Finite Group Databasebeta

← C127×#21764 #44585437

Subgroups — C127×#21764 #44585437

Order 32512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121×C127C211yesmaximal, characteristic
2C2×#407×C127C211yesmaximal, characteristic, complemented
3C2×#380×C127C211yesmaximal, characteristic, complemented
4C127×#1895C214yesmaximal, complemented
5C127×#1874C214yesmaximal, complemented
6C127×#1824C212yesmaximal, complemented
7C127×#1786C212yesmaximal, complemented
8C23×D4×C127—21no—
9C23×D4×C127C2211yescharacteristic
10C24×C4×C127C2211yescharacteristic
11C22×M16×C127C2212yescomplemented
12C2×#121×C127—24no—
13C2×#121×C127—21no—
14C2×#121×C127—21no—
15C2×#121×C127C418yescomplemented
16C2×#121×C127C2212yes—
17C2×#121×C127C2212yes—
18C22×C4⋊C4×C127C2211yescharacteristic
19C22×C4⋊C4×C127C2212yescomplemented
20C22×C22⋊C4×C127—21no—
21C22×C22⋊C4×C127—21no—
22C2×#102×C127C2218yescomplemented
23#407×C127—24no—
24C2×C22⋊C8×C127C2212yes—
25C2×C22⋊C8×C127C2212yescomplemented
26#380×C127—21no—
27#380×C127—21no—
28#380×C127C2212yes—
29#380×C127C2212yescomplemented
30C2×#95×C127—21no—
31#340×C127C2218yescomplemented
32C25×C127—21no—
33C25×C127D411yescharacteristic
34C22×D4×C127—28no—
35C22×D4×C127—22no—
36C22×D4×C127—22no—
37C22×D4×C127—21no—
38C22×D4×C127—22no—
39C22×D4×C127—44no—
40C22×D4×C127—48no—
41C22×D4×C127C2×C414yescomplemented
42C22×D4×C127C2312yes—
43C23×C4×C127—21no—
44C23×C4×C127—22no—
45C23×C4×C127—44no—
46C23×C4×C127C2×C414yes—
47C23×C4×C127D411yescharacteristic
48C23×C4×C127D411yescharacteristic
49C23×C4×C127D411yescharacteristic
50C23×C4×C127D414yes—
51C23×C4×C127C2311yescharacteristic
52C23×C4×C127C2311yescharacteristic
53C23×C4×C127C2311yescharacteristic
54C2×M16×C127—24no—
55C2×M16×C127—44no—
56C22×C8×C127C2314yes—
57#121×C127—28no—
58#121×C127—28no—
59#121×C127—24no—
60#121×C127—24no—
61#121×C127C2×C418yes—
62#121×C127C2×C418yescomplemented
63C2×C4⋊C4×C127—21no—
64C2×C4⋊C4×C127—22no—
65C2×C4⋊C4×C127—22no—
66C2×C4⋊C4×C127—44no—
67C2×C4⋊C4×C127C2×C414yescomplemented
68C2×C4⋊C4×C127C2314yes—
69C2×C4⋊C4×C127C2312yes—
70C2×C22⋊C4×C127—28no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—22no—
73C2×C22⋊C4×C127—22no—
74C2×C22⋊C4×C127—22no—
75C2×C22⋊C4×C127—44no—
76#102×C127—48no—
77C22⋊C8×C127—24no—
78C22⋊C8×C127—24no—
79#95×C127—22no—
80#95×C127—22no—
81C24×C127—22no—
82C24×C127—24no—
83C24×C127—22no—
84C24×C127—24no—
85C24×C127—22no—
86C24×C127—42no—
87C24×C127—44no—
88C24×C127—44no—
89C24×C127—42no—
90C24×C127—42no—
91C24×C127C22⋊C414yes—
92C24×C127C2×D411yescharacteristic
93C24×C127C2×D411yescharacteristic
94C24×C127C2×D411yescharacteristic
95C2×D4×C127—28no—
96C2×D4×C127—28no—
97C2×D4×C127—24no—
98C2×D4×C127—22no—
99C2×D4×C127—28no—
100C2×D4×C127—48no—
101C2×D4×C127—44no—
102C2×D4×C127—48no—
103C2×D4×C127—432no—
104C2×D4×C127—48no—
105C2×D4×C127—48no—
106C2×D4×C127C22×C414yes—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127—24no—
111C22×C4×C127—22no—
112C22×C4×C127—24no—
113C22×C4×C127—48no—
114C22×C4×C127—48no—
