Finite Group Databasebeta

← C127×#8347 #44572020

Subgroups — C127×#8347 #44572020

Order 32512.

906 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3492C211yesmaximal, characteristic
2C127×#1896C212yesmaximal, complemented
3C127×#1504C212yesmaximal, complemented
4C127×#1463C211yesmaximal, characteristic, complemented
5C127×#1436C211yesmaximal, characteristic, complemented
6#554×C127—22no—
7#553×C127—22no—
8#552×C127—22no—
9#547×C127—22no—
10#544×C127C411yescharacteristic, complemented
11#539×C127—21no—
12#539×C127C412yescomplemented
13#537×C127C411yescharacteristic, complemented
14#525×C127—21no—
15#525×C127—21no—
16#525×C127C2212yes—
17#520×C127—21no—
18C22×C4⋊C4×C127C2211yescharacteristic
19C22×C42×C127C2211yescharacteristic
20C2×#104×C127—22no—
21#407×C127C2212yes—
22C2×#95×C127—21no—
23C2×#95×C127—21no—
24C2×#95×C127C2211yescharacteristic
25#339×C127—22no—
26#328×C127—22no—
27#327×C127—22no—
28#324×C127—21no—
29#324×C127—21no—
30C2×#41×C127—21no—
31C2×#41×C127—21no—
32C2×#41×C127D412yes—
33C23×C4×C127—21no—
34C23×C4×C127—21no—
35C23×C4×C127D411yescharacteristic
36C23×C4×C127D411yescharacteristic
37C23×C4×C127C2311yescharacteristic
38C2×M16×C127—22no—
39C4⋊Q8×C127—22no—
40#126×C127—22no—
41#126×C127—22no—
42#125×C127—22no—
43#125×C127—22no—
44#125×C127—22no—
45#123×C127—22no—
46#123×C127—22no—
47#123×C127—22no—
48#123×C127—22no—
49#123×C127—22no—
50C22⋊Q8×C127—21no—
51C22⋊Q8×C127—21no—
52C22⋊Q8×C127—21no—
53C22⋊Q8×C127—22no—
54C22⋊Q8×C127—22no—
55C22⋊Q8×C127—21no—
56C22⋊Q8×C127—22no—
57C22⋊Q8×C127C2×C412yes—
58C22⋊Q8×C127D412yes—
59#121×C127—21no—
60#121×C127—21no—
61#121×C127—21no—
62#121×C127—22no—
63#121×C127—21no—
64#121×C127C2×C412yes—
65#121×C127D412yes—
66#120×C127—22no—
67C4×Q8×C127—22no—
68C4×D4×C127—22no—
69C4×D4×C127—22no—
70C4×D4×C127—22no—
71C4×D4×C127—22no—
72C4×D4×C127—22no—
73C4×D4×C127—22no—
74C4×D4×C127—22no—
75#117×C127—22no—
76#117×C127—21no—
77#117×C127—21no—
78#117×C127D412yes—
79C2×C4⋊C4×C127—22no—
80C2×C4⋊C4×C127—22no—
81C2×C4⋊C4×C127—22no—
82C2×C4⋊C4×C127C2×C411yescharacteristic
83C2×C4⋊C4×C127C2×C411yescharacteristic
84C2×C4⋊C4×C127D411yescharacteristic
85C2×C4⋊C4×C127D411yescharacteristic
86C2×C42×C127—22no—
87C2×C42×C127—42no—
88#104×C127—44no—
89C22⋊C8×C127—22no—
90C22⋊C8×C127—22no—
91#95×C127—24no—
92#95×C127—22no—
93#95×C127—42no—
94C24×C127C2×D411yescharacteristic
95#41×C127—42no—
96#41×C127—42no—
97#41×C127—42no—
98#41×C127—42no—
99#41×C127—81no—
100#41×C127—81no—
101C2×Q8×C127—21no—
102C2×Q8×C127—21no—
103C2×Q8×C127C22⋊C412yes—
104C2×D4×C127—21no—
105C2×D4×C127—22no—
106C2×D4×C127—22no—
107C2×D4×C127—22no—
108C2×D4×C127—21no—
109C2×D4×C127C22⋊C412yes—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—21no—
115C22×C4×C127—21no—
116C22×C4×C127—21no—
117C22×C4×C127—22no—
118C22×C4×C127—21no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—41no—
123C22×C4×C127—41no—
124C22×C4×C127—41no—
125C22×C4×C127—41no—
126C22×C4×C127—41no—
127C22×C4×C127—41no—
128C22×C4×C127—41no—
129C22×C4×C127—41no—
130C22×C4×C127—41no—
131C22×C4×C127—41no—
132C22×C4×C127—41no—
133C22×C4×C127—41no—
134C22×C4×C127—41no—
135C22×C4×C127—41no—
136C22×C4×C127—41no—
137C22×C4×C127C22⋊C411yescharacteristic
138C22×C4×C127C22⋊C411yescharacteristic
139C22×C4×C127C22⋊C411yescharacteristic
140C22×C4×C127C22⋊C411yescharacteristic
141C22×C4×C127C22×C411yescharacteristic
142C22×C4×C127C2×D412yes—
143C22×C4×C127C2×D412yes—
144C22×C4×C127C2×D411yescharacteristic
145M16×C127—42no—
146M16×C127—42no—
147C2×C8×C127—42no—
148C4⋊C4×C127—21no—
149C4⋊C4×C127—22no—
150C4⋊C4×C127—21no—
151C4⋊C4×C127—21no—
152C4⋊C4×C127—21no—
