Finite Group Databasebeta

← C127×#8399 #44572072

Subgroups — C127×#8399 #44572072

Order 32512.

868 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3459C211yesmaximal, characteristic
2C127×#1871C212yesmaximal, complemented
3C127×#1870C211yesmaximal, characteristic, complemented
4C4×#99×C127C211yesmaximal, characteristic, complemented
5C127×#1501C212yesmaximal, complemented
6C22×M16×C127C2211yescharacteristic, complemented
7#543×C127C411yescharacteristic, complemented
8#539×C127—22no—
9#539×C127C412yescomplemented
10#538×C127—22no—
11#538×C127—21no—
12#534×C127—21no—
13#534×C127C411yescharacteristic, complemented
14#529×C127—22no—
15#529×C127—21no—
16#525×C127—22no—
17#525×C127C2212yes—
18C2×C4×D4×C127—21no—
19C2×C4×D4×C127C2211yescharacteristic
20C2×#104×C127—22no—
21C2×#99×C127—21no—
22C2×C22⋊C8×C127—21no—
23C2×C22⋊C8×C127C2211yescharacteristic, complemented
24C4×C22⋊C4×C127C2212yescomplemented
25#342×C127—21no—
26#339×C127—22no—
27#336×C127—21no—
28#328×C127—22no—
29#327×C127—22no—
30C2×#41×C127D412yes—
31C22×D4×C127—21no—
32C22×D4×C127D411yescharacteristic
33C23×C4×C127—21no—
34C23×C4×C127D411yescharacteristic
35C23×C4×C127C2311yescharacteristic
36C2×M16×C127—22no—
37C2×M16×C127—42no—
38C22×C8×C127—21no—
39C22×C8×C127—21no—
40#126×C127—22no—
41#126×C127—22no—
42#126×C127—22no—
43#125×C127—21no—
44#125×C127—22no—
45#124×C127—22no—
46#124×C127—21no—
47#123×C127—22no—
48#123×C127—22no—
49#123×C127—22no—
50#123×C127—22no—
51#123×C127—22no—
52#123×C127—22no—
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—22no—
60#121×C127—22no—
61#121×C127—22no—
62#121×C127—21no—
63#121×C127C2×C412yes—
64#121×C127D412yes—
65#120×C127—22no—
66#120×C127—22no—
67#120×C127—22no—
68C4×Q8×C127—21no—
69C4×D4×C127—22no—
70C4×D4×C127—22no—
71C4×D4×C127—21no—
72C4×D4×C127—22no—
73C4×D4×C127—22no—
74C4×D4×C127—22no—
75C4×D4×C127—22no—
76C4×D4×C127—22no—
77#117×C127—22no—
78#117×C127D412yes—
79C2×C4⋊C4×C127—21no—
80C2×C4⋊C4×C127C2×C411yescharacteristic
81C2×C22⋊C4×C127—21no—
82C2×C22⋊C4×C127—21no—
83C2×C22⋊C4×C127—22no—
84C2×C22⋊C4×C127C2×C411yescharacteristic
85C2×C22⋊C4×C127D411yescharacteristic
86C2×C42×C127—21no—
87C2×C42×C127—22no—
88C2×C42×C127D411yescharacteristic
89#104×C127—44no—
90#99×C127—24no—
91C22⋊C8×C127—22no—
92C22⋊C8×C127—42no—
93#95×C127—22no—
94C24×C127—21no—
95C24×C127C22⋊C411yescharacteristic
96C24×C127C2×D411yescharacteristic
97#41×C127—42no—
98#41×C127—42no—
99C2×Q8×C127—21no—
100C2×Q8×C127C22⋊C412yes—
101C2×D4×C127—22no—
102C2×D4×C127—22no—
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127—21no—
106C2×D4×C127—22no—
107C2×D4×C127—22no—
108C2×D4×C127—41no—
109C2×D4×C127—42no—
110C2×D4×C127—42no—
111C2×D4×C127C22⋊C412yes—
112C22×C4×C127—21no—
113C22×C4×C127—21no—
114C22×C4×C127—21no—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—22no—
118C22×C4×C127—21no—
119C22×C4×C127—41no—
120C22×C4×C127—41no—
121C22×C4×C127—41no—
122C22×C4×C127—41no—
123C22×C4×C127—42no—
124C22×C4×C127—81no—
125C22×C4×C127—81no—
126C22×C4×C127C22⋊C411yescharacteristic
127C22×C4×C127C22⋊C411yescharacteristic
128C22×C4×C127C22⋊C411yescharacteristic
129C22×C4×C127C22×C411yescharacteristic
130C22×C4×C127C2×D412yes—
131C22×C4×C127C2×D411yescharacteristic
132C22×C4×C127C2×D412yes—
133M16×C127—24no—
134M16×C127—42no—
135C2×C8×C127—22no—
136C2×C8×C127—41no—
137C2×C8×C127—41no—
138C2×C8×C127—41no—
139C2×C8×C127—41no—
140C2×C8×C127—41no—
141C4⋊C4×C127—22no—
142C4⋊C4×C127—22no—
143C4⋊C4×C127—22no—
144C4⋊C4×C127—21no—
145C4⋊C4×C127—21no—
146C4⋊C4×C127—22no—
147C4⋊C4×C127—22no—
