Finite Group Databasebeta

← C127×#18140 #44581813

Subgroups — C127×#18140 #44581813

Order 32512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3459C211yesmaximal, characteristic
2C127×#3092C212yesmaximal, complemented
3C127×#2486C211yesmaximal, characteristic, complemented
4C127×#1609C214yesmaximal, complemented
5C127×#1608C212yesmaximal, complemented
6C127×#1607C212yesmaximal, complemented
7C127×#1513C212yesmaximal, complemented
8C127×#1465C211yesmaximal, characteristic, complemented
9#543×C127C2211yescharacteristic
10#539×C127—22no—
11#539×C127C2212yes—
12#538×C127—22no—
13#538×C127—21no—
14#534×C127—21no—
15#534×C127C2211yescharacteristic
16#529×C127—22no—
17#529×C127—21no—
18#525×C127—22no—
19#525×C127C2212yes—
20C2×#123×C127—21no—
21C2×C4×D4×C127—21no—
22C2×C4×D4×C127C2211yescharacteristic
23C22×C4⋊C4×C127C2211yescharacteristic, complemented
24#484×C127—22no—
25#483×C127C2214yescomplemented
26#481×C127—22no—
27#480×C127C2214yes—
28#428×C127—22no—
29#417×C127—22no—
30#407×C127C2212yescomplemented
31#398×C127C2212yescomplemented
32#397×C127C2212yes—
33#389×C127—21no—
34#381×C127—21no—
35#380×C127—21no—
36#380×C127C2211yescharacteristic, complemented
37#328×C127C2214yescomplemented
38#327×C127C2214yes—
39#324×C127C2212yescomplemented
40#323×C127C2212yes—
41C2×#41×C127D412yes—
42C22×D4×C127—21no—
43C22×D4×C127D411yescharacteristic
44C23×C4×C127—21no—
45C23×C4×C127—21no—
46C23×C4×C127D411yescharacteristic
47C23×C4×C127C2311yescharacteristic
48C2×M16×C127—22no—
49#126×C127—22no—
50#126×C127—22no—
51#126×C127—22no—
52#125×C127—21no—
53#125×C127—22no—
54#124×C127—22no—
55#124×C127—21no—
56#123×C127—22no—
57#123×C127—22no—
58#123×C127—22no—
59#123×C127—22no—
60#123×C127—22no—
61#123×C127—22no—
62#123×C127—44no—
63C22⋊Q8×C127—22no—
64C22⋊Q8×C127—22no—
65C22⋊Q8×C127—21no—
66C22⋊Q8×C127—22no—
67C22⋊Q8×C127D412yes—
68C22⋊Q8×C127C2312yes—
69#121×C127—22no—
70#121×C127—22no—
71#121×C127—22no—
72#121×C127—21no—
73#121×C127D412yes—
74#121×C127C2312yes—
75#120×C127—22no—
76#120×C127—22no—
77#120×C127—22no—
78C4×Q8×C127—21no—
79C4×D4×C127—22no—
80C4×D4×C127—22no—
81C4×D4×C127—21no—
82C4×D4×C127—22no—
83C4×D4×C127—22no—
84C4×D4×C127—22no—
85C4×D4×C127—22no—
86C4×D4×C127—22no—
87#117×C127—22no—
88#117×C127D412yes—
89C2×C4⋊C4×C127—21no—
90C2×C4⋊C4×C127—22no—
91C2×C4⋊C4×C127—22no—
92C2×C4⋊C4×C127—21no—
93C2×C4⋊C4×C127—22no—
94C2×C4⋊C4×C127—21no—
95C2×C4⋊C4×C127C2312yes—
96C2×C4⋊C4×C127C2311yescharacteristic
97C2×C22⋊C4×C127—21no—
98C2×C22⋊C4×C127—21no—
99C2×C22⋊C4×C127—22no—
100C2×C22⋊C4×C127—21no—
101C2×C22⋊C4×C127D411yescharacteristic
102C2×C22⋊C4×C127C2311yescharacteristic
103C2×C42×C127—21no—
104C2×C42×C127D411yescharacteristic
105#107×C127—24no—
106#106×C127—24no—
107#103×C127—24no—
108#102×C127—24no—
109C22⋊C8×C127—22no—
110C22⋊C8×C127C2314yes—
111#95×C127—22no—
