Finite Group Databasebeta

← C4×#469×C127 #44585206

Subgroups — C4×#469×C127 #44585206

Order 32512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469×C127C211yesmaximal, characteristic
2C2×C4×D8×C127C211yesmaximal, characteristic, complemented
3C2×C4×M16×C127C211yesmaximal, characteristic, complemented
4C4×#121×C127C212yesmaximal, complemented
5C127×#1972C211yesmaximal, characteristic, complemented
6C127×#1944C211yesmaximal, characteristic, complemented
7C127×#1936C212yesmaximal, complemented
8C127×#1895C212yesmaximal, complemented
9C127×#1809C211yesmaximal, characteristic, complemented
10C4×#106×C127C211yesmaximal, characteristic, complemented
11C4×#102×C127C212yesmaximal, complemented
12C22×D8×C127C2211yescharacteristic
13C22×M16×C127C2211yescharacteristic
14C22×M16×C127C2211yescharacteristic
15C2×#121×C127C2212yes—
16C2×C4×D4×C127—22no—
17C2×C4×D4×C127C2212yes—
18C22×C42×C127C2211yescharacteristic
19#469×C127C418yesdirect product factor
20C4×D8×C127—24no—
21C2×#107×C127C2211yescharacteristic
22C2×#106×C127C2211yescharacteristic
23C2×#106×C127C2211yescharacteristic
24C2×#102×C127C2212yes—
25C2×#102×C127C2212yes—
26C2×#102×C127C2212yes—
27C4×M16×C127—24no—
28C2×#97×C127C2211yescharacteristic
29C2×#97×C127C2211yescharacteristic
30C2×C4×C8×C127C2211yescharacteristic
31C2×C4×C8×C127C2211yescharacteristic
32#390×C127C2212yes—
33#389×C127C2212yes—
34#388×C127—22no—
35#386×C127—22no—
36#380×C127C2212yes—
37C4×C4⋊C4×C127C2212yes—
38C4×C22⋊C4×C127—22no—
39#340×C127C2212yes—
40#340×C127C2212yes—
41#340×C127C2212yes—
42C22×D4×C127—22no—
43C22×D4×C127C2312yes—
44C23×C4×C127—22no—
45C23×C4×C127D411yescharacteristic
46C23×C4×C127D411yescharacteristic
47C23×C4×C127C2311yescharacteristic
48C2×D8×C127—22no—
49C2×D8×C127—22no—
50C2×D8×C127C2×C414yescomplemented
51C2×M16×C127—22no—
52C2×M16×C127—22no—
53C2×M16×C127—22no—
54C2×M16×C127—22no—
55C2×M16×C127—22no—
56C2×M16×C127C2×C414yescomplemented
57C22×C8×C127C2311yescharacteristic
58C22×C8×C127C2311yescharacteristic
59C22×C8×C127C2311yescharacteristic
60C22×C8×C127C2311yescharacteristic
61#121×C127—24no—
62#121×C127C2×C418yescomplemented
63C4×D4×C127—24no—
64C4×D4×C127—24no—
65C4×D4×C127—48no—
66C2×C4⋊C4×C127—22no—
67C2×C4⋊C4×C127C2312yes—
68C2×C4⋊C4×C127C2312yes—
69C2×C4⋊C4×C127C2312yes—
70C2×C22⋊C4×C127—22no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—22no—
73C2×C22⋊C4×C127—22no—
74C2×C22⋊C4×C127—22no—
75C2×C42×C127—22no—
76C2×C42×C127—22no—
77C2×C42×C127—22no—
78C2×C42×C127D411yescharacteristic
79C2×C42×C127D411yescharacteristic
80C2×C42×C127C2311yescharacteristic
81C2×C42×C127C2311yescharacteristic
82#107×C127—22no—
83#106×C127—22no—
84#106×C127C2×C414yescomplemented
85#102×C127—24no—
86#102×C127—24no—
87#102×C127C2×C418yescomplemented
88#97×C127—22no—
89#97×C127—22no—
90C4×C8×C127—22no—
91C4×C8×C127D414yescomplemented
92#95×C127—22no—
93#95×C127—22no—
94#95×C127—22no—
95C24×C127—22no—
96C24×C127C2×D411yescharacteristic
97C2×D4×C127—28no—
98C2×D4×C127—22no—
99C2×D4×C127—22no—
100C2×D4×C127—22no—
101C2×D4×C127—22no—
102C2×D4×C127—22no—
103C2×D4×C127—44no—
104C2×D4×C127—44no—
105C2×D4×C127C22×C414yes—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—24no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—42no—
