42no—
185C23×C127—44no—
186C23×C127—44no—
187C23×C127—44no—
188C23×C127—44no—
189C23×C127—44no—
190C23×C127—44no—
191C23×C127—44no—
192C23×C127—44no—
193C23×C127—44no—
194C23×C127—42no—
195C23×C127—44no—
196C23×C127—42no—
197C23×C127—42no—
198C23×C127—42no—
199C23×C127—44no—
200C23×C127—44no—
201C23×C127—44no—
202C23×C127—44no—
203C23×C127—44no—
204C23×C127C22×D411yescharacteristic
205C23×C127C22×D411yescharacteristic
206C23×C127C22×D412yes—
207C23×C127C2×#4111yescharacteristic
208C23×C127C2×#4112yes—
209C23×C127H(5,𝔽2)12yes—
210C23×C127H(5,𝔽2)12yes—
211C23×C127H(5,𝔽2)12yes—
212C23×C127H(5,𝔽2)12yes—
213D4×C127—416no—
214D4×C127—416no—
215D4×C127—464no—
216C2×C4×C127—22no—
217C2×C4×C127—22no—
218C2×C4×C127—22no—
219C2×C4×C127—24no—
220C2×C4×C127—22no—
221C2×C4×C127—24no—
222C2×C4×C127—28no—
223C2×C4×C127—22no—
224C2×C4×C127—22no—
225C2×C4×C127—22no—
226C2×C4×C127—28no—
227C2×C4×C127—28no—
228C2×C4×C127—28no—
229C2×C4×C127—44no—
230C2×C4×C127—44no—
231C2×C4×C127—44no—
232C2×C4×C127—44no—
233C2×C4×C127—44no—
234C2×C4×C127—42no—
235C2×C4×C127—42no—
236C2×C4×C127—42no—
237C2×C4×C127—44no—
238C2×C4×C127—42no—
239C2×C4×C127—42no—
240C2×C4×C127—42no—
241C2×C4×C127—44no—
242C2×C4×C127—44no—
243C2×C4×C127—44no—
244C2×C4×C127#12114yes—
245C2×C4×C127#12714yes—
246C22×C127—44no—
247C22×C127—44no—
248C22×C127—48no—
249C22×C127—48no—
250C22×C127—48no—
251C22×C127—48no—
252C22×C127—44no—
253C22×C127—44no—
254C22×C127—44no—
255C22×C127—44no—
256C22×C127—82no—
257C22×C127—82no—
258C22×C127—82no—
259C22×C127—82no—
260C22×C127—84no—
261C22×C127—84no—
262C22×C127—84no—
263C22×C127—84no—
264C22×C127—82no—
265C22×C127—82no—
266C22×C127—82no—
267C22×C127—82no—
268C22×C127C2×#12111yescharacteristic
269C22×C127#52511yescharacteristic
270C22×C127C2×#12711yescharacteristic
271C22×C127#53412yes—
272C22×C127#53411yescharacteristic
273C22×C127#53511yescharacteristic
274C22×C127#53511yescharacteristic
275C22×C127#53514yes—
276C22×C127#53511yescharacteristic
277C22×C127#53812yes—
278C22×C127#53814yes—
279C22×C127D4212yes—
280C22×C127#54612yes—
281C22×C127#54614yes—
282C22×C127#55012yes—
283C22×C127#55312yes—
284C22×C127#56014yes—
285C4×C127—24no—
286C4×C127—24no—
287C4×C127—48no—
288C4×C127—42no—
289C4×C127—82no—
290C4×C127—82no—
291#16810C12711yesmaximal, characteristic, direct product factor, Sylow
292C2×C127—84no—
293C2×C127—84no—
294C2×C127#253511yescharacteristic
295C2×C127#253911yescharacteristic
296C2×C127#254111yescharacteristic
297C2×C127#261711yescharacteristic
298C2×C127#263111yescharacteristic
299C2×C127#265711yescharacteristic
300C2×C127#267814yes—
301C2×C127#271414yes—
302C2×C127#281511yescharacteristic
303C2×#397C2×C12712yescomplemented
304C2×#392C2×C12712yescomplemented
305C2×#392C2×C12712yescomplemented
306C2×#390C2×C12711yescharacteristic, complemented
307C2×#390C2×C12711yescharacteristic, complemented
308#1439C2×C12712yescomplemented
309#1438C2×C12714yescomplemented
310#1434C2×C12711yescharacteristic, complemented
311C127#1681011yescharacteristic, direct product factor, Sylow
312C23×D4C22×C12711yescharacteristic
313C23×D4C22×C12714yes—
314C23×D4C22×C12711yescharacteristic
315C22×C4⋊C4C22×C12711yescharacteristic
316C22×C4⋊C4C22×C12711yescharacteristic
317C22×C22⋊C4C22×C12712yes—
318C22×C22⋊C4C22×C12712yes—
319C22×C22⋊C4C22×C12714yes—
320C22×C22⋊C4C22×C12714yes—
321C22×C22⋊C4C22×C12712yes—
322C22×C22⋊C4C22×C12712yes—
323C22×C42C22×C12711yescharacteristic
324#397—28no—
325#392—28no—
326#392—28no—
327#390—24no—
328#390—24no—
329C2×#95C22×C12712yes—
330C2×#95C22×C12714yes—
331C2×#95C22×C12712yes—
332C2×#95C22×C12712yes—
333C25C23×C12712yes—
