Finite Group Databasebeta

← C127×#10971 #44574644

Subgroups — C127×#10971 #44574644

Order 32512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525×C127C211yesmaximal, characteristic
2C22×#104×C127C212yesmaximal, complemented
3C2×#407×C127C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
5C2×#339×C127C212yesmaximal, complemented
6C22×#41×C127C2211yescharacteristic
7C24×C4×C127C2211yescharacteristic
8C22×M16×C127C2212yes—
9#525×C127—24no—
10#525×C127—24no—
11C2×#123×C127—21no—
12C2×C22⋊Q8×C127C412yescomplemented
13C2×#121×C127C412yescomplemented
14C2×#120×C127—21no—
15C2×C4×D4×C127—22no—
16C2×#117×C127C2211yescharacteristic
17C22×C22⋊C4×C127—21no—
18C22×C42×C127C2212yes—
19C2×#104×C127—24no—
20C2×#104×C127—24no—
21C2×#104×C127—24no—
22#407×C127—24no—
23C2×C22⋊C8×C127—22no—
24C4×C22⋊C4×C127—24no—
25C2×#95×C127—21no—
26#339×C127—24no—
27C25×C127D411yescharacteristic
28C2×#41×C127—22no—
29C2×#41×C127—22no—
30C2×#41×C127—22no—
31C2×#41×C127—22no—
32C2×#41×C127—22no—
33C2×#41×C127D414yes—
34C22×Q8×C127C2×C411yescharacteristic
35C22×D4×C127—22no—
36C22×D4×C127—21no—
37C22×D4×C127C2×C411yescharacteristic
38C23×C4×C127—22no—
39C23×C4×C127—22no—
40C23×C4×C127—22no—
41C23×C4×C127—22no—
42C23×C4×C127—22no—
43C23×C4×C127—21no—
44C23×C4×C127—22no—
45C23×C4×C127—22no—
46C23×C4×C127C2×C412yes—
47C23×C4×C127D411yescharacteristic
48C23×C4×C127D412yes—
49C23×C4×C127C2311yescharacteristic
50C2×M16×C127—22no—
51C2×M16×C127—22no—
52C2×M16×C127—22no—
53C2×M16×C127—22no—
54C2×M16×C127—22no—
55C2×M16×C127D414yescomplemented
56C22×C8×C127—22no—
57#123×C127—28no—
58C22⋊Q8×C127—24no—
59C22⋊Q8×C127—24no—
60#121×C127—24no—
61#121×C127—24no—
62#120×C127—28no—
63C4×D4×C127—216no—
64#117×C127—22no—
65#117×C127D414yes—
66C2×C4⋊C4×C127—22no—
67C2×C4⋊C4×C127C2×C412yes—
68C2×C22⋊C4×C127—28no—
69C2×C22⋊C4×C127—24no—
70C2×C22⋊C4×C127—22no—
71C2×C22⋊C4×C127—21no—
72C2×C22⋊C4×C127—22no—
73C2×C42×C127—21no—
74C2×C42×C127—22no—
75C2×C42×C127—22no—
76C2×C42×C127—22no—
77C2×C42×C127—22no—
78C2×C42×C127—22no—
79C2×C42×C127D414yescomplemented
80#104×C127—216no—
81C22⋊C8×C127—28no—
82#95×C127—24no—
83C24×C127—24no—
84C24×C127—21no—
85C24×C127—21no—
86C24×C127—21no—
87C24×C127—22no—
88C24×C127—22no—
89C24×C127—42no—
90C24×C127C22⋊C412yes—
91C24×C127C2×D411yescharacteristic
92#41×C127—24no—
93#41×C127—24no—
94#41×C127—24no—
95#41×C127—24no—
96#41×C127—42no—
97#41×C127—42no—
98#41×C127—42no—
99#41×C127—42no—
100C2×Q8×C127—21no—
101C2×Q8×C127—22no—
102C2×Q8×C127—22no—
103C2×Q8×C127C22⋊C412yescomplemented
104C2×D4×C127—28no—
105C2×D4×C127—24no—
106C2×D4×C127—21no—
107C2×D4×C127—22no—
108C2×D4×C127—22no—
109C2×D4×C127—44no—
110C2×D4×C127—44no—
111C2×D4×C127—41no—
112C2×D4×C127—42no—
113C2×D4×C127—41no—
114C2×D4×C127C22⋊C412yescomplemented
115C22×C4×C127—28no—
116C22×C4×C127—22no—
117C22×C4×C127—21no—
118C22×C4×C127—21no—
119C22×C4×C127—24no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—21no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—24no—
126C22×C4×C127—24no—
