Finite Group Databasebeta

← C127×#20876 #44584549

Subgroups — C127×#20876 #44584549

Order 32512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518×C127C211yesmaximal, characteristic
2C2×#428×C127C211yesmaximal, characteristic, complemented
3C2×#418×C127C211yesmaximal, characteristic
4C22×#102×C127C211yesmaximal, characteristic
5C127×#2317C211yesmaximal, characteristic, complemented
6C127×#1864C214yesmaximal, complemented
7C127×#1862C214yesmaximal, complemented
8C127×#1735C211yesmaximal, characteristic, complemented
9C127×#1729C211yesmaximal, characteristic, complemented
10C23×D4×C127C2211yescharacteristic
11C22×M16×C127C2211yescharacteristic
12C23×C8×C127C2211yescharacteristic
13#518×C127—24no—
14#518×C127—24no—
15C2×C4×D4×C127—21no—
16C2×C4×D4×C127—21no—
17C2×C4×D4×C127C2212yes—
18C2×C4×D4×C127C2212yes—
19C2×#117×C127C2211yescharacteristic
20C2×#117×C127C2211yescharacteristic
21C2×#117×C127C2211yescharacteristic
22C22×C4⋊C4×C127C2211yescharacteristic
23C22×C22⋊C4×C127—22no—
24#428×C127—24no—
25C2×#107×C127C2212yescomplemented
26C2×#106×C127C2212yescomplemented
27#418×C127C418yescomplemented
28C2×#102×C127—24no—
29C2×#102×C127C2214yes—
30C2×#102×C127C2214yes—
31#389×C127C2214yescomplemented
32#385×C127—22no—
33#384×C127—21no—
34#382×C127—21no—
35#340×C127C2212yescomplemented
36#340×C127C2212yescomplemented
37#336×C127C2214yescomplemented
38C25×C127—21no—
39C22×D4×C127—21no—
40C22×D4×C127—22no—
41C22×D4×C127—22no—
42C22×D4×C127—22no—
43C22×D4×C127C2×C414yes—
44C22×D4×C127D414yes—
45C22×D4×C127Q814yes—
46C22×D4×C127C2312yes—
47C23×C4×C127—22no—
48C23×C4×C127—22no—
49C23×C4×C127—21no—
50C23×C4×C127C2311yescharacteristic
51C2×M16×C127—22no—
52C2×M16×C127—22no—
53C2×M16×C127C2×C414yescomplemented
54C22×C8×C127—22no—
55C22×C8×C127—22no—
56C22×C8×C127—22no—
57C22×C8×C127C2312yes—
58C22×C8×C127C2312yes—
59C4×D4×C127—28no—
60C4×D4×C127—28no—
61C4×D4×C127—24no—
62C4×D4×C127—24no—
63C4×D4×C127—24no—
64C4×D4×C127—24no—
65#117×C127—22no—
66#117×C127—22no—
67#117×C127—22no—
68#117×C127—22no—
69#117×C127C2×C414yescomplemented
70#117×C127D414yes—
71C2×C4⋊C4×C127—22no—
72C2×C4⋊C4×C127—22no—
73C2×C4⋊C4×C127—21no—
74C2×C4⋊C4×C127—24no—
75C2×C4⋊C4×C127D411yescharacteristic
76C2×C4⋊C4×C127D411yescharacteristic
77C2×C4⋊C4×C127C2312yes—
78C2×C4⋊C4×C127C2312yes—
79C2×C4⋊C4×C127C2311yescharacteristic
80C2×C4⋊C4×C127C2311yescharacteristic
81C2×C22⋊C4×C127—216no—
82C2×C22⋊C4×C127—21no—
83C2×C22⋊C4×C127—21no—
84C2×C22⋊C4×C127—22no—
85C2×C22⋊C4×C127—22no—
86C2×C22⋊C4×C127D412yes—
87C2×C42×C127—21no—
88C2×C42×C127—21no—
89C2×C42×C127C2312yes—
90#107×C127—24no—
91#106×C127—24no—
92#102×C127—28no—
93#102×C127C2×C4116yescomplemented
94#95×C127—24no—
95C24×C127—22no—
96C24×C127—24no—
97C24×C127—24no—
98C24×C127—24no—
99C24×C127—82no—
100C24×C127—82no—
101C24×C127C2×D411yescharacteristic
102C2×D4×C127—28no—
103C2×D4×C127—24no—
104C2×D4×C127—24no—
105C2×D4×C127—28no—
106C2×D4×C127—24no—
107C2×D4×C127—28no—
108C2×D4×C127—22no—
109C2×D4×C127—22no—
110C2×D4×C127—22no—
111C2×D4×C127C4⋊C4116yescomplemented
112C2×D4×C127C22×C414yes—
113C2×D4×C127C2×D414yes—