117C22×C4×C127—44no—
118C22×C4×C127—44no—
119C22×C4×C127—44no—
120C22×C4×C127C22×C412yes—
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123C22×C4×C127C2×D411yescharacteristic
124C22×C4×C127C2×D411yescharacteristic
125C22×C4×C127C2×D411yescharacteristic
126C22×C4×C127#4112yes—
127C22×C4×C127C2411yescharacteristic
128D8×C127—28no—
129M16×C127—28no—
130M16×C127—28no—
131C2×C8×C127—21no—
132C2×C8×C127—21no—
133C2×C8×C127—21no—
134C2×C8×C127—21no—
135C2×C8×C127—21no—
136C2×C8×C127—21no—
137C2×C8×C127C22×C412yes—
138C2×C8×C127C22×C412yes—
139C2×C8×C127C2×D412yes—
140C2×C8×C127C2×D412yes—
141C2×C8×C127#4112yes—
142C2×C8×C127#4112yes—
143C4⋊C4×C127—22no—
144C4⋊C4×C127—22no—
145C4⋊C4×C127—22no—
146C4⋊C4×C127—22no—
147C4⋊C4×C127—22no—
148C4⋊C4×C127—44no—
149C4⋊C4×C127C22×C414yes—
150C22⋊C4×C127—28no—
151C22⋊C4×C127—44no—
152C22⋊C4×C127—44no—
153C22⋊C4×C127—44no—
154C22⋊C4×C127—44no—
155C42×C127—22no—
156C42×C127—22no—
157C42×C127—21no—
158C42×C127—21no—
159C42×C127—44no—
160C42×C127C2×D412yes—
161C42×C127#4112yes—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127—22no—
165C23×C127—22no—
166C23×C127—24no—
167C23×C127—42no—
168C23×C127—42no—
169C23×C127—42no—
170C23×C127—42no—
171C23×C127—42no—
172C23×C127—42no—
173C23×C127C4×D412yes—
174C23×C127C22×D411yescharacteristic
175D4×C127—24no—
176D4×C127—24no—
177D4×C127—416no—
178D4×C127—42no—
179D4×C127—42no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—22no—
183C2×C4×C127—22no—
184C2×C4×C127—22no—
185C2×C4×C127—22no—
186C2×C4×C127—22no—
187C2×C4×C127—24no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—22no—
191C2×C4×C127—22no—
192C2×C4×C127—44no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—44no—
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—88no—
203C2×C4×C127C4×D412yes—
204C2×C4×C127#12112yes—
205C2×C4×C127#12112yes—
206C2×C4×C127AGL(1,ℤ/8)14yes—
207C2×C4×C127C23×C411yescharacteristic
208C2×C4×C127C22×D411yescharacteristic
209C2×C4×C127C2×#4111yescharacteristic
210C2×C4×C127C2×#4111yescharacteristic
211C2×C4×C127C2×#4111yescharacteristic
212C2×C4×C127C2×#4111yescharacteristic
213C8×C127—22no—
214C8×C127—22no—
215C8×C127—22no—
216C8×C127C4×D414yescomplemented
217C22×C127—24no—
218C22×C127—22no—
219C22×C127—42no—
220C22×C127—41no—
221C22×C127—41no—
222C22×C127—44no—
223C22×C127—44no—
224C22×C127—44no—
225C22×C127—82no—
226C22×C127—82no—
227C22×C127—82no—
228C22×C127—82no—
229C22×C127C2×C4×D411yescharacteristic
230C22×C127C2×#12111yescharacteristic
231C22×C127#53211yescharacteristic
232C22×C127C2×AGL(1,ℤ/8)12yes—
233C22×C127#57512yes—
234C4×C127—22no—
235C4×C127—22no—
236C4×C127—44no—
237C4×C127—41no—
238C4×C127—41no—
239C4×C127—82no—
240C4×C127—82no—
241C4×C127#46914yesdirect product factor
242C4×C127C2×C4×D411yescharacteristic
243C4×C127C4×#4111yescharacteristic
244C4×C127#52511yescharacteristic
245C4×C127#52911yescharacteristic
246C4×#469C12711yesmaximal, characteristic, direct product factor, Sylow
247C2×C127—42no—
248C2×C127—84no—
249C2×C127C4×#12111yescharacteristic
250C2×C127C4×AGL(1,ℤ/8)12yes—
251C2×C127C2×#46911yescharacteristic
252C2×C127#305411yescharacteristic
