Finite Group Databasebeta

← C2×C127×#1907 #44584674

Subgroups — C2×C127×#1907 #44584674

Order 32512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
2C22×#131×C127C211yesmaximal, characteristic, complemented
3C2×#417×C127C211yesmaximal, characteristic, complemented
4C22×#102×C127C211yesmaximal, characteristic, complemented
5C22×#100×C127C211yesmaximal, characteristic, complemented
6C127×#1907C218yesmaximal, direct product factor
7C2×#343×C127C211yesmaximal, characteristic, complemented
8C2×#341×C127C211yesmaximal, characteristic, complemented
9C22×#41×C127C2211yescharacteristic
10C23×D4×C127C2211yescharacteristic
11C2×AGL(1,ℤ/8)×C127—24no—
12C2×AGL(1,ℤ/8)×C127—22no—
13C2×AGL(1,ℤ/8)×C127C418yescomplemented
14C2×AGL(1,ℤ/8)×C127C2214yes—
15C22×SD16×C127—21no—
16C22×D8×C127—21no—
17C2×#131×C127—24no—
18C2×#131×C127—24no—
19C2×#131×C127—22no—
20C2×#131×C127C2214yescomplemented
21C22×M16×C127—21no—
22C22×M16×C127C2211yescharacteristic
23C22×M16×C127C2211yescharacteristic
24C22×M16×C127C2211yescharacteristic
25C23×C8×C127—21no—
26C23×C8×C127C2211yescharacteristic
27C22×C4⋊C4×C127C2211yescharacteristic
28#417×C127—22no—
29#417×C127C2214yescomplemented
30C2×#103×C127—21no—
31C2×#102×C127—21no—
32C2×#102×C127—21no—
33C2×#102×C127—21no—
34C2×#102×C127—22no—
35C2×#102×C127C2214yescomplemented
36C2×#100×C127—24no—
37C2×#100×C127C2214yescomplemented
38#343×C127C2214yescomplemented
39#341×C127C2214yescomplemented
40C25×C127—21no—
41C2×#41×C127—22no—
42C2×#41×C127—21no—
43C2×#41×C127—22no—
44C2×#41×C127—21no—
45C2×#41×C127C2×C414yes—
46C2×#41×C127D414yes—
47C2×#41×C127D412yes—
48C2×#41×C127C2312yes—
49C22×Q8×C127D411yescharacteristic
50C22×D4×C127—21no—
51C22×D4×C127—21no—
52C22×D4×C127—22no—
53C22×D4×C127—21no—
54C22×D4×C127—21no—
55C22×D4×C127—21no—
56C22×D4×C127—22no—
57C22×D4×C127—22no—
58C22×D4×C127—21no—
59C22×D4×C127C2×C414yes—
60C22×D4×C127D411yescharacteristic
61C22×D4×C127C2312yes—
62C23×C4×C127—21no—
63C23×C4×C127—21no—
64C23×C4×C127C2311yescharacteristic
65AGL(1,ℤ/8)×C127—216no—
66AGL(1,ℤ/8)×C127—28no—
67AGL(1,ℤ/8)×C127C2×C4116yescomplemented
68C2×SD16×C127—28no—
69C2×SD16×C127—24no—
70C2×D8×C127—28no—
71C2×D8×C127—24no—
72#131×C127—216no—
73#131×C127—28no—
74#131×C127—28no—
75C2×M16×C127—28no—
76C2×M16×C127—22no—
77C2×M16×C127—21no—
78C2×M16×C127—22no—
79C2×M16×C127—21no—
80C2×M16×C127—21no—
81C2×M16×C127—24no—
82C2×M16×C127—22no—
83C2×M16×C127—22no—
84C2×M16×C127C2×C414yes—
85C2×M16×C127D414yes—
86C2×M16×C127C2312yes—
87C2×M16×C127C2312yes—
88C2×M16×C127C2312yes—
89C22×C8×C127—28no—
90C22×C8×C127—21no—
91C22×C8×C127—22no—
92C22×C8×C127—22no—
93C22×C8×C127—21no—
94C22×C8×C127—24no—
95C22×C8×C127—21no—
96C22×C8×C127D411yescharacteristic
97C22×C8×C127D411yescharacteristic
98C22×C8×C127D412yes—
99C22×C8×C127C2312yes—
100C2×C4⋊C4×C127—21no—
101C2×C4⋊C4×C127—21no—
102C2×C4⋊C4×C127—21no—
103C2×C4⋊C4×C127—21no—
104C2×C4⋊C4×C127—21no—
105C2×C4⋊C4×C127C2312yes—
106#103×C127—24no—
107#102×C127—24no—
108#102×C127—24no—
109#102×C127—24no—
110#100×C127—28no—
