Finite Group Databasebeta

← C127×#21006 #44584679

Subgroups — C127×#21006 #44584679

Order 32512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#137×C127C211yesmaximal, characteristic
2C2×#568×C127C211yesmaximal, characteristic
3C127×#2896C211yesmaximal, characteristic
4C2×#419×C127C211yesmaximal, characteristic, complemented
5C22×#101×C127C211yesmaximal, characteristic, complemented
6C127×#1908C218yesmaximal, complemented
7C127×#1742C211yesmaximal, characteristic, complemented
8C127×#1738C211yesmaximal, characteristic, complemented
9C22×#41×C127C2211yescharacteristic
10C23×Q8×C127C2211yescharacteristic
11C2×#137×C127—24no—
12C2×#137×C127—22no—
13C2×#137×C127C418yescomplemented
14C2×#137×C127C2214yes—
15C22×Q16×C127—21no—
16C22×SD16×C127—21no—
17#568×C127—24no—
18#568×C127—24no—
19C2×#131×C127—22no—
20C2×#131×C127C2214yes—
21C22×M16×C127—22no—
22C22×M16×C127C2211yescharacteristic
23C22×M16×C127C2211yescharacteristic
24C22×M16×C127C2211yescharacteristic
25C22×M16×C127C2211yescharacteristic
26C2×#117×C127C2211yescharacteristic
27#419×C127—21no—
28#419×C127—22no—
29#419×C127—22no—
30#418×C127—21no—
31#417×C127C2214yes—
32#416×C127—22no—
33C2×#103×C127C2214yescomplemented
34C2×#101×C127—24no—
35C2×#101×C127C2214yescomplemented
36#343×C127C2214yescomplemented
37#341×C127C2214yescomplemented
38C2×#41×C127—21no—
39C2×#41×C127—22no—
40C2×#41×C127—22no—
41C2×#41×C127—21no—
42C2×#41×C127C2×C414yes—
43C2×#41×C127D414yes—
44C2×#41×C127D412yes—
45C2×#41×C127C2312yes—
46C22×Q8×C127—22no—
47C22×Q8×C127—21no—
48C22×Q8×C127—21no—
49C22×Q8×C127—21no—
50C22×Q8×C127—22no—
51C22×Q8×C127—21no—
52C22×Q8×C127—21no—
53C22×Q8×C127—22no—
54C22×Q8×C127C2×C414yes—
55C22×Q8×C127D411yescharacteristic
56C22×Q8×C127C2312yes—
57C22×D4×C127—21no—
58C22×D4×C127D411yescharacteristic
59C23×C4×C127—21no—
60C23×C4×C127—21no—
61C23×C4×C127C2311yescharacteristic
62#137×C127—216no—
63#137×C127—28no—
64#137×C127C2×C4116yes—
65C2×Q16×C127—28no—
66C2×Q16×C127—24no—
67C2×SD16×C127—28no—
68C2×SD16×C127—24no—
69#131×C127—216no—
70#131×C127—28no—
71#131×C127—28no—
72C2×M16×C127—216no—
73C2×M16×C127—21no—
74C2×M16×C127—22no—
75C2×M16×C127—22no—
76C2×M16×C127—22no—
77C2×M16×C127—22no—
78C2×M16×C127—21no—
79C2×M16×C127—21no—
80C2×M16×C127—24no—
81C2×M16×C127—22no—
82C2×M16×C127—22no—
83C2×M16×C127C2×C414yes—
84C2×M16×C127D414yes—
85C2×M16×C127D412yes—
86C2×M16×C127D412yes—
87C2×M16×C127C2312yes—
88C2×M16×C127C2312yes—
89C2×M16×C127C2312yes—
90C22×C8×C127—21no—
91C22×C8×C127—21no—
92C22×C8×C127—24no—
93C22×C8×C127—21no—
94C22×C8×C127C2312yes—
95#117×C127—21no—
96#117×C127—21no—
97#117×C127—21no—
98#117×C127—21no—
99C2×C4⋊C4×C127C2312yes—
100C2×C22⋊C4×C127—21no—
101C2×C42×C127—21no—
102#103×C127—24no—
103#103×C127—28no—
104#102×C127—24no—
105#101×C127—28no—
106C24×C127—21no—
107C24×C127#4111yescharacteristic
108#41×C127—24no—
109#41×C127—24no—
110#41×C127—24no—
111#41×C127—24no—
112#41×C127—24no—
113#41×C127—22no—
114#41×C127—22no—
115#41×C127C22⋊C418yes—
116#41×C127C22×C414yes—
117#41×C127C2×D414yes—
118C2×Q8×C127—24no—
119C2×Q8×C127—21no—
120C2×Q8×C127—22no—
121C2×Q8×C127—22no—
122C2×Q8×C127—24no—
123C2×Q8×C127—24no—
124C2×Q8×C127—24no—
125C2×Q8×C127—24no—
126C2×Q8×C127—22no—
127C2×Q8×C127—22no—
128C2×Q8×C127—22no—
129C2×Q8×C127—22no—
130C2×Q8×C127—42no—