127C22×C4×C127—22no—
128C22×C4×C127—24no—
129C22×C4×C127—24no—
130C22×C4×C127—24no—
131C22×C4×C127—22no—
132C22×C4×C127—44no—
133C22×C4×C127—42no—
134C22×C4×C127—42no—
135C22×C4×C127—42no—
136C22×C4×C127—41no—
137C22×C4×C127—42no—
138C22×C4×C127—42no—
139C22×C4×C127—42no—
140C22×C4×C127—42no—
141C22×C4×C127—42no—
142C22×C4×C127—42no—
143C22×C4×C127—41no—
144C22×C4×C127—42no—
145C22×C4×C127—42no—
146C22×C4×C127C22⋊C412yes—
147C22×C4×C127C22×C411yescharacteristic
148C22×C4×C127C2×D412yes—
149C22×C4×C127C2×D412yes—
150C22×C4×C127C2×D412yes—
151C22×C4×C127C2×D412yes—
152C22×C4×C127C2×D411yescharacteristic
153C22×C4×C127#4112yes—
154C22×C4×C127#4112yes—
155M16×C127—24no—
156M16×C127—24no—
157M16×C127—42no—
158M16×C127—42no—
159C2×C8×C127—24no—
160C2×C8×C127—42no—
161C2×C8×C127—42no—
162C4⋊C4×C127—28no—
163C4⋊C4×C127—22no—
164C4⋊C4×C127C22⋊C414yescomplemented
165C22⋊C4×C127—216no—
166C22⋊C4×C127—28no—
167C22⋊C4×C127—28no—
168C22⋊C4×C127—24no—
169C42×C127—24no—
170C42×C127—24no—
171C42×C127—24no—
172C42×C127—42no—
173C42×C127—42no—
174C23×C127—22no—
175C23×C127—22no—
176C23×C127—24no—
177C23×C127—24no—
178C23×C127—22no—
179C23×C127—22no—
180C23×C127—22no—
181C23×C127—22no—
182C23×C127—22no—
183C23×C127—22no—
184C23×C127—22no—
185C23×C127—48no—
186C23×C127—48no—
187C23×C127—42no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127—41no—
192C23×C127—41no—
193C23×C127—42no—
194C23×C127—42no—
195C23×C127#10414yescomplemented
196C23×C127C2×C22⋊C411yescharacteristic
197C23×C127#12012yes—
198C23×C127#12011yescharacteristic
199C23×C127#12112yes—
200C23×C127#12311yescharacteristic
201Q8×C127—24no—
202Q8×C127—42no—
203D4×C127—24no—
204D4×C127—416no—
205D4×C127—44no—
206D4×C127—44no—
207D4×C127—42no—
208C2×C4×C127—28no—
209C2×C4×C127—24no—
210C2×C4×C127—22no—
211C2×C4×C127—22no—
212C2×C4×C127—22no—
213C2×C4×C127—22no—
214C2×C4×C127—48no—
215C2×C4×C127—44no—
216C2×C4×C127—48no—
217C2×C4×C127—41no—
218C2×C4×C127—41no—
219C2×C4×C127—44no—
220C2×C4×C127—42no—
221C2×C4×C127—42no—
222C2×C4×C127—42no—
223C2×C4×C127—42no—
224C2×C4×C127—44no—
225C2×C4×C127—44no—
226C2×C4×C127—44no—
227C2×C4×C127—44no—
228C2×C4×C127—44no—
229C2×C4×C127—44no—
230C2×C4×C127C2×C22⋊C411yescharacteristic
231C2×C4×C127C2×C22⋊C411yescharacteristic
232C2×C4×C127C4×D412yes—
233C2×C4×C127C4×D412yes—
234C2×C4×C127#12012yes—
235C2×C4×C127#12011yescharacteristic
236C2×C4×C127#12012yes—
237C2×C4×C127#12112yes—
238C2×C4×C127#12112yes—
239C2×C4×C127#12112yes—
240C2×C4×C127#12112yes—
241C2×C4×C127#12112yes—
242C2×C4×C127#12312yes—
243C2×C4×C127#12311yescharacteristic
244C2×C4×C127#12412yes—
245C2×C4×C127#12712yes—
246C8×C127—44no—
247C22×C127—24no—
248C22×C127—21no—
249C22×C127—21no—
250C22×C127—21no—
251C22×C127—21no—
252C22×C127—24no—
253C22×C127—24no—
254C22×C127—24no—
255C22×C127—24no—
256C22×C127—48no—
257C22×C127—48no—
258C22×C127—42no—
259C22×C127—42no—
260C22×C127—42no—
261C22×C127—42no—
262C22×C127—42no—
263C22×C127—44no—
264C22×C127—82no—
265C22×C127—81no—
266C22×C127—81no—