111C24×C127—21no—
112C24×C127—24no—
113C24×C127—22no—
114C24×C127—41no—
115C24×C127—41no—
116C24×C127—42no—
117C24×C127—82no—
118C24×C127—82no—
119C24×C127#4111yescharacteristic
120#41×C127—24no—
121#41×C127—24no—
122#41×C127—24no—
123#41×C127—24no—
124#41×C127—24no—
125#41×C127—22no—
126#41×C127—22no—
127#41×C127C22⋊C418yes—
128#41×C127C22×C414yes—
129#41×C127C2×D414yes—
130C2×Q8×C127—21no—
131C2×Q8×C127—22no—
132C2×Q8×C127C22⋊C414yes—
133C2×Q8×C127C2×D412yes—
134C2×D4×C127—24no—
135C2×D4×C127—22no—
136C2×D4×C127—24no—
137C2×D4×C127—22no—
138C2×D4×C127—24no—
139C2×D4×C127—24no—
140C2×D4×C127—24no—
141C2×D4×C127—24no—
142C2×D4×C127—24no—
143C2×D4×C127—22no—
144C2×D4×C127—21no—
145C2×D4×C127—22no—
146C2×D4×C127—22no—
147C2×D4×C127—22no—
148C2×D4×C127—22no—
149C2×D4×C127—22no—
150C2×D4×C127—42no—
151C2×D4×C127—42no—
152C2×D4×C127—42no—
153C2×D4×C127—42no—
154C2×D4×C127—41no—
155C2×D4×C127—41no—
156C2×D4×C127—42no—
157C2×D4×C127C22⋊C414yes—
158C2×D4×C127C22×C414yes—
159C2×D4×C127C2×D412yes—
160C22×C4×C127—21no—
161C22×C4×C127—24no—
162C22×C4×C127—24no—
163C22×C4×C127—22no—
164C22×C4×C127—22no—
165C22×C4×C127—22no—
166C22×C4×C127—41no—
167C22×C4×C127—41no—
168C22×C4×C127—82no—
169C22×C4×C127C22×C412yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127C2×D411yescharacteristic
172C22×C4×C127C2×D411yescharacteristic
173C22×C4×C127C2×D411yescharacteristic
174C22×C4×C127C2×D411yescharacteristic
175C22×C4×C127C2×D412yes—
176C22×C4×C127#4111yescharacteristic
177C22×C4×C127#4112yes—
178C22×C4×C127C2411yescharacteristic
179SD16×C127—216no—
180D8×C127—216no—
181M16×C127—216no—
182M16×C127—24no—
183M16×C127—24no—
184M16×C127—22no—
185M16×C127—22no—
186M16×C127—22no—
187M16×C127—22no—
188M16×C127—42no—
189M16×C127C22×C414yes—
190M16×C127C2×D414yes—
191C2×C8×C127—216no—
192C2×C8×C127—24no—
193C2×C8×C127—22no—
194C2×C8×C127—24no—
195C2×C8×C127—22no—
196C2×C8×C127—24no—
197C2×C8×C127—22no—
198C2×C8×C127—22no—
199C2×C8×C127—22no—
200C2×C8×C127—22no—
201C2×C8×C127—22no—
202C2×C8×C127—42no—
203C2×C8×C127C2×D412yes—
204C2×C8×C127C2×D412yes—
205C4⋊C4×C127—22no—
206C4⋊C4×C127—22no—
207C4⋊C4×C127—41no—
208C4⋊C4×C127—41no—
209C23×C127—22no—
210C23×C127—22no—
211C23×C127—22no—
212C23×C127—24no—
213C23×C127—24no—
214C23×C127—42no—
215C23×C127—42no—
216C23×C127—42no—
217C23×C127—42no—
218C23×C127—44no—
219C23×C127—44no—
220C23×C127—88no—
221C23×C127—82no—
222C23×C127—82no—
223C23×C127—88no—
224C23×C127—82no—
225C23×C127—82no—
226C23×C127—82no—
227C23×C127—82no—
228C23×C127C4×D412yes—
229C23×C127#12111yescharacteristic
230C23×C127#12111yescharacteristic
231C23×C127#12312yes—
232C23×C127C2×#4111yescharacteristic
233Q8×C127—22no—
234Q8×C127—24no—
235Q8×C127C2×C22⋊C414yes—
236D4×C127—24no—
237D4×C127—24no—
238D4×C127—28no—
239D4×C127—24no—
240D4×C127—24no—
241D4×C127—22no—
242D4×C127—24no—
243D4×C127—42no—
244D4×C127—42no—
245D4×C127—44no—
246D4×C127—44no—
247D4×C127C2×C22⋊C414yes—
248C2×C4×C127—22no—
249C2×C4×C127—24no—
250C2×C4×C127—24no—
251C2×C4×C127—24no—
252C2×C4×C127—28no—
