Finite Group Databasebeta

← C127×#20601 #44584274

Subgroups — C127×#20601 #44584274

Order 32512.

4038 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C127C211yesmaximal, characteristic
2C2×#417×C127C214yesmaximal, complemented
3C2×#407×C127C211yesmaximal, characteristic, complemented
4C2×#380×C127C211yesmaximal, characteristic, complemented
5C127×#1788C214yesmaximal, complemented
6C127×#1787C212yesmaximal, complemented
7C127×#1786C212yesmaximal, complemented
8C22×#41×C127—216no—
9C22×#41×C127—24no—
10C22×#41×C127—21no—
11C22×#41×C127—21no—
12C22×#41×C127C418yescomplemented
13C22×#41×C127C2212yes—
14C22×#41×C127C2212yes—
15C23×Q8×C127C2211yescharacteristic
16C23×D4×C127—21no—
17C23×D4×C127C2211yescharacteristic
18C24×C4×C127—21no—
19C24×C4×C127C2211yescharacteristic
20C22×M16×C127C2212yescomplemented
21C22×C4⋊C4×C127C2212yescomplemented
22C22×C22⋊C4×C127—21no—
23#417×C127—216no—
24C2×#103×C127C2218yescomplemented
25C2×#102×C127C2218yescomplemented
26#407×C127—24no—
27C2×C22⋊C8×C127C2212yes—
28C2×C22⋊C8×C127C2212yescomplemented
29#380×C127—21no—
30#380×C127—21no—
31#380×C127C2212yes—
32#380×C127C2212yescomplemented
33C2×#95×C127—21no—
34C25×C127—21no—
35C25×C127D411yescharacteristic
36C2×#41×C127—216no—
37C2×#41×C127—264no—
38C2×#41×C127—216no—
39C2×#41×C127—264no—
40C2×#41×C127—28no—
41C2×#41×C127—28no—
42C2×#41×C127—28no—
43C2×#41×C127—28no—
44C2×#41×C127—24no—
45C2×#41×C127—24no—
46C2×#41×C127—28no—
47C2×#41×C127C2×C418yes—
48C2×#41×C127C2×C418yescomplemented
49C2×#41×C127D4116yes—
50C22×Q8×C127—28no—
51C22×Q8×C127—22no—
52C22×Q8×C127—21no—
53C22×Q8×C127C2×C414yescomplemented
54C22×Q8×C127C2312yes—
55C22×D4×C127—216no—
56C22×D4×C127—28no—
57C22×D4×C127—22no—
58C22×D4×C127—22no—
59C22×D4×C127—28no—
60C22×D4×C127—21no—
61C22×D4×C127—22no—
62C22×D4×C127C2×C414yescomplemented
63C22×D4×C127C2312yes—
64C23×C4×C127—216no—
65C23×C4×C127—21no—
66C23×C4×C127—28no—
67C23×C4×C127—22no—
68C23×C4×C127—22no—
69C23×C4×C127—28no—
70C23×C4×C127—22no—
71C23×C4×C127C2×C414yes—
72C23×C4×C127D411yescharacteristic
73C23×C4×C127D411yescharacteristic
74C23×C4×C127D411yescharacteristic
75C23×C4×C127D414yes—
76C23×C4×C127C2311yescharacteristic
77C23×C4×C127C2311yescharacteristic
78C23×C4×C127C2311yescharacteristic
79C2×M16×C127—24no—
80C2×M16×C127—28no—
81C22×C8×C127C2314yes—
82C2×C4⋊C4×C127—28no—
83C2×C4⋊C4×C127—22no—
84C2×C4⋊C4×C127C2314yes—
85C2×C22⋊C4×C127—22no—
86C2×C22⋊C4×C127—22no—
87C2×C22⋊C4×C127—44no—
88#103×C127—216no—
89#102×C127—216no—
90C22⋊C8×C127—24no—
91C22⋊C8×C127—24no—
92#95×C127—22no—
93#95×C127—22no—
94C24×C127—28no—
95C24×C127—24no—
96C24×C127—24no—
97C24×C127—22no—
98C24×C127—22no—
99C24×C127—22no—
100C24×C127—42no—
101C24×C127—44no—
102C24×C127—42no—
103C24×C127—42no—
104C24×C127C22⋊C414yes—
