Finite Group Databasebeta

← C127×#21524 #44585197

Subgroups — C127×#21524 #44585197

Order 32512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466×C127C211yesmaximal, characteristic
2C2×C4×D8×C127C211yesmaximal, characteristic, complemented
3C2×#405×C127C211yesmaximal, characteristic, complemented
4C127×#2324C212yesmaximal, complemented
5C127×#1970C211yesmaximal, characteristic, complemented
6C127×#1942C211yesmaximal, characteristic, complemented
7C127×#1936C212yesmaximal, complemented
8C127×#1895C212yesmaximal, complemented
9C127×#1808C211yesmaximal, characteristic, complemented
10C4×#107×C127C211yesmaximal, characteristic, complemented
11C127×#1762C212yesmaximal, complemented
12C22×D8×C127C2211yescharacteristic
13C22×M16×C127C2211yescharacteristic
14C23×C8×C127C2211yescharacteristic
15C2×#121×C127C2212yes—
16C2×C4×D4×C127C2212yes—
17C2×#117×C127C2211yescharacteristic
18#466×C127C418yescomplemented
19C4×D8×C127—24no—
20C2×#107×C127C2211yescharacteristic
21C2×#107×C127C2211yescharacteristic
22C2×#107×C127C2211yescharacteristic
23C2×#102×C127C2212yes—
24C2×#102×C127C2212yes—
25C2×#102×C127C2212yes—
26#405×C127—24no—
27C2×#97×C127C2211yescharacteristic
28C2×#97×C127C2211yescharacteristic
29C2×C4×C8×C127C2211yescharacteristic
30C2×C4×C8×C127C2211yescharacteristic
31#390×C127—22no—
32#390×C127C2212yes—
33#389×C127C2212yes—
34#388×C127—22no—
35#386×C127—22no—
36#385×C127—22no—
37#380×C127C2212yes—
38C4×C4⋊C4×C127C2212yes—
39#340×C127C2212yes—
40#340×C127C2212yes—
41#340×C127C2212yes—
42C22×D4×C127—22no—
43C22×D4×C127C2312yes—
44C23×C4×C127—22no—
45C23×C4×C127C2311yescharacteristic
46C2×D8×C127—22no—
47C2×D8×C127—22no—
48C2×D8×C127C2×C414yes—
49C2×M16×C127—22no—
50C2×M16×C127—22no—
51C2×M16×C127—22no—
52C22×C8×C127—22no—
53C22×C8×C127—22no—
54C22×C8×C127C2×C414yescomplemented
55C22×C8×C127C2311yescharacteristic
56C22×C8×C127C2311yescharacteristic
57C22×C8×C127C2311yescharacteristic
58C22×C8×C127C2311yescharacteristic
59#121×C127—24no—
60#121×C127C2×C418yescomplemented
61C4×D4×C127—24no—
62C4×D4×C127—24no—
63#117×C127—22no—
64#117×C127—22no—
65C2×C4⋊C4×C127—22no—
66C2×C4⋊C4×C127D411yescharacteristic
67C2×C4⋊C4×C127D411yescharacteristic
68C2×C4⋊C4×C127C2312yes—
69C2×C4⋊C4×C127C2312yes—
70C2×C4⋊C4×C127C2312yes—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—22no—
73C2×C22⋊C4×C127—22no—
74C2×C22⋊C4×C127—22no—
75C2×C22⋊C4×C127—22no—
76C2×C22⋊C4×C127D411yescharacteristic
77C2×C22⋊C4×C127D411yescharacteristic
78C2×C42×C127—22no—
79C2×C42×C127C2311yescharacteristic
80C2×C42×C127C2311yescharacteristic
81#107×C127—22no—
82#107×C127—22no—
83#107×C127C2×C414yescomplemented
84#102×C127—24no—
85#102×C127—24no—
86#102×C127C2×C418yescomplemented
87#97×C127—22no—
88#97×C127—22no—
89C4×C8×C127—22no—
90C4×C8×C127D414yescomplemented
91#95×C127—22no—
92#95×C127—22no—
93#95×C127—22no—
94C24×C127—22no—
95C24×C127C2×D411yescharacteristic
96C2×D4×C127—28no—
97C2×D4×C127—22no—
98C2×D4×C127—22no—
99C2×D4×C127—22no—
100C2×D4×C127—22no—
101C2×D4×C127—22no—
102C2×D4×C127—82no—
103C2×D4×C127—82no—
104C2×D4×C127C22×C414yes—
105C22×C4×C127—21no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—44no—
113C22×C4×C127—82no—
114C22×C4×C127—82no—