114C2×D4×C127C2×Q814yes—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127—28no—
119C22×C4×C127—28no—
120C22×C4×C127—22no—
121C22×C4×C127—42no—
122C22×C4×C127—84no—
123C22×C4×C127—84no—
124C22×C4×C127—82no—
125C22×C4×C127C22×C412yes—
126C22×C4×C127C2×D412yes—
127C22×C4×C127C2×D412yes—
128C22×C4×C127C2×D412yes—
129C22×C4×C127C2×D411yescharacteristic
130C22×C4×C127C2×D412yes—
131C22×C4×C127C2×Q812yes—
132C22×C4×C127#4112yes—
133C22×C4×C127#4112yes—
134C22×C4×C127C2411yescharacteristic
135M16×C127—28no—
136C2×C8×C127—22no—
137C2×C8×C127—22no—
138C2×C8×C127—22no—
139C2×C8×C127—22no—
140C2×C8×C127—22no—
141C2×C8×C127—44no—
142C2×C8×C127C22×C414yes—
143C4⋊C4×C127—22no—
144C4⋊C4×C127—22no—
145C4⋊C4×C127—21no—
146C4⋊C4×C127—21no—
147C4⋊C4×C127—22no—
148C4⋊C4×C127—22no—
149C4⋊C4×C127—22no—
150C4⋊C4×C127—42no—
151C4⋊C4×C127—84no—
152C4⋊C4×C127C22×C414yes—
153C4⋊C4×C127C2×D412yes—
154C4⋊C4×C127C2×D412yes—
155C22⋊C4×C127—28no—
156C22⋊C4×C127—28no—
157C22⋊C4×C127—24no—
158C22⋊C4×C127—24no—
159C22⋊C4×C127—24no—
160C42×C127—24no—
161C42×C127—24no—
162C42×C127—22no—
163C42×C127—22no—
164C23×C127—22no—
165C23×C127—22no—
166C23×C127—24no—
167C23×C127—24no—
168C23×C127—24no—
169C23×C127—216no—
170C23×C127—44no—
171C23×C127—44no—
172C23×C127—88no—
173C23×C127—88no—
174C23×C127—84no—
175C23×C127—84no—
176C23×C127C4×D412yes—
177C23×C127#12012yes—
178C23×C127C22⋊Q812yes—
179C23×C127C22×D411yescharacteristic
180D4×C127—24no—
181D4×C127—24no—
182D4×C127—24no—
183D4×C127—216no—
184D4×C127C2×C4⋊C418yes—
185C2×C4×C127—24no—
186C2×C4×C127—22no—
187C2×C4×C127—22no—
188C2×C4×C127—22no—
189C2×C4×C127—22no—
190C2×C4×C127—24no—
191C2×C4×C127—24no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—42no—
196C2×C4×C127—42no—
197C2×C4×C127—44no—
198C2×C4×C127—88no—
199C2×C4×C127—88no—
200C2×C4×C127—84no—
201C2×C4×C127C2×C4⋊C414yes—
202C2×C4×C127C4×D414yes—
203C2×C4×C127C4×D412yes—
204C2×C4×C127#12012yes—
205C2×C4×C127C22⋊Q814yes—
206C2×C4×C127C22⋊Q812yes—
207C2×C4×C127#12314yes—
208C2×C4×C127C23×C411yescharacteristic
209C2×C4×C127C22×D411yescharacteristic
210C2×C4×C127C22×D411yescharacteristic
211C2×C4×C127C22×Q811yescharacteristic
212C2×C4×C127C2×#4111yescharacteristic
213C2×C4×C127C2×#4111yescharacteristic
214C8×C127—24no—
215C8×C127—42no—
216C22×C127—24no—
217C22×C127—24no—
218C22×C127—22no—
219C22×C127—22no—
220C22×C127—28no—
221C22×C127—48no—
222C22×C127—48no—
223C22×C127—44no—
224C22×C127—44no—
225C22×C127—816no—
226C22×C127#38514yes—
227C22×C127C2×C4×D411yescharacteristic
228C22×C127C2×#12011yescharacteristic
229C22×C127C2×C22⋊Q811yescharacteristic
230C22×C127#57612yes—
231C22×C127#57712yes—
232C4×C127—44no—
233C4×C127—42no—
234C4×C127—42no—
235C4×C127—42no—
236C4×C127—42no—
237C4×C127—82no—
238C4×C127#38514yes—
239C4×C127C22×C4⋊C411yescharacteristic
240C4×C127C2×C4×D411yescharacteristic
241C4×C127C2×C22⋊Q811yescharacteristic
242C4×C127C2×#12311yescharacteristic
243#20876C12711yesmaximal, characteristic, direct product factor, Sylow
244C2×C127—24no—
245C2×C127—48no—
246C2×C127C2×#38511yescharacteristic
247C2×C127#294612yes—