105C24×C127C2×D411yescharacteristic
106C24×C127C2×D411yescharacteristic
107C24×C127C2×D411yescharacteristic
108#41×C127—264no—
109#41×C127—232no—
110#41×C127—264no—
111#41×C127—216no—
112#41×C127—216no—
113#41×C127—216no—
114#41×C127—232no—
115#41×C127C22⋊C4132yescomplemented
116C2×Q8×C127—216no—
117C2×Q8×C127—28no—
118C2×Q8×C127—216no—
119C2×Q8×C127—24no—
120C2×Q8×C127—22no—
121C2×Q8×C127—24no—
122C2×Q8×C127C22×C414yes—
123C2×Q8×C127C2×D418yes—
124C2×D4×C127—264no—
125C2×D4×C127—216no—
126C2×D4×C127—28no—
127C2×D4×C127—216no—
128C2×D4×C127—28no—
129C2×D4×C127—28no—
130C2×D4×C127—216no—
131C2×D4×C127—216no—
132C2×D4×C127—24no—
133C2×D4×C127—24no—
134C2×D4×C127—22no—
135C2×D4×C127C22×C414yes—
136C2×D4×C127C2×D418yes—
137C22×C4×C127—28no—
138C22×C4×C127—264no—
139C22×C4×C127—216no—
140C22×C4×C127—24no—
141C22×C4×C127—24no—
142C22×C4×C127—22no—
143C22×C4×C127—28no—
144C22×C4×C127—28no—
145C22×C4×C127—216no—
146C22×C4×C127—24no—
147C22×C4×C127—216no—
148C22×C4×C127—24no—
149C22×C4×C127—216no—
150C22×C4×C127—24no—
151C22×C4×C127—24no—
152C22×C4×C127—24no—
153C22×C4×C127—44no—
154C22×C4×C127—44no—
155C22×C4×C127—44no—
156C22×C4×C127C22⋊C414yes—
157C22×C4×C127C22⋊C414yes—
158C22×C4×C127C22⋊C414yes—
159C22×C4×C127C22×C412yes—
160C22×C4×C127C22×C412yes—
161C22×C4×C127C22×C412yes—
162C22×C4×C127C2×D411yescharacteristic
163C22×C4×C127C2×D411yescharacteristic
164C22×C4×C127C2×D412yes—
165C22×C4×C127C2×D412yes—
166C22×C4×C127C2×D411yescharacteristic
167C22×C4×C127C2×D418yes—
168C22×C4×C127C2×D412yes—
169C22×C4×C127C2411yescharacteristic
170M16×C127—48no—
171C2×C8×C127—24no—
172C2×C8×C127—28no—
173C4⋊C4×C127—28no—
174C4⋊C4×C127—44no—
175C22⋊C4×C127—44no—
176C22⋊C4×C127—44no—
177C23×C127—22no—
178C23×C127—21no—
179C23×C127—21no—
180C23×C127—22no—
181C23×C127—22no—
182C23×C127—216no—
183C23×C127—24no—
184C23×C127—22no—
185C23×C127—28no—
186C23×C127—21no—
187C23×C127—21no—
188C23×C127—22no—
189C23×C127—28no—
190C23×C127—28no—
191C23×C127—44no—
192C23×C127—48no—
193C23×C127—48no—
194C23×C127—48no—
195C23×C127—416no—
196C23×C127—416no—
197C23×C127—42no—
198C23×C127—42no—
199C23×C127—44no—
200C23×C127—48no—
201C23×C127C2×C22⋊C412yes—
202C23×C127C2×C22⋊C412yes—
203C23×C127C2×C22⋊C412yes—
204C23×C127#12014yes—
205C23×C127AGL(1,ℤ/8)12yes—
206C23×C127#13712yes—
207C23×C127C22×D411yescharacteristic
208Q8×C127—216no—
209Q8×C127—24no—
210Q8×C127—28no—
211Q8×C127C2×C22⋊C418yescomplemented
212D4×C127—232no—
213D4×C127—216no—
214D4×C127—232no—
215D4×C127—24no—
216D4×C127—28no—
217D4×C127C2×C22⋊C418yescomplemented
218C2×C4×C127—216no—
219C2×C4×C127—232no—
220C2×C4×C127—24no—
221C2×C4×C127—22no—
222C2×C4×C127—28no—
223C2×C4×C127—22no—
224C2×C4×C127—28no—
225C2×C4×C127—24no—
226C2×C4×C127—24no—
227C2×C4×C127—24no—
228C2×C4×C127—24no—
229C2×C4×C127—28no—
230C2×C4×C127—232no—