115C22×C4×C127—42no—
116C22×C4×C127—42no—
117C22×C4×C127—42no—
118C22×C4×C127—44no—
119C22×C4×C127—44no—
120C22×C4×C127—44no—
121C22×C4×C127—44no—
122C22×C4×C127C22⋊C414yes—
123C22×C4×C127C22⋊C414yes—
124C22×C4×C127C22⋊C414yes—
125C22×C4×C127C22×C412yes—
126C22×C4×C127C22×C412yes—
127C22×C4×C127C22×C412yes—
128C22×C4×C127C2×D411yescharacteristic
129C22×C4×C127C2×D411yescharacteristic
130C22×C4×C127C2×D411yescharacteristic
131C22×C4×C127C2×D412yes—
132C22×C4×C127C2×D412yes—
133C22×C4×C127C2×D412yes—
134C22×C4×C127C2411yescharacteristic
135M16×C127—48no—
136C2×C8×C127—24no—
137C2×C8×C127—28no—
138C4⋊C4×C127—24no—
139C4⋊C4×C127—22no—
140C4⋊C4×C127—28no—
141C4⋊C4×C127—44no—
142C4⋊C4×C127C22×C414yes—
143C22⋊C4×C127—28no—
144C22⋊C4×C127—28no—
145C22⋊C4×C127—44no—
146C22⋊C4×C127—44no—
147C23×C127—22no—
148C23×C127—22no—
149C23×C127—21no—
150C23×C127—22no—
151C23×C127—21no—
152C23×C127—22no—
153C23×C127—21no—
154C23×C127—22no—
155C23×C127—21no—
156C23×C127—28no—
157C23×C127—28no—
158C23×C127—24no—
159C23×C127—44no—
160C23×C127—48no—
161C23×C127—48no—
162C23×C127—48no—
163C23×C127—48no—
164C23×C127—44no—
165C23×C127—44no—
166C23×C127—42no—
167C23×C127—42no—
168C23×C127—48no—
169C23×C127—88no—
170C23×C127—88no—
171C23×C127C2×C22⋊C412yes—
172C23×C127C2×C22⋊C412yes—
173C23×C127C2×C22⋊C412yes—
174C23×C127AGL(1,ℤ/8)12yes—
175C23×C127AGL(1,ℤ/8)14yes—
176C23×C127#13712yes—
177C23×C127C22×D411yescharacteristic
178D4×C127—28no—
179D4×C127—48no—
180D4×C127—416no—
181D4×C127—416no—
182D4×C127—44no—
183C2×C4×C127—22no—
184C2×C4×C127—22no—
185C2×C4×C127—22no—
186C2×C4×C127—24no—
187C2×C4×C127—24no—
188C2×C4×C127—24no—
189C2×C4×C127—24no—
190C2×C4×C127—24no—
191C2×C4×C127—24no—
192C2×C4×C127—24no—
193C2×C4×C127—44no—
194C2×C4×C127—44no—
195C2×C4×C127—44no—
196C2×C4×C127—48no—
197C2×C4×C127—44no—
198C2×C4×C127—44no—
199C2×C4×C127—44no—
200C2×C4×C127—44no—
201C2×C4×C127—88no—
202C2×C4×C127C2×C22⋊C412yes—
203C2×C4×C127C2×C22⋊C412yes—
204C2×C4×C127C2×C22⋊C412yes—
205C2×C4×C127C23×C411yescharacteristic
206C2×C4×C127C22×D411yescharacteristic
207C2×C4×C127H(5,𝔽2)14yes—
208C8×C127—44no—
209C22×C127—22no—
210C22×C127—22no—
211C22×C127—22no—
212C22×C127—22no—
213C22×C127—24no—
214C22×C127—24no—
215C22×C127—24no—
216C22×C127—24no—
217C22×C127—22no—
218C22×C127—22no—
219C22×C127—22no—
220C22×C127—416no—
221C22×C127—42no—
222C22×C127—42no—
223C22×C127—44no—
224C22×C127—44no—
225C22×C127—44no—
226C22×C127—816no—
227C22×C127—84no—
228C22×C127—84no—
229C22×C127#41714yes—
230C22×C127#41814yes—
231C22×C127C22×C22⋊C411yescharacteristic
232C22×C127#53412yes—
233C22×C127C2×AGL(1,ℤ/8)12yes—
234C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
235C22×C127C2×#13711yescharacteristic
236C4×C127—22no—
237C4×C127—22no—
238C4×C127—22no—
239C4×C127—44no—
240C4×C127—84no—
241C4×C127#51812yes—
242C4×C127#53412yes—
243#21764C12711yesmaximal, characteristic, direct product factor, Sylow
244C2×C127—24no—
245C2×C127—24no—
246C2×C127—42no—
247C2×C127—84no—