248C2×C127#300811yescharacteristic
249C2×C127#300911yescharacteristic
250C2×C127#308112yes—
251C2×#518C2×C12711yescharacteristic
252C2×#428C2×C12711yescharacteristic, complemented
253C2×#418C2×C12711yescharacteristic
254C22×#102C2×C12711yescharacteristic
255#2317C2×C12711yescharacteristic, complemented
256#1864C2×C12714yescomplemented
257#1862C2×C12714yescomplemented
258#1735C2×C12711yescharacteristic, complemented
259#1729C2×C12711yescharacteristic, complemented
260C127#2087611yescharacteristic, direct product factor, Sylow
261C23×D4C22×C12711yescharacteristic
262C22×M16C22×C12711yescharacteristic
263C23×C8C22×C12711yescharacteristic
264#518—24no—
265#518—24no—
266C2×C4×D4—21no—
267C2×C4×D4—21no—
268C2×C4×D4C22×C12712yes—
269C2×C4×D4C22×C12712yes—
270C2×#117C22×C12711yescharacteristic
271C2×#117C22×C12711yescharacteristic
272C2×#117C22×C12711yescharacteristic
273C22×C4⋊C4C22×C12711yescharacteristic
274C22×C22⋊C4—22no—
275#428—24no—
276C2×#107C22×C12712yescomplemented
277C2×#106C22×C12712yescomplemented
278#418C4×C12718yescomplemented
279C2×#102—24no—
280C2×#102C22×C12714yes—
281C2×#102C22×C12714yes—
282#389C22×C12714yescomplemented
283#385—22no—
284#384—21no—
285#382—21no—
286#340C22×C12712yescomplemented
287#340C22×C12712yescomplemented
288#336C22×C12714yescomplemented
289C25—21no—
290C22×D4—21no—
291C22×D4—22no—
292C22×D4—22no—
293C22×D4—22no—
294C22×D4C2×C4×C12714yes—
295C22×D4D4×C12714yes—
296C22×D4Q8×C12714yes—
297C22×D4C23×C12712yes—
298C23×C4—22no—
299C23×C4—22no—
300C23×C4—21no—
301C23×C4C23×C12711yescharacteristic
302C2×M16—22no—
303C2×M16—22no—
304C2×M16C2×C4×C12714yescomplemented
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8C23×C12712yes—
309C22×C8C23×C12712yes—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117C2×C4×C12714yescomplemented
321#117D4×C12714yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4D4×C12711yescharacteristic
327C2×C4⋊C4D4×C12711yescharacteristic
328C2×C4⋊C4C23×C12712yes—
329C2×C4⋊C4C23×C12712yes—
330C2×C4⋊C4C23×C12711yescharacteristic
331C2×C4⋊C4C23×C12711yescharacteristic
332C2×C22⋊C4—216no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4D4×C12712yes—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C23×C12712yes—
341#107—24no—
342#106—24no—
343#102—28no—
344#102C2×C4×C127116yescomplemented
345#95—24no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—82no—
351C24—82no—
352C24C2×D4×C12711yescharacteristic
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—28no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4C4⋊C4×C127116yescomplemented
363C2×D4C22×C4×C12714yes—
364C2×D4C2×D4×C12714yes—
365C2×D4C2×Q8×C12714yes—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—82no—
376C22×C4C22×C4×C12712yes—
377C22×C4C2×D4×C12712yes—
378C22×C4C2×D4×C12712yes—
379C22×C4C2×D4×C12712yes—
380C22×C4C2×D4×C12711yescharacteristic
381C22×C4C2×D4×C12712yes—
382C22×C4C2×Q8×C12712yes—
383C22×C4#41×C12712yes—
384C22×C4#41×C12712yes—
385C22×C4C24×C12711yescharacteristic
386M16—28no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—44no—