253C2×C4×C127—42no—
254C2×C4×C127—82no—
255C2×C4×C127—82no—
256C2×C4×C127—82no—
257C2×C4×C127C2×C22⋊C412yes—
258C2×C4×C127C2×C22⋊C412yes—
259C2×C4×C127C4×D412yes—
260C2×C4×C127C4×D412yes—
261C2×C4×C127C4×D412yes—
262C2×C4×C127#12012yes—
263C2×C4×C127#12012yes—
264C2×C4×C127#12112yes—
265C2×C4×C127#12112yes—
266C2×C4×C127#12111yescharacteristic
267C2×C4×C127#12111yescharacteristic
268C2×C4×C127#12312yes—
269C2×C4×C127C23×C411yescharacteristic
270C2×C4×C127C22×D411yescharacteristic
271C2×C4×C127C22×D411yescharacteristic
272C2×C4×C127C22×D411yescharacteristic
273C2×C4×C127C22×D411yescharacteristic
274C2×C4×C127C2×#4111yescharacteristic
275C8×C127—24no—
276C8×C127—22no—
277C8×C127—24no—
278C8×C127—22no—
279C8×C127—42no—
280C22×C127—24no—
281C22×C127—24no—
282C22×C127—21no—
283C22×C127—21no—
284C22×C127—21no—
285C22×C127—21no—
286C22×C127—24no—
287C22×C127—48no—
288C22×C127—42no—
289C22×C127—42no—
290C22×C127—44no—
291C22×C127—84no—
292C22×C127—84no—
293C22×C127—88no—
294C22×C127—88no—
295C22×C127—82no—
296C22×C127—82no—
297C22×C127#38612yes—
298C22×C127#38612yes—
299C22×C127#47112yes—
300C22×C127#47212yes—
301C22×C127C2×C4×D411yescharacteristic
302C22×C127C2×#12111yescharacteristic
303C22×C127C2×#12311yescharacteristic
304C4×C127—24no—
305C4×C127—82no—
306C4×C127#38612yes—
307C4×C127#38612yes—
308C4×C127C22×C22⋊C411yescharacteristic
309C4×C127C2×C4×D411yescharacteristic
310C4×C127C2×#12011yescharacteristic
311C4×C127C2×#12111yescharacteristic
312C2×#1907C12711yesmaximal, characteristic, direct product factor, Sylow
313C2×C127—21no—
314C2×C127—21no—
315C2×C127—22no—
316C2×C127—44no—
317C2×C127—84no—
318C2×C127#190714yesdirect product factor
319C2×C127C2×#38611yescharacteristic
320C2×C127C2×#47111yescharacteristic
321C2×C127C2×#47211yescharacteristic
322C22×AGL(1,ℤ/8)C2×C12711yescharacteristic
323C22×#131C2×C12711yescharacteristic, complemented
324C2×#417C2×C12711yescharacteristic, complemented
325C22×#102C2×C12711yescharacteristic, complemented
326C22×#100C2×C12711yescharacteristic, complemented
327#1907C2×C12718yesdirect product factor
328C2×#343C2×C12711yescharacteristic, complemented
329C2×#341C2×C12711yescharacteristic, complemented
330C127C2×#190711yescharacteristic, direct product factor, Sylow
331C22×#41C22×C12711yescharacteristic
332C23×D4C22×C12711yescharacteristic
333C2×AGL(1,ℤ/8)—24no—
334C2×AGL(1,ℤ/8)—22no—
335C2×AGL(1,ℤ/8)C4×C12718yescomplemented
336C2×AGL(1,ℤ/8)C22×C12714yes—
337C22×SD16—21no—
338C22×D8—21no—
339C2×#131—24no—
340C2×#131—24no—
341C2×#131—22no—
342C2×#131C22×C12714yescomplemented
343C22×M16—21no—
344C22×M16C22×C12711yescharacteristic
345C22×M16C22×C12711yescharacteristic
346C22×M16C22×C12711yescharacteristic
347C23×C8—21no—
348C23×C8C22×C12711yescharacteristic
349C22×C4⋊C4C22×C12711yescharacteristic
350#417—22no—
351#417C22×C12714yescomplemented
352C2×#103—21no—
353C2×#102—21no—
354C2×#102—21no—
355C2×#102—21no—
356C2×#102—22no—
357C2×#102C22×C12714yescomplemented
358C2×#100—24no—
359C2×#100C22×C12714yescomplemented
360#343C22×C12714yescomplemented