148C4⋊C4×C127—22no—
149C4⋊C4×C127—22no—
150C4⋊C4×C127C22⋊C412yes—
151C4⋊C4×C127C22⋊C412yes—
152C22⋊C4×C127—22no—
153C22⋊C4×C127—22no—
154C22⋊C4×C127—22no—
155C22⋊C4×C127—22no—
156C22⋊C4×C127—22no—
157C22⋊C4×C127—22no—
158C22⋊C4×C127—22no—
159C22⋊C4×C127—22no—
160C22⋊C4×C127—22no—
161C22⋊C4×C127—22no—
162C22⋊C4×C127—22no—
163C22⋊C4×C127—22no—
164C22⋊C4×C127—24no—
165C22⋊C4×C127—42no—
166C42×C127—22no—
167C42×C127—21no—
168C42×C127—22no—
169C42×C127—21no—
170C42×C127—42no—
171C42×C127—42no—
172C23×C127—21no—
173C23×C127—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127—22no—
177C23×C127—41no—
178C23×C127—41no—
179C23×C127—41no—
180C23×C127—41no—
181C23×C127—42no—
182C23×C127—81no—
183C23×C127—81no—
184C23×C127#10412yescomplemented
185C23×C127C2×C22⋊C411yescharacteristic
186C23×C127#12012yes—
187Q8×C127—42no—
188Q8×C127—81no—
189D4×C127—42no—
190D4×C127—44no—
191D4×C127—44no—
192D4×C127—42no—
193D4×C127—42no—
194D4×C127—82no—
195D4×C127—82no—
196D4×C127—81no—
197C2×C4×C127—22no—
198C2×C4×C127—21no—
199C2×C4×C127—22no—
200C2×C4×C127—21no—
201C2×C4×C127—21no—
202C2×C4×C127—22no—
203C2×C4×C127—21no—
204C2×C4×C127—22no—
205C2×C4×C127—21no—
206C2×C4×C127—22no—
207C2×C4×C127—21no—
208C2×C4×C127—21no—
209C2×C4×C127—41no—
210C2×C4×C127—42no—
211C2×C4×C127—42no—
212C2×C4×C127—41no—
213C2×C4×C127—42no—
214C2×C4×C127—41no—
215C2×C4×C127—42no—
216C2×C4×C127—41no—
217C2×C4×C127—41no—
218C2×C4×C127—44no—
219C2×C4×C127—81no—
220C2×C4×C127—81no—
221C2×C4×C127—82no—
222C2×C4×C127—81no—
223C2×C4×C127—82no—
224C2×C4×C127—82no—
225C2×C4×C127C2×C22⋊C412yes—
226C2×C4×C127#12011yescharacteristic
227C2×C4×C127#12011yescharacteristic
228C8×C127—42no—
229C8×C127—81no—
230C22×C127—21no—
231C22×C127—21no—
232C22×C127—22no—
233C22×C127—22no—
234C22×C127—22no—
235C22×C127—22no—
236C22×C127—42no—
237C22×C127—41no—
238C22×C127—41no—
239C22×C127—42no—
240C22×C127—42no—
241C22×C127—42no—
242C22×C127—81no—
243C22×C127—81no—
244C22×C127—81no—
245C22×C127—84no—
246C22×C127#37911yescharacteristic
247C22×C127C2×#10411yescharacteristic
248C22×C127#42111yescharacteristic
249C4×C127—21no—
250C4×C127—22no—
251C4×C127—41no—
252C4×C127—42no—
253C4×C127—41no—
254C4×C127—42no—
255C4×C127—41no—
256C4×C127—42no—
257C4×C127—84no—
258C4×C127B(4,𝔽2)11yescharacteristic
259C4×C127#45811yescharacteristic
260#8399C12711yesmaximal, characteristic, direct product factor, Sylow
261C2×C127—21no—
262C2×C127—21no—
263C2×C127—42no—
264C2×C127—42no—
265C2×C127—82no—
266C2×C127#178911yescharacteristic
267C2×C127#179011yescharacteristic
268C2×C127#179411yescharacteristic
269#3459C2×C12711yescharacteristic
270#1871C2×C12712yescomplemented
271#1870C2×C12711yescharacteristic, complemented
272C4×#99C2×C12711yescharacteristic, complemented
273#1501C2×C12712yescomplemented
274C127#839911yescharacteristic, direct product factor, Sylow
275C22×M16C22×C12711yescharacteristic, complemented
276#543C4×C12711yescharacteristic, complemented
277#539—22no—
278#539C4×C12712yescomplemented
279#538—22no—
280#538—21no—
281#534—21no—
282#534C4×C12711yescharacteristic, complemented
283#529—22no—
284#529—21no—
285#525—22no—
286#525C22×C12712yes—
287C2×C4×D4—21no—
288C2×C4×D4C22×C12711yescharacteristic
289C2×#104—22no—
290C2×#99—21no—
291C2×C22⋊C8—21no—
292C2×C22⋊C8C22×C12711yescharacteristic, complemented
293C4×C22⋊C4C22×C12712yescomplemented
294#342—21no—
295#339—22no—
296#336—21no—
297#328—22no—
298#327—22no—
299C2×#41D4×C12712yes—
300C22×D4—21no—
301C22×D4D4×C12711yescharacteristic
302C23×C4—21no—
303C23×C4D4×C12711yescharacteristic
304C23×C4C23×C12711yescharacteristic