248C2×C127#230811yescharacteristic
249C2×C127C2×#41811yescharacteristic
250C2×C127C2×#41711yescharacteristic
251C2×C127#319212yes—
252C2×C127#319312yes—
253C22×#121C2×C12711yescharacteristic
254C2×#407C2×C12711yescharacteristic, complemented
255C2×#380C2×C12711yescharacteristic, complemented
256#1895C2×C12714yescomplemented
257#1874C2×C12714yescomplemented
258#1824C2×C12712yescomplemented
259#1786C2×C12712yescomplemented
260C127#2176411yescharacteristic, direct product factor, Sylow
261C23×D4—21no—
262C23×D4C22×C12711yescharacteristic
263C24×C4C22×C12711yescharacteristic
264C22×M16C22×C12712yescomplemented
265C2×#121—24no—
266C2×#121—21no—
267C2×#121—21no—
268C2×#121C4×C12718yescomplemented
269C2×#121C22×C12712yes—
270C2×#121C22×C12712yes—
271C22×C4⋊C4C22×C12711yescharacteristic
272C22×C4⋊C4C22×C12712yescomplemented
273C22×C22⋊C4—21no—
274C22×C22⋊C4—21no—
275C2×#102C22×C12718yescomplemented
276#407—24no—
277C2×C22⋊C8C22×C12712yes—
278C2×C22⋊C8C22×C12712yescomplemented
279#380—21no—
280#380—21no—
281#380C22×C12712yes—
282#380C22×C12712yescomplemented
283C2×#95—21no—
284#340C22×C12718yescomplemented
285C25—21no—
286C25D4×C12711yescharacteristic
287C22×D4—28no—
288C22×D4—22no—
289C22×D4—22no—
290C22×D4—21no—
291C22×D4—22no—
292C22×D4—44no—
293C22×D4—48no—
294C22×D4C2×C4×C12714yescomplemented
295C22×D4C23×C12712yes—
296C23×C4—21no—
297C23×C4—22no—
298C23×C4—44no—
299C23×C4C2×C4×C12714yes—
300C23×C4D4×C12711yescharacteristic
301C23×C4D4×C12711yescharacteristic
302C23×C4D4×C12711yescharacteristic
303C23×C4D4×C12714yes—
304C23×C4C23×C12711yescharacteristic
305C23×C4C23×C12711yescharacteristic
306C23×C4C23×C12711yescharacteristic
307C2×M16—24no—
308C2×M16—44no—
309C22×C8C23×C12714yes—
310#121—28no—
311#121—28no—
312#121—24no—
313#121—24no—
314#121C2×C4×C12718yes—
315#121C2×C4×C12718yescomplemented
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×C4×C12714yescomplemented
321C2×C4⋊C4C23×C12714yes—
322C2×C4⋊C4C23×C12712yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—44no—
329#102—48no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332#95—22no—
333#95—22no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—42no—
340C24—44no—
341C24—44no—
342C24—42no—
343C24—42no—
344C24C22⋊C4×C12714yes—
345C24C2×D4×C12711yescharacteristic
346C24C2×D4×C12711yescharacteristic
347C24C2×D4×C12711yescharacteristic
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—28no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—432no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4C22×C4×C12714yes—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4C22⋊C4×C12714yes—
376C22×C4C22⋊C4×C12714yes—
377C22×C4C22⋊C4×C12714yes—
378C22×C4C22×C4×C12712yes—
379C22×C4C22×C4×C12712yes—
380C22×C4C22×C4×C12712yes—
381C22×C4C2×D4×C12711yescharacteristic
382C22×C4C2×D4×C12711yescharacteristic
383C22×C4C2×D4×C12711yescharacteristic
384C22×C4C2×D4×C12712yes—
385C22×C4C2×D4×C12712yes—
386C22×C4C2×D4×C12712yes—
387C22×C4C24×C12711yescharacteristic
388M16—48no—
