Finite Group Databasebeta

← C127×#18533 #44582206

Subgroups — C127×#18533 #44582206

Order 32512.

1200 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C127C211yesmaximal, characteristic, complemented
2C127×#3141C211yesmaximal, characteristic, complemented
3C127×#3109C211yesmaximal, characteristic, complemented
4C127×#3109C211yesmaximal, characteristic, complemented
5C127×#1635C214yesmaximal, complemented
6C127×#1625C214yesmaximal, complemented
7C127×#1556C211yesmaximal, characteristic, complemented
8C127×#1490C211yesmaximal, characteristic, complemented
9C127×#1490C211yesmaximal, characteristic, complemented
10C22×#41×C127—21no—
11#547×C127—22no—
12#547×C127—22no—
13#547×C127—22no—
14#547×C127C2214yes—
15#532×C127C2211yescharacteristic
16C2×C4⋊Q8×C127C2211yescharacteristic
17C2×#123×C127—22no—
18C2×C22⋊Q8×C127—21no—
19C2×#121×C127—21no—
20C2×C4×D4×C127C2211yescharacteristic
21C2×C4×D4×C127C2211yescharacteristic
22C22×C4⋊C4×C127—21no—
23#489×C127—22no—
24#487×C127C2212yescomplemented
25#486×C127C2212yescomplemented
26#481×C127—21no—
27#481×C127—21no—
28#476×C127C2212yescomplemented
29#476×C127C2212yescomplemented
30#459×C127C2212yescomplemented
31#459×C127C2212yescomplemented
32#447×C127—21no—
33#447×C127—21no—
34#442×C127—22no—
35#442×C127—22no—
36#432×C127C2211yescharacteristic
37#432×C127C2211yescharacteristic
38#417×C127—22no—
39C4×M16×C127C2211yescharacteristic
40#332×C127C2214yescomplemented
41D4⋊C8×C127C2214yescomplemented
42D4⋊C8×C127C2214yescomplemented
43C2×#41×C127—24no—
44C2×#41×C127—24no—
45C2×#41×C127—21no—
46C2×#41×C127—48no—
47C22×Q8×C127—21no—
48C22×D4×C127—22no—
49C22×D4×C127D411yescharacteristic
50C22×D4×C127D411yescharacteristic
51C23×C4×C127—22no—
52C23×C4×C127—21no—
53C23×C4×C127—21no—
54C2×D8×C127—22no—
55C2×D8×C127—22no—
56C2×M16×C127—21no—
57C4⋊Q8×C127—21no—
58C4⋊Q8×C127—21no—
59C4⋊Q8×C127D412yes—
60C4⋊Q8×C127C2311yescharacteristic
61C4⋊Q8×C127C2312yes—
62#127×C127C2311yescharacteristic
63#124×C127—21no—
64#123×C127—28no—
65#123×C127—28no—
66C22⋊Q8×C127—24no—
67C22⋊Q8×C127—24no—
68#121×C127—24no—
69#121×C127—24no—
70#121×C127—21no—
71#121×C127—21no—
72C4×D4×C127—21no—
73C4×D4×C127—21no—
74C4×D4×C127—21no—
75C4×D4×C127—21no—
76C4×D4×C127—21no—
77C4×D4×C127—21no—
78C4×D4×C127D412yes—
79C4×D4×C127D412yes—
80C4×D4×C127C2312yes—
81C4×D4×C127C2312yes—
82C2×C4⋊C4×C127—22no—
83C2×C4⋊C4×C127—24no—
84C2×C4⋊C4×C127—21no—
85C2×C4⋊C4×C127—24no—
86C2×C4⋊C4×C127D411yescharacteristic
87C2×C4⋊C4×C127D411yescharacteristic
88C2×C4⋊C4×C127D412yes—
89C2×C22⋊C4×C127—22no—
90C2×C22⋊C4×C127—21no—
91C2×C22⋊C4×C127—21no—
92C2×C42×C127C2311yescharacteristic
93#106×C127—22no—
94#106×C127—22no—
95C4⋊C8×C127C2312yes—
96C4⋊C8×C127C2312yes—
97#103×C127—24no—
98#102×C127—22no—
99#102×C127—22no—
100#102×C127—24no—
101#102×C127—22no—
102#102×C127—22no—
103C22⋊C8×C127—21no—
104C22⋊C8×C127—21no—
105#97×C127—21no—
106#97×C127—21no—
107#97×C127—21no—
108C4×C8×C127C2312yes—
109C24×C127—21no—
