Finite Group Databasebeta

← C127×#18317 #44581990

Subgroups — C127×#18317 #44581990

Order 32512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#534×C127C211yesmaximal, characteristic
2C127×#3193C212yesmaximal, complemented
3C127×#2561C211yesmaximal, characteristic, complemented
4C127×#1607C214yesmaximal, complemented
5C127×#1601C214yesmaximal, complemented
6C127×#1507C212yesmaximal, complemented
7C127×#1462C211yesmaximal, characteristic, complemented
8C23×D4×C127—21no—
9#534×C127—24no—
10#534×C127—22no—
11#534×C127C2214yes—
12C2×#123×C127—22no—
13C2×C22⋊Q8×C127C2211yescharacteristic, complemented
14C2×#121×C127—21no—
15C2×#121×C127C2212yes—
16C2×#120×C127—22no—
17C22×C22⋊C4×C127C2211yescharacteristic
18#484×C127—22no—
19#480×C127C2214yescomplemented
20#468×C127—24no—
21#450×C127—22no—
22#448×C127C2214yescomplemented
23#407×C127C2212yescomplemented
24#397×C127C2212yescomplemented
25#393×C127C2212yescomplemented
26#388×C127—22no—
27#381×C127—21no—
28#379×C127—21no—
29C4×C22⋊C4×C127C2211yescharacteristic, complemented
30#327×C127C2218yescomplemented
31#323×C127C2214yescomplemented
32C25×C127—21no—
33C25×C127D411yescharacteristic
34C22×Q8×C127C2311yescharacteristic, complemented
35C22×D4×C127—22no—
36C22×D4×C127—24no—
37C22×D4×C127—22no—
38C22×D4×C127—24no—
39C22×D4×C127—21no—
40C22×D4×C127—48no—
41C22×D4×C127D412yes—
42C23×C4×C127—21no—
43C23×C4×C127D411yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C2×SD16×C127—24no—
46C2×M16×C127—22no—
47#123×C127—28no—
48#123×C127—28no—
49C22⋊Q8×C127—22no—
50C22⋊Q8×C127—22no—
51#121×C127—24no—
52#121×C127—24no—
53#121×C127—22no—
54#121×C127—22no—
55#121×C127D414yes—
56#121×C127C2314yes—
57#120×C127—28no—
58#120×C127—28no—
59C2×C4⋊C4×C127—21no—
60C2×C4⋊C4×C127—21no—
61C2×C4⋊C4×C127D412yes—
62C2×C4⋊C4×C127C2311yescharacteristic, complemented
63C2×C22⋊C4×C127—22no—
64C2×C22⋊C4×C127—22no—
65C2×C22⋊C4×C127—22no—
66C2×C22⋊C4×C127—22no—
67C2×C22⋊C4×C127—21no—
68C2×C22⋊C4×C127—21no—
69C2×C22⋊C4×C127—22no—
70C2×C22⋊C4×C127—22no—
71C2×C22⋊C4×C127D412yes—
72C2×C22⋊C4×C127C2312yes—
73C2×C42×C127—21no—
74#107×C127—24no—
75#103×C127—24no—
76#102×C127—24no—
77C22⋊C8×C127—22no—
78C22⋊C8×C127C2314yescomplemented
79#95×C127—22no—
80#95×C127—22no—
81#95×C127C2312yescomplemented
82C24×C127—21no—
83C24×C127—21no—
84C24×C127—24no—
85C24×C127—24no—
86C24×C127—22no—
87C24×C127—44no—
88C24×C127—42no—
89C24×C127—81no—
90C24×C127—82no—
91C24×C127—82no—
92C24×C127C2×D411yescharacteristic
93C24×C127C2×D412yes—
94C2×Q8×C127—22no—
95C2×Q8×C127—81no—
96C2×D4×C127—28no—
97C2×D4×C127—24no—
98C2×D4×C127—24no—
99C2×D4×C127—24no—
100C2×D4×C127—22no—
101C2×D4×C127—48no—
102C2×D4×C127—416no—
103C2×D4×C127—48no—
104C2×D4×C127—48no—
105C2×D4×C127—48no—
106C2×D4×C127—82no—
107C2×D4×C127—82no—
108C2×D4×C127C2×D414yes—
109C22×C4×C127—24no—
110C22×C4×C127—21no—
111C22×C4×C127—22no—
112C22×C4×C127—21no—
113C22×C4×C127—22no—
114C22×C4×C127—21no—
115C22×C4×C127—21no—
116C22×C4×C127—44no—
117C22×C4×C127—42no—
118C22×C4×C127—81no—
119C22×C4×C127C2×D412yes—
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D412yes—
122C22×C4×C127C2×D412yes—
123C22×C4×C127C2411yescharacteristic
124SD16×C127—84no—
125M16×C127—82no—
126C2×C8×C127—24no—
127C4⋊C4×C127—24no—
128C4⋊C4×C127—22no—
129C4⋊C4×C127—22no—
130C4⋊C4×C127—42no—
131C4⋊C4×C127C2×D414yes—
132C22⋊C4×C127—28no—
133C22⋊C4×C127—24no—
134C22⋊C4×C127—24no—
135C22⋊C4×C127—24no—
136C22⋊C4×C127—24no—