361#341C22×C12714yescomplemented
362C25—21no—
363C2×#41—22no—
364C2×#41—21no—
365C2×#41—22no—
366C2×#41—21no—
367C2×#41C2×C4×C12714yes—
368C2×#41D4×C12714yes—
369C2×#41D4×C12712yes—
370C2×#41C23×C12712yes—
371C22×Q8D4×C12711yescharacteristic
372C22×D4—21no—
373C22×D4—21no—
374C22×D4—22no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—22no—
380C22×D4—21no—
381C22×D4C2×C4×C12714yes—
382C22×D4D4×C12711yescharacteristic
383C22×D4C23×C12712yes—
384C23×C4—21no—
385C23×C4—21no—
386C23×C4C23×C12711yescharacteristic
387AGL(1,ℤ/8)—216no—
388AGL(1,ℤ/8)—28no—
389AGL(1,ℤ/8)C2×C4×C127116yescomplemented
390C2×SD16—28no—
391C2×SD16—24no—
392C2×D8—28no—
393C2×D8—24no—
394#131—216no—
395#131—28no—
396#131—28no—
397C2×M16—28no—
398C2×M16—22no—
399C2×M16—21no—
400C2×M16—22no—
401C2×M16—21no—
402C2×M16—21no—
403C2×M16—24no—
404C2×M16—22no—
405C2×M16—22no—
406C2×M16C2×C4×C12714yes—
407C2×M16D4×C12714yes—
408C2×M16C23×C12712yes—
409C2×M16C23×C12712yes—
410C2×M16C23×C12712yes—
411C22×C8—28no—
412C22×C8—21no—
413C22×C8—22no—
414C22×C8—22no—
415C22×C8—21no—
416C22×C8—24no—
417C22×C8—21no—
418C22×C8D4×C12711yescharacteristic
419C22×C8D4×C12711yescharacteristic
420C22×C8D4×C12712yes—
421C22×C8C23×C12712yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4C23×C12712yes—
428#103—24no—
429#102—24no—
430#102—24no—
431#102—24no—
432#100—28no—
433C24—21no—
434C24—24no—
435C24—22no—
436C24—41no—
437C24—41no—
438C24—42no—
439C24—82no—
440C24—82no—
441C24#41×C12711yescharacteristic
442#41—24no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—24no—
447#41—22no—
448#41—22no—
449#41C22⋊C4×C12718yes—
450#41C22×C4×C12714yes—
451#41C2×D4×C12714yes—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8C22⋊C4×C12714yes—
455C2×Q8C2×D4×C12712yes—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—42no—
479C2×D4C22⋊C4×C12714yes—
480C2×D4C22×C4×C12714yes—
481C2×D4C2×D4×C12712yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—82no—
491C22×C4C22×C4×C12712yes—
492C22×C4C2×D4×C12712yes—
493C22×C4C2×D4×C12711yescharacteristic
494C22×C4C2×D4×C12711yescharacteristic
495C22×C4C2×D4×C12711yescharacteristic
496C22×C4C2×D4×C12711yescharacteristic
497C22×C4C2×D4×C12712yes—
498C22×C4#41×C12711yescharacteristic
499C22×C4#41×C12712yes—
500C22×C4C24×C12711yescharacteristic
501SD16—216no—
502D8—216no—
503M16—216no—
504M16—24no—
505M16—24no—
506M16—22no—
507M16—22no—
508M16—22no—
509M16—22no—
510M16—42no—
511M16C22×C4×C12714yes—
512M16C2×D4×C12714yes—
513C2×C8—216no—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—42no—
525C2×C8C2×D4×C12712yes—
526C2×C8C2×D4×C12712yes—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—88no—
543C23—82no—
544C23—82no—
545C23—88no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23C4×D4×C12712yes—
551C23#121×C12711yescharacteristic
552C23#121×C12711yescharacteristic
553C23#123×C12712yes—
554C23C2×#41×C12711yescharacteristic
555Q8—22no—
556Q8—24no—
557Q8C2×C22⋊C4×C12714yes—
558D4—24no—
559D4—24no—
560D4—28no—
561D4—24no—
562D4—24no—
563D4—22no—
564D4—24no—