110C24×C127—21no—
111#41×C127—44no—
112#41×C127—416no—
113#41×C127—48no—
114#41×C127—48no—
115C2×Q8×C127—22no—
116C2×Q8×C127—22no—
117C2×Q8×C127—21no—
118C2×Q8×C127—44no—
119C2×D4×C127—24no—
120C2×D4×C127—28no—
121C2×D4×C127—21no—
122C2×D4×C127—21no—
123C2×D4×C127—21no—
124C2×D4×C127—21no—
125C2×D4×C127—21no—
126C2×D4×C127—42no—
127C2×D4×C127—48no—
128C2×D4×C127—41no—
129C2×D4×C127C2×D412yes—
130C2×D4×C127C2×D412yes—
131C22×C4×C127—28no—
132C22×C4×C127—21no—
133C22×C4×C127—21no—
134C22×C4×C127—22no—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—44no—
140C22×C4×C127—44no—
141C22×C4×C127—48no—
142C22×C4×C127—41no—
143C22×C4×C127C2×D411yescharacteristic
144C22×C4×C127C2×D411yescharacteristic
145C22×C4×C127C2×D411yescharacteristic
146D8×C127—44no—
147D8×C127—44no—
148M16×C127—42no—
149C2×C8×C127—22no—
150C2×C8×C127—22no—
151C2×C8×C127—22no—
152C4⋊C4×C127—28no—
153C4⋊C4×C127—21no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—21no—
156C4⋊C4×C127—22no—
157C4⋊C4×C127—22no—
158C4⋊C4×C127—21no—
159C4⋊C4×C127—21no—
160C4⋊C4×C127C2×D414yes—
161C4⋊C4×C127C2×D412yes—
162C4⋊C4×C127C2×D412yes—
163C22⋊C4×C127—28no—
164C22⋊C4×C127—22no—
165C22⋊C4×C127—22no—
166C22⋊C4×C127—22no—
167C22⋊C4×C127—22no—
168C22⋊C4×C127—41no—
169C22⋊C4×C127—41no—
170C42×C127C2×D411yescharacteristic
171C42×C127C2×D411yescharacteristic
172C42×C127C2×D411yescharacteristic
173C42×C127C2411yescharacteristic
174C23×C127—24no—
175C23×C127—22no—
176C23×C127—22no—
177C23×C127—42no—
178C23×C127—42no—
179C23×C127—42no—
180C23×C127—42no—
181C23×C127—42no—
182C23×C127—41no—
183C23×C127#12011yescharacteristic
184Q8×C127—44no—
185Q8×C127—42no—
186Q8×C127—44no—
187D4×C127—24no—
188D4×C127—22no—
189D4×C127—22no—
190D4×C127—22no—
191D4×C127—42no—
192D4×C127—42no—
193D4×C127—416no—
194D4×C127—42no—
195D4×C127—42no—
196D4×C127—82no—
197D4×C127AGL(1,ℤ/8)14yes—
198C2×C4×C127—22no—
199C2×C4×C127—22no—
200C2×C4×C127—22no—
201C2×C4×C127—22no—
202C2×C4×C127—22no—
203C2×C4×C127—24no—
204C2×C4×C127—44no—
205C2×C4×C127—44no—
206C2×C4×C127—42no—
207C2×C4×C127—42no—
208C2×C4×C127—44no—
209C2×C4×C127—416no—
210C2×C4×C127—44no—
211C2×C4×C127—44no—
212C2×C4×C127—41no—
213C2×C4×C127—82no—
214C2×C4×C127#12011yescharacteristic
215C2×C4×C127#12011yescharacteristic
216C2×C4×C127#12011yescharacteristic
217C2×C4×C127C22×D411yescharacteristic
218C2×C4×C127C22×D411yescharacteristic
219C2×C4×C127C22×D411yescharacteristic
220C8×C127—42no—
221C8×C127—42no—
222C8×C127—42no—
223C22×C127—21no—
224C22×C127—21no—
225C22×C127—24no—
226C22×C127—48no—
227C22×C127—42no—
228C22×C127—42no—
229C22×C127—42no—
230C22×C127—42no—
231C22×C127—42no—
232C22×C127—84no—
233C22×C127—81no—
234C22×C127—81no—
235C22×C127—81no—
236C22×C127#45412yes—
237C22×C127C2×#12011yescharacteristic
238C4×C127—21no—
239C4×C127—21no—
240C4×C127—21no—
241C4×C127—21no—
242C4×C127—42no—
243C4×C127—44no—
244C4×C127—42no—
245C4×C127—81no—
246C4×C127C2×AGL(1,ℤ/8)12yes—
247C4×C127#57512yes—