115C22×C4×C127C22×C412yes—
116C22×C4×C127C2×D411yescharacteristic
117C22×C4×C127C2×D411yescharacteristic
118C22×C4×C127C2×D411yescharacteristic
119C22×C4×C127C2×D411yescharacteristic
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127#4112yes—
122C22×C4×C127C2411yescharacteristic
123D8×C127—28no—
124M16×C127—28no—
125C2×C8×C127—21no—
126C2×C8×C127—21no—
127C2×C8×C127—21no—
128C2×C8×C127—21no—
129C2×C8×C127—21no—
130C2×C8×C127—21no—
131C2×C8×C127—28no—
132C2×C8×C127C22×C412yes—
133C2×C8×C127C22×C412yes—
134C2×C8×C127C2×D412yes—
135C2×C8×C127C2×D412yes—
136C2×C8×C127#4112yes—
137C2×C8×C127#4112yes—
138C4⋊C4×C127—22no—
139C4⋊C4×C127—22no—
140C4⋊C4×C127—22no—
141C4⋊C4×C127—22no—
142C4⋊C4×C127—22no—
143C4⋊C4×C127—22no—
144C4⋊C4×C127—22no—
145C4⋊C4×C127—82no—
146C4⋊C4×C127C22×C414yes—
147C22⋊C4×C127—28no—
148C22⋊C4×C127—22no—
149C22⋊C4×C127—22no—
150C22⋊C4×C127—44no—
151C22⋊C4×C127—82no—
152C22⋊C4×C127—82no—
153C22⋊C4×C127—82no—
154C42×C127—21no—
155C42×C127—21no—
156C42×C127—82no—
157C42×C127C2×D412yes—
158C42×C127#4112yes—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—21no—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127—21no—
165C23×C127—24no—
166C23×C127—42no—
167C23×C127—42no—
168C23×C127—42no—
169C23×C127—42no—
170C23×C127—42no—
171C23×C127—42no—
172C23×C127C4×D412yes—
173C23×C127C22×D411yescharacteristic
174D4×C127—24no—
175D4×C127—24no—
176D4×C127—42no—
177D4×C127—42no—
178D4×C127—88no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127—21no—
184C2×C4×C127—21no—
185C2×C4×C127—22no—
186C2×C4×C127—22no—
187C2×C4×C127—22no—
188C2×C4×C127—22no—
189C2×C4×C127—24no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127—21no—
194C2×C4×C127—21no—
195C2×C4×C127—21no—
196C2×C4×C127—42no—
197C2×C4×C127—42no—
198C2×C4×C127—42no—
199C2×C4×C127—42no—
200C2×C4×C127—42no—
201C2×C4×C127—42no—
202C2×C4×C127—42no—
203C2×C4×C127—42no—
204C2×C4×C127—88no—
205C2×C4×C127C4×D412yes—
206C2×C4×C127#12112yes—
207C2×C4×C127#12112yes—
208C2×C4×C127C23×C411yescharacteristic
209C2×C4×C127C22×D411yescharacteristic
210C2×C4×C127C2×#4111yescharacteristic
211C2×C4×C127C2×#4111yescharacteristic
212C2×C4×C127C2×#4111yescharacteristic
213C2×C4×C127C2×#4111yescharacteristic
214C8×C127—22no—
215C8×C127—22no—
216C8×C127—22no—
217C8×C127C4×D414yes—
218C22×C127—22no—
219C22×C127—22no—
220C22×C127—22no—
221C22×C127—41no—
222C22×C127—41no—
223C22×C127—41no—
224C22×C127—41no—
225C22×C127—44no—
226C22×C127—44no—
227C22×C127—44no—
228C22×C127—82no—
229C22×C127—82no—
230C22×C127—82no—
231C22×C127—82no—
232C22×C127C2×C4×D411yescharacteristic
233C22×C127C2×#12111yescharacteristic
234C22×C127#53211yescharacteristic
235C22×C127#57611yescharacteristic
236C22×C127#57611yescharacteristic
237C22×C127#57611yescharacteristic
238C22×C127#57811yescharacteristic
239C4×C127—22no—
240C4×C127—22no—
241C4×C127—22no—
242C4×C127—44no—
243C4×C127—41no—
244C4×C127—41no—
245C4×C127—82no—
246C4×C127—82no—
247C4×C127C2×C4×D411yescharacteristic
248C4×C127C4×#4111yescharacteristic
249C4×C127#52511yescharacteristic
250C4×C127#52911yescharacteristic
251#21524C12711yesmaximal, characteristic, direct product factor, Sylow
252C2×C127—42no—
253C2×C127—84no—
254C2×C127C4×#12111yescharacteristic