231C2×C4×C127—28no—
232C2×C4×C127—44no—
233C2×C4×C127—44no—
234C2×C4×C127—44no—
235C2×C4×C127—88no—
236C2×C4×C127C2×C22⋊C412yes—
237C2×C4×C127C2×C22⋊C412yes—
238C2×C4×C127C2×C22⋊C418yes—
239C2×C4×C127C2×C22⋊C412yes—
240C2×C4×C127#12018yes—
241C2×C4×C127#12014yes—
242C2×C4×C127C23×C411yescharacteristic
243C2×C4×C127C22×D414yes—
244C2×C4×C127C22×D411yescharacteristic
245C8×C127—44no—
246C22×C127—22no—
247C22×C127—22no—
248C22×C127—22no—
249C22×C127—24no—
250C22×C127—24no—
251C22×C127—24no—
252C22×C127—24no—
253C22×C127—22no—
254C22×C127—22no—
255C22×C127—22no—
256C22×C127—22no—
257C22×C127—28no—
258C22×C127—416no—
259C22×C127—432no—
260C22×C127—44no—
261C22×C127—44no—
262C22×C127—42no—
263C22×C127—48no—
264C22×C127—42no—
265C22×C127—48no—
266C22×C127#37914yes—
267C22×C127#41714yescomplemented
268C22×C127C22×C22⋊C411yescharacteristic
269C22×C127C2×#12012yes—
270C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C127C2×#13711yescharacteristic
272C22×C127#57512yes—
273C4×C127—22no—
274C4×C127—28no—
275C4×C127—22no—
276C4×C127—84no—
277C4×C127#37914yes—
278C4×C127C22×C22⋊C412yes—
279C4×C127C2×#12012yes—
280#20601C12711yesmaximal, characteristic, direct product factor, Sylow
281C2×C127—24no—
282C2×C127—24no—
283C2×C127—42no—
284C2×C127—48no—
285C2×C127C2×#37911yescharacteristic
286C2×C127C2×#41711yescharacteristic
287C2×C127#289611yescharacteristic
288C2×C127#300612yes—
289C2×C127#300712yes—
290C23×#41C2×C12711yescharacteristic
291C2×#417C2×C12714yescomplemented
292C2×#407C2×C12711yescharacteristic, complemented
293C2×#380C2×C12711yescharacteristic, complemented
294#1788C2×C12714yescomplemented
295#1787C2×C12712yescomplemented
296#1786C2×C12712yescomplemented
297C127#2060111yescharacteristic, direct product factor, Sylow
298C22×#41—216no—
299C22×#41—24no—
300C22×#41—21no—
301C22×#41—21no—
302C22×#41C4×C12718yescomplemented
303C22×#41C22×C12712yes—
304C22×#41C22×C12712yes—
305C23×Q8C22×C12711yescharacteristic
306C23×D4—21no—
307C23×D4C22×C12711yescharacteristic
308C24×C4—21no—
309C24×C4C22×C12711yescharacteristic
310C22×M16C22×C12712yescomplemented
311C22×C4⋊C4C22×C12712yescomplemented
312C22×C22⋊C4—21no—
313#417—216no—
314C2×#103C22×C12718yescomplemented
315C2×#102C22×C12718yescomplemented
316#407—24no—
317C2×C22⋊C8C22×C12712yes—
318C2×C22⋊C8C22×C12712yescomplemented
319#380—21no—
320#380—21no—
321#380C22×C12712yes—
322#380C22×C12712yescomplemented
323C2×#95—21no—
324C25—21no—
325C25D4×C12711yescharacteristic
326C2×#41—216no—
327C2×#41—264no—
328C2×#41—216no—
329C2×#41—264no—
330C2×#41—28no—
331C2×#41—28no—
332C2×#41—28no—
333C2×#41—28no—
334C2×#41—24no—
335C2×#41—24no—
336C2×#41—28no—
337C2×#41C2×C4×C12718yes—
338C2×#41C2×C4×C12718yescomplemented
339C2×#41D4×C127116yes—
340C22×Q8—28no—
341C22×Q8—22no—
342C22×Q8—21no—