253C2×C127#317112yes—
254C2×#469C2×C12711yescharacteristic
255C2×C4×D8C2×C12711yescharacteristic, complemented
256C2×C4×M16C2×C12711yescharacteristic, complemented
257C4×#121C2×C12712yescomplemented
258#1972C2×C12711yescharacteristic, complemented
259#1944C2×C12711yescharacteristic, complemented
260#1936C2×C12712yescomplemented
261#1895C2×C12712yescomplemented
262#1809C2×C12711yescharacteristic, complemented
263C4×#106C2×C12711yescharacteristic, complemented
264C4×#102C2×C12712yescomplemented
265C127C4×#46911yescharacteristic, direct product factor, Sylow
266C22×D8C22×C12711yescharacteristic
267C22×M16C22×C12711yescharacteristic
268C22×M16C22×C12711yescharacteristic
269C2×#121C22×C12712yes—
270C2×C4×D4—22no—
271C2×C4×D4C22×C12712yes—
272C22×C42C22×C12711yescharacteristic
273#469C4×C12718yesdirect product factor
274C4×D8—24no—
275C2×#107C22×C12711yescharacteristic
276C2×#106C22×C12711yescharacteristic
277C2×#106C22×C12711yescharacteristic
278C2×#102C22×C12712yes—
279C2×#102C22×C12712yes—
280C2×#102C22×C12712yes—
281C4×M16—24no—
282C2×#97C22×C12711yescharacteristic
283C2×#97C22×C12711yescharacteristic
284C2×C4×C8C22×C12711yescharacteristic
285C2×C4×C8C22×C12711yescharacteristic
286#390C22×C12712yes—
287#389C22×C12712yes—
288#388—22no—
289#386—22no—
290#380C22×C12712yes—
291C4×C4⋊C4C22×C12712yes—
292C4×C22⋊C4—22no—
293#340C22×C12712yes—
294#340C22×C12712yes—
295#340C22×C12712yes—
296C22×D4—22no—
297C22×D4C23×C12712yes—
298C23×C4—22no—
299C23×C4D4×C12711yescharacteristic
300C23×C4D4×C12711yescharacteristic
301C23×C4C23×C12711yescharacteristic
302C2×D8—22no—
303C2×D8—22no—
304C2×D8C2×C4×C12714yescomplemented
305C2×M16—22no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—22no—
309C2×M16—22no—
310C2×M16C2×C4×C12714yescomplemented
311C22×C8C23×C12711yescharacteristic
312C22×C8C23×C12711yescharacteristic
313C22×C8C23×C12711yescharacteristic
314C22×C8C23×C12711yescharacteristic
315#121—24no—
316#121C2×C4×C12718yescomplemented
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—48no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C23×C12712yes—
322C2×C4⋊C4C23×C12712yes—
323C2×C4⋊C4C23×C12712yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42D4×C12711yescharacteristic
333C2×C42D4×C12711yescharacteristic
334C2×C42C23×C12711yescharacteristic
335C2×C42C23×C12711yescharacteristic
336#107—22no—
337#106—22no—
338#106C2×C4×C12714yescomplemented
339#102—24no—
340#102—24no—
341#102C2×C4×C12718yescomplemented
342#97—22no—
343#97—22no—
344C4×C8—22no—
345C4×C8D4×C12714yescomplemented
346#95—22no—
347#95—22no—
348#95—22no—
349C24—22no—
350C24C2×D4×C12711yescharacteristic
351C2×D4—28no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4C22×C4×C12714yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4C22×C4×C12712yes—
375C22×C4C2×D4×C12711yescharacteristic
376C22×C4C2×D4×C12711yescharacteristic
377C22×C4C2×D4×C12711yescharacteristic
378C22×C4C2×D4×C12711yescharacteristic
379C22×C4C2×D4×C12711yescharacteristic
380C22×C4#41×C12712yes—
381C22×C4C24×C12711yescharacteristic
382D8—28no—
383M16—28no—
384M16—28no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8C22×C4×C12712yes—
392C2×C8C22×C4×C12712yes—
393C2×C8C2×D4×C12712yes—