248#18533C12711yesmaximal, characteristic, direct product factor, Sylow
249C2×C127—22no—
250C2×C127—44no—
251C2×C127—82no—
252C2×C127—161no—
253C2×C127#300311yescharacteristic
254C2×C127#300511yescharacteristic
255C2×C127C2×#45411yescharacteristic
256C2×#547C2×C12711yescharacteristic, complemented
257#3141C2×C12711yescharacteristic, complemented
258#3109C2×C12711yescharacteristic, complemented
259#3109C2×C12711yescharacteristic, complemented
260#1635C2×C12714yescomplemented
261#1625C2×C12714yescomplemented
262#1556C2×C12711yescharacteristic, complemented
263#1490C2×C12711yescharacteristic, complemented
264#1490C2×C12711yescharacteristic, complemented
265C127#1853311yescharacteristic, direct product factor, Sylow
266C22×#41—21no—
267#547—22no—
268#547—22no—
269#547—22no—
270#547C22×C12714yes—
271#532C22×C12711yescharacteristic
272C2×C4⋊Q8C22×C12711yescharacteristic
273C2×#123—22no—
274C2×C22⋊Q8—21no—
275C2×#121—21no—
276C2×C4×D4C22×C12711yescharacteristic
277C2×C4×D4C22×C12711yescharacteristic
278C22×C4⋊C4—21no—
279#489—22no—
280#487C22×C12712yescomplemented
281#486C22×C12712yescomplemented
282#481—21no—
283#481—21no—
284#476C22×C12712yescomplemented
285#476C22×C12712yescomplemented
286#459C22×C12712yescomplemented
287#459C22×C12712yescomplemented
288#447—21no—
289#447—21no—
290#442—22no—
291#442—22no—
292#432C22×C12711yescharacteristic
293#432C22×C12711yescharacteristic
294#417—22no—
295C4×M16C22×C12711yescharacteristic
296#332C22×C12714yescomplemented
297D4⋊C8C22×C12714yescomplemented
298D4⋊C8C22×C12714yescomplemented
299C2×#41—24no—
300C2×#41—24no—
301C2×#41—21no—
302C2×#41—48no—
303C22×Q8—21no—
304C22×D4—22no—
305C22×D4D4×C12711yescharacteristic
306C22×D4D4×C12711yescharacteristic
307C23×C4—22no—
308C23×C4—21no—
309C23×C4—21no—
310C2×D8—22no—
311C2×D8—22no—
312C2×M16—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8D4×C12712yes—
316C4⋊Q8C23×C12711yescharacteristic
317C4⋊Q8C23×C12712yes—
318#127C23×C12711yescharacteristic
319#124—21no—
320#123—28no—
321#123—28no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324#121—24no—
325#121—24no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4D4×C12712yes—
335C4×D4D4×C12712yes—
336C4×D4C23×C12712yes—
337C4×D4C23×C12712yes—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4D4×C12711yescharacteristic
343C2×C4⋊C4D4×C12711yescharacteristic
344C2×C4⋊C4D4×C12712yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C42C23×C12711yescharacteristic
349#106—22no—
350#106—22no—
351C4⋊C8C23×C12712yes—
352C4⋊C8C23×C12712yes—
353#103—24no—
354#102—22no—
355#102—22no—
356#102—24no—
357#102—22no—
358#102—22no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361#97—21no—
362#97—21no—
363#97—21no—
364C4×C8C23×C12712yes—
365C24—21no—
366C24—21no—
367#41—44no—
368#41—416no—
369#41—48no—
370#41—48no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—44no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—48no—
384C2×D4—41no—
385C2×D4C2×D4×C12712yes—