137C22⋊C4×C127—42no—
138C22⋊C4×C127—41no—
139C22⋊C4×C127—41no—
140C22⋊C4×C127—44no—
141C22⋊C4×C127—82no—
142C42×C127—81no—
143C23×C127—24no—
144C23×C127—24no—
145C23×C127—22no—
146C23×C127—22no—
147C23×C127—22no—
148C23×C127—22no—
149C23×C127—24no—
150C23×C127—41no—
151C23×C127—42no—
152C23×C127—42no—
153C23×C127—42no—
154C23×C127—44no—
155C23×C127—48no—
156C23×C127—44no—
157C23×C127—42no—
158C23×C127—44no—
159C23×C127—81no—
160C23×C127—81no—
161C23×C127—88no—
162C23×C127—84no—
163C23×C127—84no—
164C23×C127—816no—
165C23×C127#12011yescharacteristic
166C23×C127#12011yescharacteristic
167C23×C127AGL(1,ℤ/8)14yes—
168C23×C127C22×D411yescharacteristic
169Q8×C127—44no—
170D4×C127—44no—
171D4×C127—416no—
172D4×C127—48no—
173D4×C127—48no—
174D4×C127—82no—
175D4×C127—84no—
176C2×C4×C127—22no—
177C2×C4×C127—24no—
178C2×C4×C127—41no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—41no—
182C2×C4×C127—44no—
183C2×C4×C127—48no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—42no—
189C2×C4×C127—41no—
190C2×C4×C127—41no—
191C2×C4×C127—42no—
192C2×C4×C127—42no—
193C2×C4×C127—81no—
194C2×C4×C127#12012yes—
195C2×C4×C127C22×D412yes—
196C8×C127—44no—
197C22×C127—21no—
198C22×C127—21no—
199C22×C127—21no—
200C22×C127—21no—
201C22×C127—24no—
202C22×C127—21no—
203C22×C127—21no—
204C22×C127—21no—
205C22×C127—44no—
206C22×C127—44no—
207C22×C127—48no—
208C22×C127—42no—
209C22×C127—42no—
210C22×C127—44no—
211C22×C127—44no—
212C22×C127—44no—
213C22×C127—82no—
214C22×C127—88no—
215C22×C127—84no—
216C22×C127—816no—
217C22×C127#45512yes—
218C22×C127C2×#12011yescharacteristic
219C22×C127C2×AGL(1,ℤ/8)12yes—
220C4×C127—22no—
221C4×C127—22no—
222C4×C127—44no—
223C4×C127—42no—
224C4×C127—82no—
225C4×C127—81no—
226C4×C127—81no—
227#18317C12711yesmaximal, characteristic, direct product factor, Sylow
228C2×C127—22no—
229C2×C127—22no—
230C2×C127—41no—
231C2×C127—44no—
232C2×C127—84no—
233C2×C127#300311yescharacteristic
234C2×C127C2×#45511yescharacteristic
235C2×C127#302611yescharacteristic
236C2×#534C2×C12711yescharacteristic
237#3193C2×C12712yescomplemented
238#2561C2×C12711yescharacteristic, complemented
239#1607C2×C12714yescomplemented
240#1601C2×C12714yescomplemented
241#1507C2×C12712yescomplemented
242#1462C2×C12711yescharacteristic, complemented
243C127#1831711yescharacteristic, direct product factor, Sylow
244C23×D4—21no—
245#534—24no—
246#534—22no—
247#534C22×C12714yes—
248C2×#123—22no—
249C2×C22⋊Q8C22×C12711yescharacteristic, complemented
250C2×#121—21no—
251C2×#121C22×C12712yes—
252C2×#120—22no—
253C22×C22⋊C4C22×C12711yescharacteristic
254#484—22no—
255#480C22×C12714yescomplemented
256#468—24no—
257#450—22no—
258#448C22×C12714yescomplemented
259#407C22×C12712yescomplemented
260#397C22×C12712yescomplemented
261#393C22×C12712yescomplemented
262#388—22no—
263#381—21no—
264#379—21no—
265C4×C22⋊C4C22×C12711yescharacteristic, complemented
266#327C22×C12718yescomplemented
267#323C22×C12714yescomplemented
268C25—21no—
269C25D4×C12711yescharacteristic
270C22×Q8C23×C12711yescharacteristic, complemented
271C22×D4—22no—
272C22×D4—24no—
273C22×D4—22no—
274C22×D4—24no—
275C22×D4—21no—
276C22×D4—48no—
277C22×D4D4×C12712yes—
278C23×C4—21no—
279C23×C4D4×C12711yescharacteristic
280C23×C4C23×C12711yescharacteristic
281C2×SD16—24no—
282C2×M16—22no—
283#123—28no—
284#123—28no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287#121—24no—
288#121—24no—
289#121—22no—
290#121—22no—