389C2×C8—24no—
390C2×C8—28no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—28no—
394C4⋊C4—44no—
395C4⋊C4C22×C4×C12714yes—
396C22⋊C4—28no—
397C22⋊C4—28no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C23—22no—
401C23—22no—
402C23—21no—
403C23—22no—
404C23—21no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23—21no—
409C23—28no—
410C23—28no—
411C23—24no—
412C23—44no—
413C23—48no—
414C23—48no—
415C23—48no—
416C23—48no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—48no—
422C23—88no—
423C23—88no—
424C23C2×C22⋊C4×C12712yes—
425C23C2×C22⋊C4×C12712yes—
426C23C2×C22⋊C4×C12712yes—
427C23AGL(1,ℤ/8)×C12712yes—
428C23AGL(1,ℤ/8)×C12714yes—
429C23#137×C12712yes—
430C23C22×D4×C12711yescharacteristic
431D4—28no—
432D4—48no—
433D4—416no—
434D4—416no—
435D4—44no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—88no—
455C2×C4C2×C22⋊C4×C12712yes—
456C2×C4C2×C22⋊C4×C12712yes—
457C2×C4C2×C22⋊C4×C12712yes—
458C2×C4C23×C4×C12711yescharacteristic
459C2×C4C22×D4×C12711yescharacteristic
460C2×C4H(5,𝔽2)×C12714yes—
461C8—44no—
462C22—22no—
463C22—22no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—416no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—816no—
480C22—84no—
481C22—84no—
482C22#417×C12714yes—
483C22#418×C12714yes—
484C22C22×C22⋊C4×C12711yescharacteristic
485C22#534×C12712yes—
486C22C2×AGL(1,ℤ/8)×C12712yes—
487C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
488C22C2×#137×C12711yescharacteristic
489C4—22no—
490C4—22no—
491C4—22no—
492C4—44no—
493C4—84no—
494C4#518×C12712yes—
495C4#534×C12712yes—
496C2—24no—
497C2—24no—
498C2—42no—
499C2—84no—
500C2C127×#230811yescharacteristic
501C2C2×#418×C12711yescharacteristic
502C2C2×#417×C12711yescharacteristic
503C2C127×#319212yes—
504C2C127×#319312yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×#121×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#407×C127C2semidirect
3. C2×#380×C127C2semidirect
4. C127×#1895C2semidirect
5. C127×#1874C2semidirect
6. C127×#1824C2semidirect
7. C127×#1786C2semidirect
11. C22×M16×C127C22semidirect
15. C2×#121×C127C4semidirect
19. C22×C4⋊C4×C127C22semidirect
22. C2×#102×C127C22semidirect
25. C2×C22⋊C8×C127C22semidirect
29. #380×C127C22semidirect
31. #340×C127C22semidirect
41. C22×D4×C127C2×C4semidirect
62. #121×C127C2×C4semidirect
67. C2×C4⋊C4×C127C2×C4semidirect
243. #21764C127direct (N × H)
254. C2×#407C2×C127semidirect
255. C2×#380C2×C127semidirect
256. #1895C2×C127semidirect
257. #1874C2×C127semidirect
258. #1824C2×C127semidirect
259. #1786C2×C127semidirect
260. C127#21764direct (N × H)
264. C22×M16C22×C127semidirect
268. C2×#121C4×C127semidirect
272. C22×C4⋊C4C22×C127semidirect
275. C2×#102C22×C127semidirect
278. C2×C22⋊C8C22×C127semidirect
282. #380C22×C127semidirect
284. #340C22×C127semidirect
294. C22×D4C2×C4×C127semidirect
315. #121C2×C4×C127semidirect
320. C2×C4⋊C4C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#121×C1272
C2×#407×C1272
C2×#380×C1272
C127×#18952
C127×#18742
C127×#18242
C127×#17862
#21764127