267C22×C127#37911yescharacteristic
268C22×C127#38611yescharacteristic
269C22×C127#38611yescharacteristic
270C22×C127#39212yes—
271C22×C127#39412yes—
272C22×C127C2×#10412yes—
273C22×C127#42112yes—
274C4×C127—48no—
275C4×C127—42no—
276C4×C127—42no—
277C4×C127—44no—
278C4×C127#38612yes—
279C4×C127#39012yes—
280C4×C127#39212yes—
281C4×C127#39412yes—
282#10971C12711yesmaximal, characteristic, direct product factor, Sylow
283C2×C127—22no—
284C2×C127—22no—
285C2×C127—44no—
286C2×C127—82no—
287C2×C127#143811yescharacteristic
288C2×C127#178912yes—
289C2×C127#187112yes—
290C2×C127#189611yescharacteristic
291C2×C127#189711yescharacteristic
292C2×#525C2×C12711yescharacteristic
293C22×#104C2×C12712yescomplemented
294C2×#407C2×C12711yescharacteristic, complemented
295C2×C4×C22⋊C4C2×C12711yescharacteristic, complemented
296C2×#339C2×C12712yescomplemented
297C127#1097111yescharacteristic, direct product factor, Sylow
298C22×#41C22×C12711yescharacteristic
299C24×C4C22×C12711yescharacteristic
300C22×M16C22×C12712yes—
301#525—24no—
302#525—24no—
303C2×#123—21no—
304C2×C22⋊Q8C4×C12712yescomplemented
305C2×#121C4×C12712yescomplemented
306C2×#120—21no—
307C2×C4×D4—22no—
308C2×#117C22×C12711yescharacteristic
309C22×C22⋊C4—21no—
310C22×C42C22×C12712yes—
311C2×#104—24no—
312C2×#104—24no—
313C2×#104—24no—
314#407—24no—
315C2×C22⋊C8—22no—
316C4×C22⋊C4—24no—
317C2×#95—21no—
318#339—24no—
319C25D4×C12711yescharacteristic
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41D4×C12714yes—
326C22×Q8C2×C4×C12711yescharacteristic
327C22×D4—22no—
328C22×D4—21no—
329C22×D4C2×C4×C12711yescharacteristic
330C23×C4—22no—
331C23×C4—21no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—22no—
338C23×C4C2×C4×C12712yes—
339C23×C4D4×C12711yescharacteristic
340C23×C4D4×C12712yes—
341C23×C4C23×C12711yescharacteristic
342C2×M16—22no—
343C2×M16—22no—
344C2×M16—22no—
345C2×M16—22no—
346C2×M16—22no—
347C2×M16D4×C12714yescomplemented
348C22×C8—22no—
349#123—28no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352#121—24no—
353#121—24no—
354#120—28no—
355C4×D4—216no—
356#117—22no—
357#117D4×C12714yes—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C2×C4×C12712yes—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42D4×C12714yescomplemented
372#104—216no—
373C22⋊C8—28no—
374#95—24no—
375C24—24no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24—42no—
382C24C22⋊C4×C12712yes—
383C24C2×D4×C12711yescharacteristic
384#41—24no—
385#41—24no—
386#41—24no—
387#41—24no—
388#41—42no—
389#41—42no—
390#41—42no—
391#41—42no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22⋊C4×C12712yescomplemented
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—41no—
406C2×D4C22⋊C4×C12712yescomplemented
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4C22⋊C4×C12712yes—
439C22×C4C22×C4×C12711yescharacteristic
440C22×C4C2×D4×C12712yes—
441C22×C4C2×D4×C12712yes—
442C22×C4C2×D4×C12712yes—
443C22×C4C2×D4×C12712yes—
444C22×C4C2×D4×C12711yescharacteristic
445C22×C4#41×C12712yes—
446C22×C4#41×C12712yes—