153C4⋊C4×C127—22no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—22no—
156C4⋊C4×C127—22no—
157C4⋊C4×C127—22no—
158C4⋊C4×C127—22no—
159C4⋊C4×C127—22no—
160C4⋊C4×C127—22no—
161C4⋊C4×C127—22no—
162C4⋊C4×C127C22⋊C412yes—
163C4⋊C4×C127C22⋊C412yes—
164C22⋊C4×C127—22no—
165C22⋊C4×C127—22no—
166C22⋊C4×C127—22no—
167C22⋊C4×C127—22no—
168C22⋊C4×C127—22no—
169C22⋊C4×C127—22no—
170C22⋊C4×C127—22no—
171C22⋊C4×C127—22no—
172C22⋊C4×C127—22no—
173C42×C127—22no—
174C42×C127—22no—
175C42×C127—22no—
176C42×C127—24no—
177C42×C127—42no—
178C23×C127—21no—
179C23×C127—21no—
180C23×C127—21no—
181C23×C127—21no—
182C23×C127—21no—
183C23×C127—21no—
184C23×C127—22no—
185C23×C127C2×C22⋊C411yescharacteristic
186C23×C127#12011yescharacteristic
187C23×C127#12011yescharacteristic
188Q8×C127—42no—
189Q8×C127—42no—
190Q8×C127—81no—
191D4×C127—42no—
192D4×C127—42no—
193D4×C127—42no—
194D4×C127—42no—
195D4×C127—42no—
196D4×C127—42no—
197D4×C127—42no—
198D4×C127—42no—
199D4×C127—81no—
200C2×C4×C127—21no—
201C2×C4×C127—21no—
202C2×C4×C127—22no—
203C2×C4×C127—22no—
204C2×C4×C127—24no—
205C2×C4×C127—22no—
206C2×C4×C127—22no—
207C2×C4×C127—21no—
208C2×C4×C127—22no—
209C2×C4×C127—21no—
210C2×C4×C127—21no—
211C2×C4×C127—21no—
212C2×C4×C127—21no—
213C2×C4×C127—41no—
214C2×C4×C127—41no—
215C2×C4×C127—42no—
216C2×C4×C127—42no—
217C2×C4×C127—42no—
218C2×C4×C127—44no—
219C2×C4×C127—42no—
220C2×C4×C127—42no—
221C2×C4×C127—42no—
222C2×C4×C127—42no—
223C2×C4×C127—42no—
224C2×C4×C127—42no—
225C2×C4×C127—41no—
226C2×C4×C127—41no—
227C2×C4×C127—41no—
228C2×C4×C127—41no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—82no—
233C2×C4×C127—81no—
234C2×C4×C127—82no—
235C2×C4×C127—81no—
236C2×C4×C127—81no—
237C2×C4×C127—81no—
238C2×C4×C127—81no—
239C2×C4×C127#10412yescomplemented
240C2×C4×C127C2×C22⋊C412yes—
241C2×C4×C127#12012yes—
242C8×C127—82no—
243C22×C127—21no—
244C22×C127—21no—
245C22×C127—21no—
246C22×C127—21no—
247C22×C127—21no—
248C22×C127—21no—
249C22×C127—21no—
250C22×C127—21no—
251C22×C127—21no—
252C22×C127—21no—
253C22×C127—21no—
254C22×C127—21no—
255C22×C127—21no—
256C22×C127—21no—
257C22×C127—42no—
258C22×C127—42no—
259C22×C127—42no—
260C22×C127#37911yescharacteristic
261C22×C127C2×#10411yescharacteristic
262C22×C127#42111yescharacteristic
263C22×C127#45411yescharacteristic
264C22×C127#45611yescharacteristic
265C22×C127B(4,𝔽2)11yescharacteristic
266C22×C127#45811yescharacteristic
267C4×C127—22no—
268C4×C127—22no—
269C4×C127—22no—
270C4×C127—44no—
271C4×C127—42no—
272C4×C127—41no—
273C4×C127—42no—
274C4×C127—41no—
275C4×C127—41no—
276C4×C127—42no—
277C4×C127—82no—
278#8347C12711yesmaximal, characteristic, direct product factor, Sylow
279C2×C127—21no—
280C2×C127—21no—
281C2×C127—21no—
282C2×C127—21no—
283C2×C127—21no—
284C2×C127—21no—
285C2×C127—82no—
286C2×C127#178911yescharacteristic
287C2×C127#179011yescharacteristic
288C2×C127#179311yescharacteristic
289#3492C2×C12711yescharacteristic
290#1896C2×C12712yescomplemented
291#1504C2×C12712yescomplemented
292#1463C2×C12711yescharacteristic, complemented
293#1436C2×C12711yescharacteristic, complemented
294C127#834711yescharacteristic, direct product factor, Sylow
295#554—22no—
296#553—22no—
297#552—22no—
298#547—22no—
299#544C4×C12711yescharacteristic, complemented
300#539—21no—
301#539C4×C12712yescomplemented
302#537C4×C12711yescharacteristic, complemented
303#525—21no—
304#525—21no—
305#525C22×C12712yes—
306#520—21no—
307C22×C4⋊C4C22×C12711yescharacteristic
308C22×C42C22×C12711yescharacteristic
309C2×#104—22no—
310#407C22×C12712yes—
311C2×#95—21no—
312C2×#95—21no—
313C2×#95C22×C12711yescharacteristic
314#339—22no—