291#121D4×C12714yes—
292#121C23×C12714yes—
293#120—28no—
294#120—28no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4D4×C12712yes—
298C2×C4⋊C4C23×C12711yescharacteristic, complemented
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4D4×C12712yes—
308C2×C22⋊C4C23×C12712yes—
309C2×C42—21no—
310#107—24no—
311#103—24no—
312#102—24no—
313C22⋊C8—22no—
314C22⋊C8C23×C12714yescomplemented
315#95—22no—
316#95—22no—
317#95C23×C12712yescomplemented
318C24—21no—
319C24—21no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—44no—
324C24—42no—
325C24—81no—
326C24—82no—
327C24—82no—
328C24C2×D4×C12711yescharacteristic
329C24C2×D4×C12712yes—
330C2×Q8—22no—
331C2×Q8—81no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4C2×D4×C12714yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—81no—
355C22×C4C2×D4×C12712yes—
356C22×C4C2×D4×C12711yescharacteristic
357C22×C4C2×D4×C12712yes—
358C22×C4C2×D4×C12712yes—
359C22×C4C24×C12711yescharacteristic
360SD16—84no—
361M16—82no—
362C2×C8—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4C2×D4×C12714yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C42—81no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—48no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—81no—
396C23—81no—
397C23—88no—
398C23—84no—
399C23—84no—
400C23—816no—
401C23#120×C12711yescharacteristic
402C23#120×C12711yescharacteristic
403C23AGL(1,ℤ/8)×C12714yes—
404C23C22×D4×C12711yescharacteristic
405Q8—44no—
406D4—44no—
407D4—416no—
408D4—48no—
409D4—48no—
410D4—82no—
411D4—84no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4#120×C12712yes—
431C2×C4C22×D4×C12712yes—
432C8—44no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—88no—
451C22—84no—
452C22—816no—
453C22#455×C12712yes—
454C22C2×#120×C12711yescharacteristic
455C22C2×AGL(1,ℤ/8)×C12712yes—
456C4—22no—
457C4—22no—
458C4—44no—
459C4—42no—
460C4—82no—
461C4—81no—
462C4—81no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—84no—
468C2C127×#300311yescharacteristic
469C2C2×#455×C12711yescharacteristic
470C2C127×#302611yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radicalC2×#534×C127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3193C2semidirect
3. C127×#2561C2semidirect
4. C127×#1607C2semidirect
5. C127×#1601C2semidirect
6. C127×#1507C2semidirect
7. C127×#1462C2semidirect
13. C2×C22⋊Q8×C127C22semidirect
19. #480×C127C22semidirect
22. #448×C127C22semidirect
23. #407×C127C22semidirect
24. #397×C127C22semidirect
25. #393×C127C22semidirect
29. C4×C22⋊C4×C127C22semidirect
30. #327×C127C22semidirect
31. #323×C127C22semidirect
34. C22×Q8×C127C23semidirect
62. C2×C4⋊C4×C127C23semidirect
78. C22⋊C8×C127C23semidirect
81. #95×C127C23semidirect
227. #18317C127direct (N × H)
237. #3193C2×C127semidirect
238. #2561C2×C127semidirect
239. #1607C2×C127semidirect
240. #1601C2×C127semidirect
241. #1507C2×C127semidirect
242. #1462C2×C127semidirect
243. C127#18317direct (N × H)
249. C2×C22⋊Q8C22×C127semidirect
255. #480C22×C127semidirect
258. #448C22×C127semidirect
259. #407C22×C127semidirect
260. #397C22×C127semidirect
261. #393C22×C127semidirect
265. C4×C22⋊C4C22×C127semidirect
266. #327C22×C127semidirect
267. #323C22×C127semidirect
270. C22×Q8C23×C127semidirect
298. C2×C4⋊C4C23×C127semidirect
314. C22⋊C8C23×C127semidirect
317. #95C23×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#534×C1272
C127×#31932
C127×#25612
C127×#16072
C127×#16012
C127×#15072
C127×#14622
#18317127