447M16—24no—
448M16—24no—
449M16—42no—
450M16—42no—
451C2×C8—24no—
452C2×C8—42no—
453C2×C8—42no—
454C4⋊C4—28no—
455C4⋊C4—22no—
456C4⋊C4C22⋊C4×C12714yescomplemented
457C22⋊C4—216no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—24no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—42no—
465C42—42no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—48no—
478C23—48no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23#104×C12714yescomplemented
488C23C2×C22⋊C4×C12711yescharacteristic
489C23#120×C12712yes—
490C23#120×C12711yescharacteristic
491C23#121×C12712yes—
492C23#123×C12711yescharacteristic
493Q8—24no—
494Q8—42no—
495D4—24no—
496D4—416no—
497D4—44no—
498D4—44no—
499D4—42no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4C2×C22⋊C4×C12711yescharacteristic
523C2×C4C2×C22⋊C4×C12711yescharacteristic
524C2×C4C4×D4×C12712yes—
525C2×C4C4×D4×C12712yes—
526C2×C4#120×C12712yes—
527C2×C4#120×C12711yescharacteristic
528C2×C4#120×C12712yes—
529C2×C4#121×C12712yes—
530C2×C4#121×C12712yes—
531C2×C4#121×C12712yes—
532C2×C4#121×C12712yes—
533C2×C4#121×C12712yes—
534C2×C4#123×C12712yes—
535C2×C4#123×C12711yescharacteristic
536C2×C4#124×C12712yes—
537C2×C4#127×C12712yes—
538C8—44no—
539C22—24no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—48no—
549C22—48no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—82no—
557C22—81no—
558C22—81no—
559C22#379×C12711yescharacteristic
560C22#386×C12711yescharacteristic
561C22#386×C12711yescharacteristic
562C22#392×C12712yes—
563C22#394×C12712yes—
564C22C2×#104×C12712yes—
565C22#421×C12712yes—
566C4—48no—
567C4—42no—
568C4—42no—
569C4—44no—
570C4#386×C12712yes—
571C4#390×C12712yes—
572C4#392×C12712yes—
573C4#394×C12712yes—
574C2—22no—
575C2—22no—
576C2—44no—
577C2—82no—
578C2C127×#143811yescharacteristic
579C2C127×#178912yes—
580C2C127×#187112yes—
581C2C127×#189611yescharacteristic
582C2C127×#189711yescharacteristic
subgroupG/H
centerC22×C4×C127C2×D4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
socleC23×C127C2×C22⋊C4
prime radicalC2×#525×C127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#104×C127C2semidirect
3. C2×#407×C127C2semidirect
4. C2×C4×C22⋊C4×C127C2semidirect
5. C2×#339×C127C2semidirect
12. C2×C22⋊Q8×C127C4semidirect
13. C2×#121×C127C4semidirect
55. C2×M16×C127D4semidirect
79. C2×C42×C127D4semidirect
103. C2×Q8×C127C22⋊C4semidirect
114. C2×D4×C127C22⋊C4semidirect
164. C4⋊C4×C127C22⋊C4semidirect
195. C23×C127#104semidirect
282. #10971C127direct (N × H)
293. C22×#104C2×C127semidirect
294. C2×#407C2×C127semidirect
295. C2×C4×C22⋊C4C2×C127semidirect
296. C2×#339C2×C127semidirect
297. C127#10971direct (N × H)
304. C2×C22⋊Q8C4×C127semidirect
305. C2×#121C4×C127semidirect
347. C2×M16D4×C127semidirect
371. C2×C42D4×C127semidirect
395. C2×Q8C22⋊C4×C127semidirect
406. C2×D4C22⋊C4×C127semidirect
456. C4⋊C4C22⋊C4×C127semidirect
487. C23#104×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#525×C1272
C22×#104×C1272
C2×#407×C1272
C2×C4×C22⋊C4×C1272
C2×#339×C1272
#10971127