315#328—22no—
316#327—22no—
317#324—21no—
318#324—21no—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41D4×C12712yes—
322C23×C4—21no—
323C23×C4—21no—
324C23×C4D4×C12711yescharacteristic
325C23×C4D4×C12711yescharacteristic
326C23×C4C23×C12711yescharacteristic
327C2×M16—22no—
328C4⋊Q8—22no—
329#126—22no—
330#126—22no—
331#125—22no—
332#125—22no—
333#125—22no—
334#123—22no—
335#123—22no—
336#123—22no—
337#123—22no—
338#123—22no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—21no—
345C22⋊Q8—22no—
346C22⋊Q8C2×C4×C12712yes—
347C22⋊Q8D4×C12712yes—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—22no—
352#121—21no—
353#121C2×C4×C12712yes—
354#121D4×C12712yes—
355#120—22no—
356C4×Q8—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364#117—22no—
365#117—21no—
366#117—21no—
367#117D4×C12712yes—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4C2×C4×C12711yescharacteristic
372C2×C4⋊C4C2×C4×C12711yescharacteristic
373C2×C4⋊C4D4×C12711yescharacteristic
374C2×C4⋊C4D4×C12711yescharacteristic
375C2×C42—22no—
376C2×C42—42no—
377#104—44no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380#95—24no—
381#95—22no—
382#95—42no—
383C24C2×D4×C12711yescharacteristic
384#41—42no—
385#41—42no—
386#41—42no—
387#41—42no—
388#41—81no—
389#41—81no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8C22⋊C4×C12712yes—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4C22⋊C4×C12712yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4C22⋊C4×C12711yescharacteristic
427C22×C4C22⋊C4×C12711yescharacteristic
428C22×C4C22⋊C4×C12711yescharacteristic
429C22×C4C22⋊C4×C12711yescharacteristic
430C22×C4C22×C4×C12711yescharacteristic
431C22×C4C2×D4×C12712yes—
432C22×C4C2×D4×C12712yes—
433C22×C4C2×D4×C12711yescharacteristic
434M16—42no—
435M16—42no—
436C2×C8—42no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4C22⋊C4×C12712yes—
452C4⋊C4C22⋊C4×C12712yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—24no—
466C42—42no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23C2×C22⋊C4×C12711yescharacteristic
475C23#120×C12711yescharacteristic
476C23#120×C12711yescharacteristic
477Q8—42no—
478Q8—42no—
479Q8—81no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—81no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4#104×C12712yescomplemented
529C2×C4C2×C22⋊C4×C12712yes—
530C2×C4#120×C12712yes—
531C8—82no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22#379×C12711yescharacteristic
550C22C2×#104×C12711yescharacteristic
551C22#421×C12711yescharacteristic
552C22#454×C12711yescharacteristic
553C22#456×C12711yescharacteristic
554C22B(4,𝔽2)×C12711yescharacteristic
555C22#458×C12711yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—44no—
560C4—41no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—82no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—82no—
574C2C127×#178911yescharacteristic
575C2C127×#179011yescharacteristic
576C2C127×#179311yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3492C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1896C2semidirect
3. C127×#1504C2semidirect
4. C127×#1463C2semidirect
5. C127×#1436C2semidirect
10. #544×C127C4semidirect
12. #539×C127C4semidirect
13. #537×C127C4semidirect
239. C2×C4×C127#104semidirect
278. #8347C127direct (N × H)
290. #1896C2×C127semidirect
291. #1504C2×C127semidirect
292. #1463C2×C127semidirect
293. #1436C2×C127semidirect
294. C127#8347direct (N × H)
299. #544C4×C127semidirect
301. #539C4×C127semidirect
302. #537C4×C127semidirect
528. C2×C4#104×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34922
C127×#18962
C127×#15042
C127×#14632
C127×#14362
#8347127