112#95×C127—22no—
113#95×C127C2312yes—
114C24×C127—21no—
115C24×C127C2×D411yescharacteristic
116C24×C127C2×D411yescharacteristic
117#41×C127—44no—
118C2×Q8×C127—21no—
119C2×Q8×C127C2×D412yes—
120C2×D4×C127—22no—
121C2×D4×C127—22no—
122C2×D4×C127—22no—
123C2×D4×C127—22no—
124C2×D4×C127—21no—
125C2×D4×C127—22no—
126C2×D4×C127—22no—
127C2×D4×C127—41no—
128C2×D4×C127—42no—
129C2×D4×C127—42no—
130C2×D4×C127C2×D412yes—
131C22×C4×C127—21no—
132C22×C4×C127—21no—
133C22×C4×C127—21no—
134C22×C4×C127—22no—
135C22×C4×C127—21no—
136C22×C4×C127—22no—
137C22×C4×C127—21no—
138C22×C4×C127—22no—
139C22×C4×C127—22no—
140C22×C4×C127—42no—
141C22×C4×C127—42no—
142C22×C4×C127—42no—
143C22×C4×C127—42no—
144C22×C4×C127—42no—
145C22×C4×C127—82no—
146C22×C4×C127C2×D411yescharacteristic
147C22×C4×C127C2×D411yescharacteristic
148C22×C4×C127C2×D412yes—
149C22×C4×C127C2×D411yescharacteristic
150C22×C4×C127C2×D411yescharacteristic
151C22×C4×C127C2×D412yes—
152C22×C4×C127C2411yescharacteristic
153M16×C127—44no—
154C2×C8×C127—24no—
155C4⋊C4×C127—22no—
156C4⋊C4×C127—22no—
157C4⋊C4×C127—22no—
158C4⋊C4×C127—21no—
159C4⋊C4×C127—21no—
160C4⋊C4×C127—22no—
161C4⋊C4×C127—22no—
162C4⋊C4×C127—22no—
163C4⋊C4×C127—22no—
164C4⋊C4×C127—24no—
165C4⋊C4×C127—42no—
166C4⋊C4×C127—44no—
167C4⋊C4×C127—41no—
168C4⋊C4×C127—41no—
169C4⋊C4×C127C2×D412yes—
170C4⋊C4×C127C2×D412yes—
171C22⋊C4×C127—22no—
172C22⋊C4×C127—22no—
173C22⋊C4×C127—22no—
174C22⋊C4×C127—22no—
175C22⋊C4×C127—22no—
176C22⋊C4×C127—22no—
177C22⋊C4×C127—22no—
178C22⋊C4×C127—22no—
179C22⋊C4×C127—22no—
180C22⋊C4×C127—22no—
181C22⋊C4×C127—22no—
182C22⋊C4×C127—22no—
183C22⋊C4×C127—42no—
184C22⋊C4×C127—44no—
185C42×C127—22no—
186C42×C127—22no—
187C42×C127—21no—
188C42×C127—21no—
189C23×C127—21no—
190C23×C127—21no—
191C23×C127—21no—
192C23×C127—21no—
193C23×C127—22no—
194C23×C127—41no—
195C23×C127—41no—
196C23×C127—41no—
197C23×C127—41no—
198C23×C127—42no—
199C23×C127—81no—
200C23×C127—81no—
201C23×C127#12012yes—
202C23×C127AGL(1,ℤ/8)12yes—
203C23×C127C22×D411yescharacteristic
204Q8×C127—42no—
205Q8×C127—81no—
206D4×C127—42no—
207D4×C127—44no—
208D4×C127—44no—
209D4×C127—42no—
210D4×C127—42no—
211D4×C127—82no—
212D4×C127—82no—
213D4×C127—81no—
214C2×C4×C127—22no—
215C2×C4×C127—24no—
216C2×C4×C127—21no—
217C2×C4×C127—22no—
218C2×C4×C127—21no—
219C2×C4×C127—22no—
220C2×C4×C127—21no—
221C2×C4×C127—22no—
222C2×C4×C127—21no—
223C2×C4×C127—21no—
224C2×C4×C127—42no—
225C2×C4×C127—41no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—44no—
229C2×C4×C127—41no—
230C2×C4×C127—41no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127—42no—
235C2×C4×C127—42no—
236C2×C4×C127—42no—
237C2×C4×C127—81no—
238C2×C4×C127—81no—
239C2×C4×C127—82no—
240C2×C4×C127—81no—
241C2×C4×C127—84no—
242C2×C4×C127#12011yescharacteristic
243C2×C4×C127#12011yescharacteristic