386C2×D4C2×D4×C12712yes—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—41no—
399C22×C4C2×D4×C12711yescharacteristic
400C22×C4C2×D4×C12711yescharacteristic
401C22×C4C2×D4×C12711yescharacteristic
402D8—44no—
403D8—44no—
404M16—42no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—28no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4C2×D4×C12714yes—
417C4⋊C4C2×D4×C12712yes—
418C4⋊C4C2×D4×C12712yes—
419C22⋊C4—28no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C42C2×D4×C12711yescharacteristic
427C42C2×D4×C12711yescharacteristic
428C42C2×D4×C12711yescharacteristic
429C42C24×C12711yescharacteristic
430C23—24no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23#120×C12711yescharacteristic
440Q8—44no—
441Q8—42no—
442Q8—44no—
443D4—24no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—42no—
448D4—42no—
449D4—416no—
450D4—42no—
451D4—42no—
452D4—82no—
453D4AGL(1,ℤ/8)×C12714yes—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—416no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4#120×C12711yescharacteristic
471C2×C4#120×C12711yescharacteristic
472C2×C4#120×C12711yescharacteristic
473C2×C4C22×D4×C12711yescharacteristic
474C2×C4C22×D4×C12711yescharacteristic
475C2×C4C22×D4×C12711yescharacteristic
476C8—42no—
477C8—42no—
478C8—42no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#454×C12712yes—
493C22C2×#120×C12711yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—44no—
500C4—42no—
501C4—81no—
502C4C2×AGL(1,ℤ/8)×C12712yes—
503C4#575×C12712yes—
504C2—22no—
505C2—44no—
506C2—82no—
507C2—161no—
508C2C127×#300311yescharacteristic
509C2C127×#300511yescharacteristic
510C2C2×#454×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC42C24×C127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#547×C127C2semidirect
2. C127×#3141C2semidirect
3. C127×#3109C2semidirect
4. C127×#3109C2semidirect
5. C127×#1635C2semidirect
6. C127×#1625C2semidirect
7. C127×#1556C2semidirect
8. C127×#1490C2semidirect
9. C127×#1490C2semidirect
24. #487×C127C22semidirect
25. #486×C127C22semidirect
28. #476×C127C22semidirect
29. #476×C127C22semidirect
30. #459×C127C22semidirect
31. #459×C127C22semidirect
40. #332×C127C22semidirect
41. D4⋊C8×C127C22semidirect
42. D4⋊C8×C127C22semidirect
248. #18533C127direct (N × H)
256. C2×#547C2×C127semidirect
257. #3141C2×C127semidirect
258. #3109C2×C127semidirect
259. #3109C2×C127semidirect
260. #1635C2×C127semidirect
261. #1625C2×C127semidirect
262. #1556C2×C127semidirect
263. #1490C2×C127semidirect
264. #1490C2×C127semidirect
265. C127#18533direct (N × H)
280. #487C22×C127semidirect
281. #486C22×C127semidirect
284. #476C22×C127semidirect
285. #476C22×C127semidirect
286. #459C22×C127semidirect
287. #459C22×C127semidirect
296. #332C22×C127semidirect
297. D4⋊C8C22×C127semidirect
298. D4⋊C8C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#547×C1272
C127×#31412
C127×#31092
C127×#16352
C127×#16252
C127×#15562
C127×#14902
#18533127