393C2×C8C22×C4×C12714yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4C22×C4×C12714yes—
404C4⋊C4C2×D4×C12712yes—
405C4⋊C4C2×D4×C12712yes—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C42—24no—
412C42—24no—
413C42—22no—
414C42—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—216no—
421C23—44no—
422C23—44no—
423C23—88no—
424C23—88no—
425C23—84no—
426C23—84no—
427C23C4×D4×C12712yes—
428C23#120×C12712yes—
429C23C22⋊Q8×C12712yes—
430C23C22×D4×C12711yescharacteristic
431D4—24no—
432D4—24no—
433D4—24no—
434D4—216no—
435D4C2×C4⋊C4×C12718yes—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—88no—
450C2×C4—88no—
451C2×C4—84no—
452C2×C4C2×C4⋊C4×C12714yes—
453C2×C4C4×D4×C12714yes—
454C2×C4C4×D4×C12712yes—
455C2×C4#120×C12712yes—
456C2×C4C22⋊Q8×C12714yes—
457C2×C4C22⋊Q8×C12712yes—
458C2×C4#123×C12714yes—
459C2×C4C23×C4×C12711yescharacteristic
460C2×C4C22×D4×C12711yescharacteristic
461C2×C4C22×D4×C12711yescharacteristic
462C2×C4C22×Q8×C12711yescharacteristic
463C2×C4C2×#41×C12711yescharacteristic
464C2×C4C2×#41×C12711yescharacteristic
465C8—24no—
466C8—42no—
467C22—24no—
468C22—24no—
469C22—22no—
470C22—22no—
471C22—28no—
472C22—48no—
473C22—48no—
474C22—44no—
475C22—44no—
476C22—816no—
477C22#385×C12714yes—
478C22C2×C4×D4×C12711yescharacteristic
479C22C2×#120×C12711yescharacteristic
480C22C2×C22⋊Q8×C12711yescharacteristic
481C22#576×C12712yes—
482C22#577×C12712yes—
483C4—44no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—82no—
489C4#385×C12714yes—
490C4C22×C4⋊C4×C12711yescharacteristic
491C4C2×C4×D4×C12711yescharacteristic
492C4C2×C22⋊Q8×C12711yescharacteristic
493C4C2×#123×C12711yescharacteristic
494C2—24no—
495C2—48no—
496C2C2×#385×C12711yescharacteristic
497C2C127×#294612yes—
498C2C127×#300811yescharacteristic
499C2C127×#300911yescharacteristic
500C2C127×#308112yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×#117×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#428×C127C2semidirect
5. C127×#2317C2semidirect
6. C127×#1864C2semidirect
7. C127×#1862C2semidirect
8. C127×#1735C2semidirect
9. C127×#1729C2semidirect
25. C2×#107×C127C22semidirect
26. C2×#106×C127C22semidirect
27. #418×C127C4semidirect
31. #389×C127C22semidirect
35. #340×C127C22semidirect
36. #340×C127C22semidirect
37. #336×C127C22semidirect
53. C2×M16×C127C2×C4semidirect
69. #117×C127C2×C4semidirect
93. #102×C127C2×C4semidirect
111. C2×D4×C127C4⋊C4semidirect
243. #20876C127direct (N × H)
252. C2×#428C2×C127semidirect
255. #2317C2×C127semidirect
256. #1864C2×C127semidirect
257. #1862C2×C127semidirect
258. #1735C2×C127semidirect
259. #1729C2×C127semidirect
260. C127#20876direct (N × H)
276. C2×#107C22×C127semidirect
277. C2×#106C22×C127semidirect
278. #418C4×C127semidirect
282. #389C22×C127semidirect
286. #340C22×C127semidirect
287. #340C22×C127semidirect
288. #336C22×C127semidirect
304. C2×M16C2×C4×C127semidirect
320. #117C2×C4×C127semidirect
344. #102C2×C4×C127semidirect
362. C2×D4C4⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#518×C1272
C2×#428×C1272
C2×#418×C1272
C22×#102×C1272
C127×#23172
C127×#18642
C127×#18622
C127×#17352
C127×#17292
#20876127