Finite Group Databasebeta

← C127×#15012 #44578685

Subgroups — C127×#15012 #44578685

Order 32512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389×C127C212yesmaximal
2C2×#389×C127C211yesmaximal, characteristic
3C2×#387×C127C211yesmaximal, characteristic
4C127×#1444C211yesmaximal, characteristic
5C127×#1444C211yesmaximal, characteristic
6C127×#1442C212yesmaximal
7C127×#1442C211yesmaximal, characteristic
8C127×#1442C211yesmaximal, characteristic
9C127×#1441C212yesmaximal
10C127×#1436C212yesmaximal
11C4×#95×C127C211yesmaximal, characteristic
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2212yes—
16C22×C4⋊C4×C127C2211yescharacteristic
17C22×C4⋊C4×C127C2211yescharacteristic
18C22×C4⋊C4×C127C2211yescharacteristic
19C22×C42×C127C2212yes—
20C22×C42×C127C2211yescharacteristic
21#389×C127—28no—
22#389×C127—24no—
23#387×C127—24no—
24C2×#95×C127C2212yes—
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2211yescharacteristic
32C2×#95×C127C2212yes—
33C2×#95×C127C2211yescharacteristic
34C2×#95×C127C2211yescharacteristic
35C2×#95×C127C2212yes—
36C2×#95×C127C2211yescharacteristic
37C2×#95×C127C2211yescharacteristic
38C23×C4×C127C2312yes—
39C23×C4×C127C2311yescharacteristic
40C23×C4×C127C2311yescharacteristic
41C23×C4×C127C2311yescharacteristic
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2312yes—
45C23×C4×C127C2312yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2311yescharacteristic
48C23×C4×C127C2312yes—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—22no—
52C2×C4⋊C4×C127—24no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—24no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127—22no—
57C2×C4⋊C4×C127—24no—
58C2×C4⋊C4×C127—22no—
59C2×C4⋊C4×C127—24no—
60C2×C4⋊C4×C127—24no—
61C2×C4⋊C4×C127—24no—
62C2×C4⋊C4×C127—24no—
63C2×C4⋊C4×C127—42no—
64C2×C4⋊C4×C127—42no—
65C2×C4⋊C4×C127—42no—
66C2×C4⋊C4×C127D414yes—
67C2×C42×C127—24no—
68C2×C42×C127—24no—
69C2×C42×C127—42no—
70C2×C42×C127—42no—
71C2×C42×C127D414yes—
72#95×C127—22no—
73#95×C127—22no—
74#95×C127—22no—
75#95×C127—22no—
76#95×C127—24no—
77#95×C127—24no—
78#95×C127—24no—
79#95×C127—24no—
80#95×C127—24no—
81#95×C127—24no—
82#95×C127—24no—
83#95×C127—42no—
84#95×C127Q814yes—
85#95×C127Q814yes—
86C24×C127C2411yescharacteristic
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—21no—
106C22×C4×C127—21no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127—22no—
119C22×C4×C127—21no—
120C22×C4×C127—21no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—21no—
133C22×C4×C127—21no—
134C22×C4×C127—21no—
135C22×C4×C127—21no—
136C22×C4×C127C2×D412yes—
137C22×C4×C127C2×D412yes—
138C22×C4×C127C2×D412yes—
139C22×C4×C127C2×D412yes—
140C22×C4×C127C2×D412yes—
141C22×C4×C127C2×D412yes—
142C22×C4×C127C2×Q814yes—
143C22×C4×C127C2×Q814yes—
144C22×C4×C127C2×Q812yes—
145C22×C4×C127C2×Q812yes—