343C22×Q8C2×C4×C12714yescomplemented
344C22×Q8C23×C12712yes—
345C22×D4—216no—
346C22×D4—28no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—28no—
350C22×D4—21no—
351C22×D4—22no—
352C22×D4C2×C4×C12714yescomplemented
353C22×D4C23×C12712yes—
354C23×C4—216no—
355C23×C4—21no—
356C23×C4—28no—
357C23×C4—22no—
358C23×C4—22no—
359C23×C4—28no—
360C23×C4—22no—
361C23×C4C2×C4×C12714yes—
362C23×C4D4×C12711yescharacteristic
363C23×C4D4×C12711yescharacteristic
364C23×C4D4×C12711yescharacteristic
365C23×C4D4×C12714yes—
366C23×C4C23×C12711yescharacteristic
367C23×C4C23×C12711yescharacteristic
368C23×C4C23×C12711yescharacteristic
369C2×M16—24no—
370C2×M16—28no—
371C22×C8C23×C12714yes—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4C23×C12714yes—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—44no—
378#103—216no—
379#102—216no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382#95—22no—
383#95—22no—
384C24—28no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—44no—
392C24—42no—
393C24—42no—
394C24C22⋊C4×C12714yes—
395C24C2×D4×C12711yescharacteristic
396C24C2×D4×C12711yescharacteristic
397C24C2×D4×C12711yescharacteristic
398#41—264no—
399#41—232no—
400#41—264no—
401#41—216no—
402#41—216no—
403#41—216no—
404#41—232no—
405#41C22⋊C4×C127132yescomplemented
406C2×Q8—216no—
407C2×Q8—28no—
408C2×Q8—216no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8C22×C4×C12714yes—
413C2×Q8C2×D4×C12718yes—
414C2×D4—264no—
415C2×D4—216no—
416C2×D4—28no—
417C2×D4—216no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4C22×C4×C12714yes—
426C2×D4C2×D4×C12718yes—
427C22×C4—28no—
428C22×C4—264no—
429C22×C4—216no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—216no—
436C22×C4—24no—
437C22×C4—216no—
438C22×C4—24no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4C22⋊C4×C12714yes—
447C22×C4C22⋊C4×C12714yes—
448C22×C4C22⋊C4×C12714yes—
449C22×C4C22×C4×C12712yes—
450C22×C4C22×C4×C12712yes—
451C22×C4C22×C4×C12712yes—
452C22×C4C2×D4×C12711yescharacteristic
453C22×C4C2×D4×C12711yescharacteristic
454C22×C4C2×D4×C12712yes—
455C22×C4C2×D4×C12712yes—
456C22×C4C2×D4×C12711yescharacteristic
457C22×C4C2×D4×C12718yes—
458C22×C4C2×D4×C12712yes—
459C22×C4C24×C12711yescharacteristic
460M16—48no—
461C2×C8—24no—
462C2×C8—28no—
463C4⋊C4—28no—
464C4⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—216no—
473C23—24no—
474C23—22no—
475C23—28no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—416no—
486C23—416no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—48no—
491C23C2×C22⋊C4×C12712yes—
492C23C2×C22⋊C4×C12712yes—
493C23C2×C22⋊C4×C12712yes—
494C23#120×C12714yes—
495C23AGL(1,ℤ/8)×C12712yes—
496C23#137×C12712yes—
497C23C22×D4×C12711yescharacteristic
498Q8—216no—
499Q8—24no—
500Q8—28no—
501Q8C2×C22⋊C4×C12718yescomplemented