334C25C23×C12712yes—
335C22×D4—24no—
336C22×D4—28no—
337C22×D4—28no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—28no—
341C22×D4—24no—
342C22×D4—22no—
343C22×D4—42no—
344C22×D4—42no—
345C22×D4—42no—
346C22×D4—44no—
347C22×D4—44no—
348C22×D4—44no—
349C22×D4—44no—
350C22×D4D4×C12714yescomplemented
351C23×C4C23×C12712yes—
352C23×C4C23×C12712yes—
353C23×C4C23×C12711yescharacteristic
354C23×C4C23×C12714yes—
355C23×C4C23×C12711yescharacteristic
356C23×C4C23×C12711yescharacteristic
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—42no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—44no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4—44no—
377C2×C42—42no—
378C2×C42D4×C12714yescomplemented
379#95—24no—
380#95—28no—
381#95—42no—
382#95D4×C12718yescomplemented
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—22no—
401C24—24no—
402C24—22no—
403C24C2×D4×C12714yes—
404C24#41×C12714yes—
405C24C24×C12711yescharacteristic
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—48no—
416C2×D4—416no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—44no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4C2×D4×C12718yes—
446C22×C4C2×D4×C12712yes—
447C22×C4C2×D4×C12712yes—
448C22×C4C2×D4×C12712yes—
449C22×C4C2×D4×C12714yes—
450C22×C4C2×D4×C12712yes—
451C22×C4#41×C12712yes—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C42—44no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C22×D4×C12711yescharacteristic
508C23C22×D4×C12711yescharacteristic
509C23C22×D4×C12712yes—
510C23C2×#41×C12711yescharacteristic
511C23C2×#41×C12712yes—
512C23H(5,𝔽2)×C12712yes—
513C23H(5,𝔽2)×C12712yes—
514C23H(5,𝔽2)×C12712yes—
515C23H(5,𝔽2)×C12712yes—
516D4—416no—
517D4—416no—
518D4—464no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4#121×C12714yes—
548C2×C4#127×C12714yes—
549C22—44no—
550C22—44no—
551C22—48no—
552C22—48no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22C2×#121×C12711yescharacteristic
572C22#525×C12711yescharacteristic
573C22C2×#127×C12711yescharacteristic
574C22#534×C12712yes—
575C22#534×C12711yescharacteristic
576C22#535×C12711yescharacteristic
577C22#535×C12711yescharacteristic
578C22#535×C12714yes—
579C22#535×C12711yescharacteristic
580C22#538×C12712yes—
581C22#538×C12714yes—
582C22D42×C12712yes—
583C22#546×C12712yes—
584C22#546×C12714yes—
585C22#550×C12712yes—
586C22#553×C12712yes—
587C22#560×C12714yes—
588C4—24no—
589C4—24no—
590C4—48no—
591C4—42no—
592C4—82no—
593C4—82no—
594C2—84no—
595C2—84no—
596C2C127×#253511yescharacteristic
597C2C127×#253911yescharacteristic
598C2C127×#254111yescharacteristic
599C2C127×#261711yescharacteristic
600C2C127×#263111yescharacteristic
601C2C127×#265711yescharacteristic
602C2C127×#267814yes—
603C2C127×#271414yes—
604C2C127×#281511yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#397×C127C2semidirect
2. C2×#392×C127C2semidirect
3. C2×#392×C127C2semidirect
4. C2×#390×C127C2semidirect
5. C2×#390×C127C2semidirect
6. C127×#1439C2semidirect
7. C127×#1438C2semidirect
8. C127×#1434C2semidirect
47. C22×D4×C127D4semidirect
75. C2×C42×C127D4semidirect
79. #95×C127D4semidirect
291. #16810C127direct (N × H)
303. C2×#397C2×C127semidirect
304. C2×#392C2×C127semidirect
305. C2×#392C2×C127semidirect
306. C2×#390C2×C127semidirect
307. C2×#390C2×C127semidirect
308. #1439C2×C127semidirect
309. #1438C2×C127semidirect
310. #1434C2×C127semidirect
311. C127#16810direct (N × H)
350. C22×D4D4×C127semidirect
378. C2×C42D4×C127semidirect
382. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#397×C1272
C2×#392×C1272
C2×#390×C1272
C127×#14392
C127×#14382
C127×#14342
#16810127