131C2×Q8×C127—42no—
132C2×Q8×C127—41no—
133C2×Q8×C127—41no—
134C2×Q8×C127—42no—
135C2×Q8×C127—42no—
136C2×Q8×C127—42no—
137C2×Q8×C127C22⋊C414yes—
138C2×Q8×C127C22×C414yes—
139C2×Q8×C127C2×D412yes—
140C2×D4×C127—24no—
141C2×D4×C127—22no—
142C2×D4×C127—24no—
143C2×D4×C127—22no—
144C2×D4×C127—22no—
145C2×D4×C127—21no—
146C2×D4×C127C22⋊C414yes—
147C2×D4×C127C2×D412yes—
148C22×C4×C127—21no—
149C22×C4×C127—24no—
150C22×C4×C127—24no—
151C22×C4×C127—22no—
152C22×C4×C127—22no—
153C22×C4×C127—24no—
154C22×C4×C127—22no—
155C22×C4×C127—22no—
156C22×C4×C127—41no—
157C22×C4×C127—41no—
158C22×C4×C127—42no—
159C22×C4×C127C22×C412yes—
160C22×C4×C127C2×D412yes—
161C22×C4×C127C2×D411yescharacteristic
162C22×C4×C127C2×D411yescharacteristic
163C22×C4×C127C2×D411yescharacteristic
164C22×C4×C127C2×D411yescharacteristic
165C22×C4×C127C2×D412yes—
166C22×C4×C127#4111yescharacteristic
167C22×C4×C127#4112yes—
168C22×C4×C127C2411yescharacteristic
169Q16×C127—216no—
170SD16×C127—216no—
171M16×C127—216no—
172M16×C127—24no—
173M16×C127—24no—
174M16×C127—24no—
175M16×C127—24no—
176M16×C127—22no—
177M16×C127—22no—
178M16×C127—22no—
179M16×C127—22no—
180M16×C127—42no—
181M16×C127C22×C414yes—
182M16×C127C2×D414yes—
183C2×C8×C127—216no—
184C2×C8×C127—24no—
185C2×C8×C127—24no—
186C2×C8×C127—22no—
187C2×C8×C127—22no—
188C2×C8×C127—22no—
189C2×C8×C127—22no—
190C2×C8×C127—22no—
191C2×C8×C127—42no—
192C2×C8×C127C2×D414yes—
193C4⋊C4×C127—22no—
194C4⋊C4×C127—22no—
195C22⋊C4×C127—41no—
196C22⋊C4×C127—41no—
197C42×C127—41no—
198C42×C127—41no—
199C23×C127—22no—
200C23×C127—24no—
201C23×C127—42no—
202C23×C127—82no—
203C23×C127C4×D412yes—
204C23×C127#12111yescharacteristic
205C23×C127#12111yescharacteristic
206C23×C127#12312yes—
207C23×C127C2×#4111yescharacteristic
208Q8×C127—22no—
209Q8×C127—24no—
210Q8×C127—24no—
211Q8×C127—24no—
212Q8×C127—42no—
213Q8×C127—42no—
214Q8×C127—44no—
215Q8×C127—44no—
216Q8×C127C2×C22⋊C414yes—
217D4×C127—24no—
218D4×C127—24no—
219D4×C127—28no—
220D4×C127—22no—
221D4×C127—24no—
222D4×C127C2×C22⋊C414yes—
223C2×C4×C127—22no—
224C2×C4×C127—24no—
225C2×C4×C127—24no—
226C2×C4×C127—24no—
227C2×C4×C127—28no—
228C2×C4×C127—24no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—44no—
232C2×C4×C127—44no—
233C2×C4×C127—42no—
234C2×C4×C127—82no—
235C2×C4×C127C2×C22⋊C412yes—
236C2×C4×C127C2×C22⋊C412yes—
237C2×C4×C127C4×D412yes—
238C2×C4×C127C4×D412yes—
239C2×C4×C127C4×D412yes—
240C2×C4×C127#12012yes—
241C2×C4×C127#12012yes—
242C2×C4×C127#12112yes—
243C2×C4×C127#12112yes—
244C2×C4×C127#12111yescharacteristic
245C2×C4×C127#12111yescharacteristic
246C2×C4×C127#12312yes—
247C2×C4×C127C23×C411yescharacteristic
248C2×C4×C127C22×D411yescharacteristic
249C2×C4×C127C22×D411yescharacteristic
250C2×C4×C127C22×D411yescharacteristic
251C2×C4×C127C22×D411yescharacteristic
252C2×C4×C127C2×#4111yescharacteristic
253C8×C127—24no—
254C8×C127—24no—
255C8×C127—24no—
256C8×C127—42no—
257C22×C127—22no—
258C22×C127—22no—
259C22×C127—22no—
260C22×C127—24no—
261C22×C127—42no—
262C22×C127—42no—
263C22×C127—44no—
264C22×C127—84no—
265C22×C127#38612yes—
266C22×C127#38612yes—
267C22×C127C2×C4×D411yescharacteristic
268C22×C127C2×#12111yescharacteristic
269C22×C127C2×#12311yescharacteristic
270C4×C127—24no—
271C4×C127—44no—
272C4×C127—82no—