565D4—42no—
566D4—42no—
567D4—44no—
568D4—44no—
569D4C2×C22⋊C4×C12714yes—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—42no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4C2×C22⋊C4×C12712yes—
580C2×C4C2×C22⋊C4×C12712yes—
581C2×C4C4×D4×C12712yes—
582C2×C4C4×D4×C12712yes—
583C2×C4C4×D4×C12712yes—
584C2×C4#120×C12712yes—
585C2×C4#120×C12712yes—
586C2×C4#121×C12712yes—
587C2×C4#121×C12712yes—
588C2×C4#121×C12711yescharacteristic
589C2×C4#121×C12711yescharacteristic
590C2×C4#123×C12712yes—
591C2×C4C23×C4×C12711yescharacteristic
592C2×C4C22×D4×C12711yescharacteristic
593C2×C4C22×D4×C12711yescharacteristic
594C2×C4C22×D4×C12711yescharacteristic
595C2×C4C22×D4×C12711yescharacteristic
596C2×C4C2×#41×C12711yescharacteristic
597C8—24no—
598C8—22no—
599C8—24no—
600C8—22no—
601C8—42no—
602C22—24no—
603C22—24no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—24no—
609C22—48no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—84no—
614C22—84no—
615C22—88no—
616C22—88no—
617C22—82no—
618C22—82no—
619C22#386×C12712yes—
620C22#386×C12712yes—
621C22#471×C12712yes—
622C22#472×C12712yes—
623C22C2×C4×D4×C12711yescharacteristic
624C22C2×#121×C12711yescharacteristic
625C22C2×#123×C12711yescharacteristic
626C4—24no—
627C4—82no—
628C4#386×C12712yes—
629C4#386×C12712yes—
630C4C22×C22⋊C4×C12711yescharacteristic
631C4C2×C4×D4×C12711yescharacteristic
632C4C2×#120×C12711yescharacteristic
633C4C2×#121×C12711yescharacteristic
634C2—21no—
635C2—21no—
636C2—22no—
637C2—44no—
638C2—84no—
639C2C127×#190714yesdirect product factor
640C2C2×#386×C12711yescharacteristic
641C2C2×#471×C12711yescharacteristic
642C2C2×#472×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#121
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×AGL(1,ℤ/8)×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#131×C127C2semidirect
3. C2×#417×C127C2semidirect
4. C22×#102×C127C2semidirect
5. C22×#100×C127C2semidirect
6. C127×#1907C2direct (N × H)
7. C2×#343×C127C2semidirect
8. C2×#341×C127C2semidirect
13. C2×AGL(1,ℤ/8)×C127C4semidirect
20. C2×#131×C127C22semidirect
29. #417×C127C22semidirect
35. C2×#102×C127C22semidirect
37. C2×#100×C127C22semidirect
38. #343×C127C22semidirect
39. #341×C127C22semidirect
67. AGL(1,ℤ/8)×C127C2×C4semidirect
312. C2×#1907C127direct (N × H)
318. C2×C127#1907direct (N × H)
323. C22×#131C2×C127semidirect
324. C2×#417C2×C127semidirect
325. C22×#102C2×C127semidirect
326. C22×#100C2×C127semidirect
327. #1907C2×C127direct (N × H)
328. C2×#343C2×C127semidirect
329. C2×#341C2×C127semidirect
330. C127C2×#1907direct (N × H)
335. C2×AGL(1,ℤ/8)C4×C127semidirect
342. C2×#131C22×C127semidirect
351. #417C22×C127semidirect
357. C2×#102C22×C127semidirect
359. C2×#100C22×C127semidirect
360. #343C22×C127semidirect
361. #341C22×C127semidirect
389. AGL(1,ℤ/8)C2×C4×C127semidirect
639. C2C127×#1907direct (N × H)

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)×C1272
C22×#131×C1272
C2×#417×C1272
C22×#102×C1272
C22×#100×C1272
C127×#19072
C2×#343×C1272
C2×#341×C1272
C2×#1907127