255C2×C127#294811yescharacteristic
256C2×C127#295111yescharacteristic
257C2×C127#305811yescharacteristic
258C2×C127#306111yescharacteristic
259C2×C127#317511yescharacteristic
260C2×C127#317611yescharacteristic
261C2×#466C2×C12711yescharacteristic
262C2×C4×D8C2×C12711yescharacteristic, complemented
263C2×#405C2×C12711yescharacteristic, complemented
264#2324C2×C12712yescomplemented
265#1970C2×C12711yescharacteristic, complemented
266#1942C2×C12711yescharacteristic, complemented
267#1936C2×C12712yescomplemented
268#1895C2×C12712yescomplemented
269#1808C2×C12711yescharacteristic, complemented
270C4×#107C2×C12711yescharacteristic, complemented
271#1762C2×C12712yescomplemented
272C127#2152411yescharacteristic, direct product factor, Sylow
273C22×D8C22×C12711yescharacteristic
274C22×M16C22×C12711yescharacteristic
275C23×C8C22×C12711yescharacteristic
276C2×#121C22×C12712yes—
277C2×C4×D4C22×C12712yes—
278C2×#117C22×C12711yescharacteristic
279#466C4×C12718yescomplemented
280C4×D8—24no—
281C2×#107C22×C12711yescharacteristic
282C2×#107C22×C12711yescharacteristic
283C2×#107C22×C12711yescharacteristic
284C2×#102C22×C12712yes—
285C2×#102C22×C12712yes—
286C2×#102C22×C12712yes—
287#405—24no—
288C2×#97C22×C12711yescharacteristic
289C2×#97C22×C12711yescharacteristic
290C2×C4×C8C22×C12711yescharacteristic
291C2×C4×C8C22×C12711yescharacteristic
292#390—22no—
293#390C22×C12712yes—
294#389C22×C12712yes—
295#388—22no—
296#386—22no—
297#385—22no—
298#380C22×C12712yes—
299C4×C4⋊C4C22×C12712yes—
300#340C22×C12712yes—
301#340C22×C12712yes—
302#340C22×C12712yes—
303C22×D4—22no—
304C22×D4C23×C12712yes—
305C23×C4—22no—
306C23×C4C23×C12711yescharacteristic
307C2×D8—22no—
308C2×D8—22no—
309C2×D8C2×C4×C12714yes—
310C2×M16—22no—
311C2×M16—22no—
312C2×M16—22no—
313C22×C8—22no—
314C22×C8—22no—
315C22×C8C2×C4×C12714yescomplemented
316C22×C8C23×C12711yescharacteristic
317C22×C8C23×C12711yescharacteristic
318C22×C8C23×C12711yescharacteristic
319C22×C8C23×C12711yescharacteristic
320#121—24no—
321#121C2×C4×C12718yescomplemented
322C4×D4—24no—
323C4×D4—24no—
324#117—22no—
325#117—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4D4×C12711yescharacteristic
328C2×C4⋊C4D4×C12711yescharacteristic
329C2×C4⋊C4C23×C12712yes—
330C2×C4⋊C4C23×C12712yes—
331C2×C4⋊C4C23×C12712yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4D4×C12711yescharacteristic
338C2×C22⋊C4D4×C12711yescharacteristic
339C2×C42—22no—
340C2×C42C23×C12711yescharacteristic
341C2×C42C23×C12711yescharacteristic
342#107—22no—
343#107—22no—
344#107C2×C4×C12714yescomplemented
345#102—24no—
346#102—24no—
347#102C2×C4×C12718yescomplemented
348#97—22no—
349#97—22no—
350C4×C8—22no—
351C4×C8D4×C12714yescomplemented
352#95—22no—
353#95—22no—
354#95—22no—
355C24—22no—
356C24C2×D4×C12711yescharacteristic
357C2×D4—28no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4C22×C4×C12714yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—44no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4C22×C4×C12712yes—
377C22×C4C2×D4×C12711yescharacteristic
378C22×C4C2×D4×C12711yescharacteristic
379C22×C4C2×D4×C12711yescharacteristic
380C22×C4C2×D4×C12711yescharacteristic
381C22×C4C2×D4×C12711yescharacteristic