273C4×C127#38612yes—
274C4×C127#38612yes—
275C4×C127C22×C22⋊C411yescharacteristic
276C4×C127C2×C4×D411yescharacteristic
277C4×C127C2×#12011yescharacteristic
278C4×C127C2×#12111yescharacteristic
279#21006C12711yesmaximal, characteristic, direct product factor, Sylow
280C2×C127—22no—
281C2×C127—21no—
282C2×C127—21no—
283C2×C127—22no—
284C2×C127—44no—
285C2×C127C2×#38611yescharacteristic
286C2×C127#306912yes—
287C22×#137C2×C12711yescharacteristic
288C2×#568C2×C12711yescharacteristic
289#2896C2×C12711yescharacteristic
290C2×#419C2×C12711yescharacteristic, complemented
291C22×#101C2×C12711yescharacteristic, complemented
292#1908C2×C12718yescomplemented
293#1742C2×C12711yescharacteristic, complemented
294#1738C2×C12711yescharacteristic, complemented
295C127#2100611yescharacteristic, direct product factor, Sylow
296C22×#41C22×C12711yescharacteristic
297C23×Q8C22×C12711yescharacteristic
298C2×#137—24no—
299C2×#137—22no—
300C2×#137C4×C12718yescomplemented
301C2×#137C22×C12714yes—
302C22×Q16—21no—
303C22×SD16—21no—
304#568—24no—
305#568—24no—
306C2×#131—22no—
307C2×#131C22×C12714yes—
308C22×M16—22no—
309C22×M16C22×C12711yescharacteristic
310C22×M16C22×C12711yescharacteristic
311C22×M16C22×C12711yescharacteristic
312C22×M16C22×C12711yescharacteristic
313C2×#117C22×C12711yescharacteristic
314#419—21no—
315#419—22no—
316#419—22no—
317#418—21no—
318#417C22×C12714yes—
319#416—22no—
320C2×#103C22×C12714yescomplemented
321C2×#101—24no—
322C2×#101C22×C12714yescomplemented
323#343C22×C12714yescomplemented
324#341C22×C12714yescomplemented
325C2×#41—21no—
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—21no—
329C2×#41C2×C4×C12714yes—
330C2×#41D4×C12714yes—
331C2×#41D4×C12712yes—
332C2×#41C23×C12712yes—
333C22×Q8—22no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—21no—
337C22×Q8—22no—
338C22×Q8—21no—
339C22×Q8—21no—
340C22×Q8—22no—
341C22×Q8C2×C4×C12714yes—
342C22×Q8D4×C12711yescharacteristic
343C22×Q8C23×C12712yes—
344C22×D4—21no—
345C22×D4D4×C12711yescharacteristic
346C23×C4—21no—
347C23×C4—21no—
348C23×C4C23×C12711yescharacteristic
349#137—216no—
350#137—28no—
351#137C2×C4×C127116yes—
352C2×Q16—28no—
353C2×Q16—24no—
354C2×SD16—28no—
355C2×SD16—24no—
356#131—216no—
357#131—28no—
358#131—28no—
359C2×M16—216no—
360C2×M16—21no—
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—24no—
368C2×M16—22no—
369C2×M16—22no—
370C2×M16C2×C4×C12714yes—
371C2×M16D4×C12714yes—
372C2×M16D4×C12712yes—
373C2×M16D4×C12712yes—
374C2×M16C23×C12712yes—
375C2×M16C23×C12712yes—
376C2×M16C23×C12712yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—24no—
380C22×C8—21no—
381C22×C8C23×C12712yes—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386C2×C4⋊C4C23×C12712yes—
387C2×C22⋊C4—21no—
388C2×C42—21no—
389#103—24no—
390#103—28no—
391#102—24no—
392#101—28no—
393C24—21no—
394C24#41×C12711yescharacteristic
395#41—24no—
396#41—24no—
397#41—24no—
398#41—24no—
399#41—24no—
400#41—22no—
401#41—22no—
402#41C22⋊C4×C12718yes—
403#41C22×C4×C12714yes—
404#41C2×D4×C12714yes—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8C22⋊C4×C12714yes—
425C2×Q8C22×C4×C12714yes—
426C2×Q8C2×D4×C12712yes—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4C22⋊C4×C12714yes—
434C2×D4C2×D4×C12712yes—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4C22×C4×C12712yes—