394C2×C8C2×D4×C12712yes—
395C2×C8#41×C12712yes—
396C2×C8#41×C12712yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—44no—
403C4⋊C4C22×C4×C12714yes—
404C22⋊C4—28no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—22no—
410C42—22no—
411C42—21no—
412C42—21no—
413C42—44no—
414C42C2×D4×C12712yes—
415C42#41×C12712yes—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23C4×D4×C12712yes—
428C23C22×D4×C12711yescharacteristic
429D4—24no—
430D4—24no—
431D4—416no—
432D4—42no—
433D4—42no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—88no—
457C2×C4C4×D4×C12712yes—
458C2×C4#121×C12712yes—
459C2×C4#121×C12712yes—
460C2×C4AGL(1,ℤ/8)×C12714yes—
461C2×C4C23×C4×C12711yescharacteristic
462C2×C4C22×D4×C12711yescharacteristic
463C2×C4C2×#41×C12711yescharacteristic
464C2×C4C2×#41×C12711yescharacteristic
465C2×C4C2×#41×C12711yescharacteristic
466C2×C4C2×#41×C12711yescharacteristic
467C8—22no—
468C8—22no—
469C8—22no—
470C8C4×D4×C12714yescomplemented
471C22—24no—
472C22—22no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22C2×C4×D4×C12711yescharacteristic
484C22C2×#121×C12711yescharacteristic
485C22#532×C12711yescharacteristic
486C22C2×AGL(1,ℤ/8)×C12712yes—
487C22#575×C12712yes—
488C4—22no—
489C4—22no—
490C4—44no—
491C4—41no—
492C4—41no—
493C4—82no—
494C4—82no—
495C4#469×C12714yesdirect product factor
496C4C2×C4×D4×C12711yescharacteristic
497C4C4×#41×C12711yescharacteristic
498C4#525×C12711yescharacteristic
499C4#529×C12711yescharacteristic
500C2—42no—
501C2—84no—
502C2C4×#121×C12711yescharacteristic
503C2C4×AGL(1,ℤ/8)×C12712yes—
504C2C2×#469×C12711yescharacteristic
505C2C127×#305411yescharacteristic
506C2C127×#317112yes—
subgroupG/H
centerC22×C4×C127C2×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C22×D4
prime radicalC2×#469×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×D8×C127C2semidirect
3. C2×C4×M16×C127C2semidirect
4. C4×#121×C127C2semidirect
5. C127×#1972C2semidirect
6. C127×#1944C2semidirect
7. C127×#1936C2semidirect
8. C127×#1895C2semidirect
9. C127×#1809C2semidirect
10. C4×#106×C127C2semidirect
11. C4×#102×C127C2semidirect
19. #469×C127C4direct (N × H)
50. C2×D8×C127C2×C4semidirect
56. C2×M16×C127C2×C4semidirect
62. #121×C127C2×C4semidirect
84. #106×C127C2×C4semidirect
87. #102×C127C2×C4semidirect
91. C4×C8×C127D4semidirect
216. C8×C127C4×D4semidirect
241. C4×C127#469direct (N × H)
246. C4×#469C127direct (N × H)
255. C2×C4×D8C2×C127semidirect
256. C2×C4×M16C2×C127semidirect
257. C4×#121C2×C127semidirect
258. #1972C2×C127semidirect
259. #1944C2×C127semidirect
260. #1936C2×C127semidirect
261. #1895C2×C127semidirect
262. #1809C2×C127semidirect
263. C4×#106C2×C127semidirect
264. C4×#102C2×C127semidirect
265. C127C4×#469direct (N × H)
273. #469C4×C127direct (N × H)
304. C2×D8C2×C4×C127semidirect
310. C2×M16C2×C4×C127semidirect
316. #121C2×C4×C127semidirect
338. #106C2×C4×C127semidirect
341. #102C2×C4×C127semidirect
345. C4×C8D4×C127semidirect
470. C8C4×D4×C127semidirect
495. C4#469×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#469×C1272
C2×C4×D8×C1272
C2×C4×M16×C1272
C4×#121×C1272
C127×#19722
C127×#19442
C127×#19362
C127×#18952
C127×#18092
C4×#106×C1272
C4×#102×C1272
C4×#469127