502D4—232no—
503D4—216no—
504D4—232no—
505D4—24no—
506D4—28no—
507D4C2×C22⋊C4×C12718yescomplemented
508C2×C4—216no—
509C2×C4—232no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—232no—
521C2×C4—28no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—88no—
526C2×C4C2×C22⋊C4×C12712yes—
527C2×C4C2×C22⋊C4×C12712yes—
528C2×C4C2×C22⋊C4×C12718yes—
529C2×C4C2×C22⋊C4×C12712yes—
530C2×C4#120×C12718yes—
531C2×C4#120×C12714yes—
532C2×C4C23×C4×C12711yescharacteristic
533C2×C4C22×D4×C12714yes—
534C2×C4C22×D4×C12711yescharacteristic
535C8—44no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—28no—
548C22—416no—
549C22—432no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—48no—
554C22—42no—
555C22—48no—
556C22#379×C12714yes—
557C22#417×C12714yescomplemented
558C22C22×C22⋊C4×C12711yescharacteristic
559C22C2×#120×C12712yes—
560C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
561C22C2×#137×C12711yescharacteristic
562C22#575×C12712yes—
563C4—22no—
564C4—28no—
565C4—22no—
566C4—84no—
567C4#379×C12714yes—
568C4C22×C22⋊C4×C12712yes—
569C4C2×#120×C12712yes—
570C2—24no—
571C2—24no—
572C2—42no—
573C2—48no—
574C2C2×#379×C12711yescharacteristic
575C2C2×#417×C12711yescharacteristic
576C2C127×#289611yescharacteristic
577C2C127×#300612yes—
578C2C127×#300712yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC23×#41×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#417×C127C2semidirect
3. C2×#407×C127C2semidirect
4. C2×#380×C127C2semidirect
5. C127×#1788C2semidirect
6. C127×#1787C2semidirect
7. C127×#1786C2semidirect
12. C22×#41×C127C4semidirect
20. C22×M16×C127C22semidirect
21. C22×C4⋊C4×C127C22semidirect
24. C2×#103×C127C22semidirect
25. C2×#102×C127C22semidirect
28. C2×C22⋊C8×C127C22semidirect
32. #380×C127C22semidirect
48. C2×#41×C127C2×C4semidirect
53. C22×Q8×C127C2×C4semidirect
62. C22×D4×C127C2×C4semidirect
115. #41×C127C22⋊C4semidirect
211. Q8×C127C2×C22⋊C4semidirect
217. D4×C127C2×C22⋊C4semidirect
267. C22×C127#417semidirect
280. #20601C127direct (N × H)
291. C2×#417C2×C127semidirect
292. C2×#407C2×C127semidirect
293. C2×#380C2×C127semidirect
294. #1788C2×C127semidirect
295. #1787C2×C127semidirect
296. #1786C2×C127semidirect
297. C127#20601direct (N × H)
302. C22×#41C4×C127semidirect
310. C22×M16C22×C127semidirect
311. C22×C4⋊C4C22×C127semidirect
314. C2×#103C22×C127semidirect
315. C2×#102C22×C127semidirect
318. C2×C22⋊C8C22×C127semidirect
322. #380C22×C127semidirect
338. C2×#41C2×C4×C127semidirect
343. C22×Q8C2×C4×C127semidirect
352. C22×D4C2×C4×C127semidirect
405. #41C22⋊C4×C127semidirect
501. Q8C2×C22⋊C4×C127semidirect
507. D4C2×C22⋊C4×C127semidirect
557. C22#417×C127semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C2×#417×C1272
C2×#407×C1272
C2×#380×C1272
C127×#17882
C127×#17872
C127×#17862
#20601127