244C2×C4×C127C22×D412yes—
245C8×C127—44no—
246C22×C127—21no—
247C22×C127—21no—
248C22×C127—22no—
249C22×C127—22no—
250C22×C127—22no—
251C22×C127—22no—
252C22×C127—42no—
253C22×C127—42no—
254C22×C127—42no—
255C22×C127—41no—
256C22×C127—41no—
257C22×C127—42no—
258C22×C127—42no—
259C22×C127—42no—
260C22×C127—81no—
261C22×C127—84no—
262C22×C127C2×#12011yescharacteristic
263C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C127#57511yescharacteristic
265C4×C127—22no—
266C4×C127—21no—
267C4×C127—41no—
268C4×C127—42no—
269C4×C127—41no—
270C4×C127—42no—
271C4×C127—41no—
272C4×C127—42no—
273C4×C127—82no—
274C4×C127—82no—
275C4×C127#45812yes—
276#18140C12711yesmaximal, characteristic, direct product factor, Sylow
277C2×C127—21no—
278C2×C127—21no—
279C2×C127—42no—
280C2×C127—42no—
281C2×C127—82no—
282C2×C127#300611yescharacteristic
283C2×C127#302211yescharacteristic
284C2×C127C2×#45811yescharacteristic
285#3459C2×C12711yescharacteristic
286#3092C2×C12712yescomplemented
287#2486C2×C12711yescharacteristic, complemented
288#1609C2×C12714yescomplemented
289#1608C2×C12712yescomplemented
290#1607C2×C12712yescomplemented
291#1513C2×C12712yescomplemented
292#1465C2×C12711yescharacteristic, complemented
293C127#1814011yescharacteristic, direct product factor, Sylow
294#543C22×C12711yescharacteristic
295#539—22no—
296#539C22×C12712yes—
297#538—22no—
298#538—21no—
299#534—21no—
300#534C22×C12711yescharacteristic
301#529—22no—
302#529—21no—
303#525—22no—
304#525C22×C12712yes—
305C2×#123—21no—
306C2×C4×D4—21no—
307C2×C4×D4C22×C12711yescharacteristic
308C22×C4⋊C4C22×C12711yescharacteristic, complemented
309#484—22no—
310#483C22×C12714yescomplemented
311#481—22no—
312#480C22×C12714yes—
313#428—22no—
314#417—22no—
315#407C22×C12712yescomplemented
316#398C22×C12712yescomplemented
317#397C22×C12712yes—
318#389—21no—
319#381—21no—
320#380—21no—
321#380C22×C12711yescharacteristic, complemented
322#328C22×C12714yescomplemented
323#327C22×C12714yes—
324#324C22×C12712yescomplemented
325#323C22×C12712yes—
326C2×#41D4×C12712yes—
327C22×D4—21no—
328C22×D4D4×C12711yescharacteristic
329C23×C4—21no—
330C23×C4—21no—
331C23×C4D4×C12711yescharacteristic
332C23×C4C23×C12711yescharacteristic
333C2×M16—22no—
334#126—22no—
335#126—22no—
336#126—22no—
337#125—21no—
338#125—22no—
339#124—22no—
340#124—21no—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—44no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8D4×C12712yes—
353C22⋊Q8C23×C12712yes—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121D4×C12712yes—
359#121C23×C12712yes—
360#120—22no—
361#120—22no—
362#120—22no—
363C4×Q8—21no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372#117—22no—
373#117D4×C12712yes—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4C23×C12712yes—
381C2×C4⋊C4C23×C12711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4D4×C12711yescharacteristic