382C22×C4#41×C12712yes—
383C22×C4C24×C12711yescharacteristic
384D8—28no—
385M16—28no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—28no—
393C2×C8C22×C4×C12712yes—
394C2×C8C22×C4×C12712yes—
395C2×C8C2×D4×C12712yes—
396C2×C8C2×D4×C12712yes—
397C2×C8#41×C12712yes—
398C2×C8#41×C12712yes—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—82no—
407C4⋊C4C22×C4×C12714yes—
408C22⋊C4—28no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—21no—
416C42—21no—
417C42—82no—
418C42C2×D4×C12712yes—
419C42#41×C12712yes—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23C4×D4×C12712yes—
434C23C22×D4×C12711yescharacteristic
435D4—24no—
436D4—24no—
437D4—42no—
438D4—42no—
439D4—88no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—88no—
466C2×C4C4×D4×C12712yes—
467C2×C4#121×C12712yes—
468C2×C4#121×C12712yes—
469C2×C4C23×C4×C12711yescharacteristic
470C2×C4C22×D4×C12711yescharacteristic
471C2×C4C2×#41×C12711yescharacteristic
472C2×C4C2×#41×C12711yescharacteristic
473C2×C4C2×#41×C12711yescharacteristic
474C2×C4C2×#41×C12711yescharacteristic
475C8—22no—
476C8—22no—
477C8—22no—
478C8C4×D4×C12714yes—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22C2×C4×D4×C12711yescharacteristic
494C22C2×#121×C12711yescharacteristic
495C22#532×C12711yescharacteristic
496C22#576×C12711yescharacteristic
497C22#576×C12711yescharacteristic
498C22#576×C12711yescharacteristic
499C22#578×C12711yescharacteristic
500C4—22no—
501C4—22no—
502C4—22no—
503C4—44no—
504C4—41no—
505C4—41no—
506C4—82no—
507C4—82no—
508C4C2×C4×D4×C12711yescharacteristic
509C4C4×#41×C12711yescharacteristic
510C4#525×C12711yescharacteristic
511C4#529×C12711yescharacteristic
512C2—42no—
513C2—84no—
514C2C4×#121×C12711yescharacteristic
515C2C127×#294811yescharacteristic
516C2C127×#295111yescharacteristic
517C2C127×#305811yescharacteristic
518C2C127×#306111yescharacteristic
519C2C127×#317511yescharacteristic
520C2C127×#317611yescharacteristic
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×#466×C127C2
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×C4×D8×C127C2semidirect
3. C2×#405×C127C2semidirect
4. C127×#2324C2semidirect
5. C127×#1970C2semidirect
6. C127×#1942C2semidirect
7. C127×#1936C2semidirect
8. C127×#1895C2semidirect
9. C127×#1808C2semidirect
10. C4×#107×C127C2semidirect
11. C127×#1762C2semidirect
18. #466×C127C4semidirect
54. C22×C8×C127C2×C4semidirect
60. #121×C127C2×C4semidirect
83. #107×C127C2×C4semidirect
86. #102×C127C2×C4semidirect
90. C4×C8×C127D4semidirect
251. #21524C127direct (N × H)
262. C2×C4×D8C2×C127semidirect
263. C2×#405C2×C127semidirect
264. #2324C2×C127semidirect
265. #1970C2×C127semidirect
266. #1942C2×C127semidirect
267. #1936C2×C127semidirect
268. #1895C2×C127semidirect
269. #1808C2×C127semidirect
270. C4×#107C2×C127semidirect
271. #1762C2×C127semidirect
272. C127#21524direct (N × H)
279. #466C4×C127semidirect
315. C22×C8C2×C4×C127semidirect
321. #121C2×C4×C127semidirect
344. #107C2×C4×C127semidirect
347. #102C2×C4×C127semidirect
351. C4×C8D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#466×C1272
C2×C4×D8×C1272
C2×#405×C1272
C127×#23242
C127×#19702
C127×#19422
C127×#19362
C127×#18952
C127×#18082
C4×#107×C1272
C127×#17622
#21524127