447C22×C4C2×D4×C12712yes—
448C22×C4C2×D4×C12711yescharacteristic
449C22×C4C2×D4×C12711yescharacteristic
450C22×C4C2×D4×C12711yescharacteristic
451C22×C4C2×D4×C12711yescharacteristic
452C22×C4C2×D4×C12712yes—
453C22×C4#41×C12711yescharacteristic
454C22×C4#41×C12712yes—
455C22×C4C24×C12711yescharacteristic
456Q16—216no—
457SD16—216no—
458M16—216no—
459M16—24no—
460M16—24no—
461M16—24no—
462M16—24no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—42no—
468M16C22×C4×C12714yes—
469M16C2×D4×C12714yes—
470C2×C8—216no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—42no—
479C2×C8C2×D4×C12714yes—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C42—41no—
485C42—41no—
486C23—22no—
487C23—24no—
488C23—42no—
489C23—82no—
490C23C4×D4×C12712yes—
491C23#121×C12711yescharacteristic
492C23#121×C12711yescharacteristic
493C23#123×C12712yes—
494C23C2×#41×C12711yescharacteristic
495Q8—22no—
496Q8—24no—
497Q8—24no—
498Q8—24no—
499Q8—42no—
500Q8—42no—
501Q8—44no—
502Q8—44no—
503Q8C2×C22⋊C4×C12714yes—
504D4—24no—
505D4—24no—
506D4—28no—
507D4—22no—
508D4—24no—
509D4C2×C22⋊C4×C12714yes—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4C2×C22⋊C4×C12712yes—
523C2×C4C2×C22⋊C4×C12712yes—
524C2×C4C4×D4×C12712yes—
525C2×C4C4×D4×C12712yes—
526C2×C4C4×D4×C12712yes—
527C2×C4#120×C12712yes—
528C2×C4#120×C12712yes—
529C2×C4#121×C12712yes—
530C2×C4#121×C12712yes—
531C2×C4#121×C12711yescharacteristic
532C2×C4#121×C12711yescharacteristic
533C2×C4#123×C12712yes—
534C2×C4C23×C4×C12711yescharacteristic
535C2×C4C22×D4×C12711yescharacteristic
536C2×C4C22×D4×C12711yescharacteristic
537C2×C4C22×D4×C12711yescharacteristic
538C2×C4C22×D4×C12711yescharacteristic
539C2×C4C2×#41×C12711yescharacteristic
540C8—24no—
541C8—24no—
542C8—24no—
543C8—42no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—84no—
552C22#386×C12712yes—
553C22#386×C12712yes—
554C22C2×C4×D4×C12711yescharacteristic
555C22C2×#121×C12711yescharacteristic
556C22C2×#123×C12711yescharacteristic
557C4—24no—
558C4—44no—
559C4—82no—
560C4#386×C12712yes—
561C4#386×C12712yes—
562C4C22×C22⋊C4×C12711yescharacteristic
563C4C2×C4×D4×C12711yescharacteristic
564C4C2×#120×C12711yescharacteristic
565C4C2×#121×C12711yescharacteristic
566C2—22no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—44no—
571C2C2×#386×C12711yescharacteristic
572C2C127×#306912yes—
subgroupG/H
center, socleC22×C127C2×#121
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×#41×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#419×C127C2semidirect
5. C22×#101×C127C2semidirect
6. C127×#1908C2semidirect
7. C127×#1742C2semidirect
8. C127×#1738C2semidirect
13. C2×#137×C127C4semidirect
33. C2×#103×C127C22semidirect
35. C2×#101×C127C22semidirect
36. #343×C127C22semidirect
37. #341×C127C22semidirect
279. #21006C127direct (N × H)
290. C2×#419C2×C127semidirect
291. C22×#101C2×C127semidirect
292. #1908C2×C127semidirect
293. #1742C2×C127semidirect
294. #1738C2×C127semidirect
295. C127#21006direct (N × H)
300. C2×#137C4×C127semidirect
320. C2×#103C22×C127semidirect
322. C2×#101C22×C127semidirect
323. #343C22×C127semidirect
324. #341C22×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#137×C1272
C2×#568×C1272
C127×#28962
C2×#419×C1272
C22×#101×C1272
C127×#19082
C127×#17422
C127×#17382
#21006127