305C2×M16—22no—
306C2×M16—42no—
307C22×C8—21no—
308C22×C8—21no—
309#126—22no—
310#126—22no—
311#126—22no—
312#125—21no—
313#125—22no—
314#124—22no—
315#124—21no—
316#123—22no—
317#123—22no—
318#123—22no—
319#123—22no—
320#123—22no—
321#123—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8C2×C4×C12712yes—
327C22⋊Q8D4×C12712yes—
328#121—22no—
329#121—22no—
330#121—22no—
331#121—21no—
332#121C2×C4×C12712yes—
333#121D4×C12712yes—
334#120—22no—
335#120—22no—
336#120—22no—
337C4×Q8—21no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346#117—22no—
347#117D4×C12712yes—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C2×C4×C12711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4C2×C4×C12711yescharacteristic
354C2×C22⋊C4D4×C12711yescharacteristic
355C2×C42—21no—
356C2×C42—22no—
357C2×C42D4×C12711yescharacteristic
358#104—44no—
359#99—24no—
360C22⋊C8—22no—
361C22⋊C8—42no—
362#95—22no—
363C24—21no—
364C24C22⋊C4×C12711yescharacteristic
365C24C2×D4×C12711yescharacteristic
366#41—42no—
367#41—42no—
368C2×Q8—21no—
369C2×Q8C22⋊C4×C12712yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4C22⋊C4×C12712yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4C22⋊C4×C12711yescharacteristic
396C22×C4C22⋊C4×C12711yescharacteristic
397C22×C4C22⋊C4×C12711yescharacteristic
398C22×C4C22×C4×C12711yescharacteristic
399C22×C4C2×D4×C12712yes—
400C22×C4C2×D4×C12711yescharacteristic
401C22×C4C2×D4×C12712yes—
402M16—24no—
403M16—42no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4C22⋊C4×C12712yes—
420C4⋊C4C22⋊C4×C12712yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42—42no—
440C42—42no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—81no—
452C23—81no—
453C23#104×C12712yescomplemented
454C23C2×C22⋊C4×C12711yescharacteristic
455C23#120×C12712yes—
456Q8—42no—
457Q8—81no—
458D4—42no—
459D4—44no—
460D4—44no—
461D4—42no—
462D4—42no—
463D4—82no—
464D4—82no—
465D4—81no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4C2×C22⋊C4×C12712yes—
495C2×C4#120×C12711yescharacteristic
496C2×C4#120×C12711yescharacteristic
497C8—42no—
498C8—81no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—84no—
515C22#379×C12711yescharacteristic
516C22C2×#104×C12711yescharacteristic
517C22#421×C12711yescharacteristic
518C4—21no—
519C4—22no—
520C4—41no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—84no—
527C4B(4,𝔽2)×C12711yescharacteristic
528C4#458×C12711yescharacteristic
529C2—21no—
530C2—21no—
531C2—42no—
532C2—42no—
533C2—82no—
534C2C127×#178911yescharacteristic
535C2C127×#179011yescharacteristic
536C2C127×#179411yescharacteristic
subgroupG/H
centerC2×C4×C127#120
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
socleC22×C127#379
prime radicalC127×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1871C2semidirect
3. C127×#1870C2semidirect
4. C4×#99×C127C2semidirect
5. C127×#1501C2semidirect
6. C22×M16×C127C22semidirect
7. #543×C127C4semidirect
9. #539×C127C4semidirect
13. #534×C127C4semidirect
23. C2×C22⋊C8×C127C22semidirect
24. C4×C22⋊C4×C127C22semidirect
184. C23×C127#104semidirect
260. #8399C127direct (N × H)
270. #1871C2×C127semidirect
271. #1870C2×C127semidirect
272. C4×#99C2×C127semidirect
273. #1501C2×C127semidirect
274. C127#8399direct (N × H)
275. C22×M16C22×C127semidirect
276. #543C4×C127semidirect
278. #539C4×C127semidirect
282. #534C4×C127semidirect
292. C2×C22⋊C8C22×C127semidirect
293. C4×C22⋊C4C22×C127semidirect
453. C23#104×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34592
C127×#18712
C127×#18702
C4×#99×C1272
C127×#15012
#8399127