387C2×C22⋊C4C23×C12711yescharacteristic
388C2×C42—21no—
389C2×C42D4×C12711yescharacteristic
390#107—24no—
391#106—24no—
392#103—24no—
393#102—24no—
394C22⋊C8—22no—
395C22⋊C8C23×C12714yes—
396#95—22no—
397#95—22no—
398#95C23×C12712yes—
399C24—21no—
400C24C2×D4×C12711yescharacteristic
401C24C2×D4×C12711yescharacteristic
402#41—44no—
403C2×Q8—21no—
404C2×Q8C2×D4×C12712yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×D4×C12712yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4C2×D4×C12711yescharacteristic
432C22×C4C2×D4×C12711yescharacteristic
433C22×C4C2×D4×C12712yes—
434C22×C4C2×D4×C12711yescharacteristic
435C22×C4C2×D4×C12711yescharacteristic
436C22×C4C2×D4×C12712yes—
437C22×C4C24×C12711yescharacteristic
438M16—44no—
439C2×C8—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4C2×D4×C12712yes—
455C4⋊C4C2×D4×C12712yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23#120×C12712yes—
487C23AGL(1,ℤ/8)×C12712yes—
488C23C22×D4×C12711yescharacteristic
489Q8—42no—
490Q8—81no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—82no—
497D4—82no—
498D4—81no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—84no—
527C2×C4#120×C12711yescharacteristic
528C2×C4#120×C12711yescharacteristic
529C2×C4C22×D4×C12712yes—
530C8—44no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—84no—
547C22C2×#120×C12711yescharacteristic
548C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
549C22#575×C12711yescharacteristic
550C4—22no—
551C4—21no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4#458×C12712yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—42no—
565C2—82no—
566C2C127×#300611yescharacteristic
567C2C127×#302211yescharacteristic
568C2C2×#458×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3459C2
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×#3092C2semidirect
3. C127×#2486C2semidirect
4. C127×#1609C2semidirect
5. C127×#1608C2semidirect
6. C127×#1607C2semidirect
7. C127×#1513C2semidirect
8. C127×#1465C2semidirect
23. C22×C4⋊C4×C127C22semidirect
25. #483×C127C22semidirect
30. #407×C127C22semidirect
31. #398×C127C22semidirect
36. #380×C127C22semidirect
37. #328×C127C22semidirect
39. #324×C127C22semidirect
276. #18140C127direct (N × H)
286. #3092C2×C127semidirect
287. #2486C2×C127semidirect
288. #1609C2×C127semidirect
289. #1608C2×C127semidirect
290. #1607C2×C127semidirect
291. #1513C2×C127semidirect
292. #1465C2×C127semidirect
293. C127#18140direct (N × H)
308. C22×C4⋊C4C22×C127semidirect
310. #483C22×C127semidirect
315. #407C22×C127semidirect
316. #398C22×C127semidirect
321. #380C22×C127semidirect
322. #328C22×C127semidirect
324. #324C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34592
C127×#30922
C127×#24862
C127×#16092
C127×#16082
C127×#16072
C127×#15132
C127×#14652
#18140127