146C22×C4×C127#4114yes—
147C22×C4×C127#4112yes—
148C22×C4×C127#4114yes—
149C22×C4×C127#4114yes—
150C4⋊C4×C127—28no—
151C4⋊C4×C127—216no—
152C4⋊C4×C127—28no—
153C4⋊C4×C127—48no—
154C4⋊C4×C127—44no—
155C4⋊C4×C127—44no—
156C4⋊C4×C127—48no—
157C42×C127—44no—
158C42×C127—48no—
159C23×C127C22×D411yescharacteristic
160C23×C127C22×D411yescharacteristic
161C23×C127C22×Q811yescharacteristic
162C23×C127C22×Q811yescharacteristic
163C23×C127C2×#4111yescharacteristic
164C23×C127C2×#4112yes—
165C23×C127C2×#4112yes—
166C23×C127C2×#4112yes—
167C23×C127H(5,𝔽2)12yes—
168C23×C1272−1+411yescharacteristic
169C23×C1272−1+411yescharacteristic
170C2×C4×C127—22no—
171C2×C4×C127—24no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—22no—
175C2×C4×C127—24no—
176C2×C4×C127—24no—
177C2×C4×C127—24no—
178C2×C4×C127—24no—
179C2×C4×C127—24no—
180C2×C4×C127—24no—
181C2×C4×C127—24no—
182C2×C4×C127—24no—
183C2×C4×C127—24no—
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—22no—
191C2×C4×C127—24no—
192C2×C4×C127—22no—
193C2×C4×C127—22no—
194C2×C4×C127—24no—
195C2×C4×C127—24no—
196C2×C4×C127—24no—
197C2×C4×C127—42no—
198C2×C4×C127—41no—
199C2×C4×C127—42no—
200C2×C4×C127—42no—
201C2×C4×C127—42no—
202C2×C4×C127—42no—
203C2×C4×C127—41no—
204C2×C4×C127—42no—
205C2×C4×C127—41no—
206C2×C4×C127—41no—
207C2×C4×C127—42no—
208C2×C4×C127—42no—
209C2×C4×C127—42no—
210C2×C4×C127—42no—
211C2×C4×C127—42no—
212C2×C4×C127—42no—
213C2×C4×C127—42no—
214C2×C4×C127—41no—
215C2×C4×C127—41no—
216C2×C4×C127—41no—
217C2×C4×C127—42no—
218C2×C4×C127—41no—
219C2×C4×C127—42no—
220C2×C4×C127—41no—
221C2×C4×C127—41no—
222C2×C4×C127—42no—
223C2×C4×C127—42no—
224C2×C4×C127—42no—
225C2×C4×C127—42no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—42no—
230C2×C4×C127#12114yes—
231C2×C4×C127#12518yes—
232C2×C4×C127C4⋊Q814yes—
233C22×C127C2×#12111yescharacteristic
234C22×C127#52511yescharacteristic
235C22×C127#52512yes—
236C22×C127C2×#12512yes—
237C22×C127#52912yes—
238C22×C127C2×C4⋊Q811yescharacteristic
239C22×C127#53211yescharacteristic
240C22×C127#53211yescharacteristic
241C22×C127#53312yes—
242C22×C127#53411yescharacteristic
243C22×C127#53611yescharacteristic
244C22×C127#53912yes—
245C22×C127#54212yes—
246C22×C127#54612yes—
247C22×C127D4×Q811yescharacteristic
248C22×C127D4×Q811yescharacteristic
249C22×C127#55212yes—
250C22×C127#55312yes—
251C22×C127#55412yes—
252C22×C127#55412yes—
253C22×C127#55612yes—
254C22×C127#55711yescharacteristic
255C22×C127#55711yescharacteristic
256C4×C127—24no—
257C4×C127—28no—
258C4×C127—24no—
259C4×C127—42no—
260C4×C127—44no—
261C4×C127—42no—
262C4×C127—44no—
263C4×C127—42no—
264C4×C127—81no—
265C4×C127—81no—
266C4×C127—82no—
267#15012C12711yesmaximal, characteristic, direct product factor, Sylow
268C2×C127#239211yescharacteristic
269C2×C127#246512yes—
270C2×C127#247312yes—
271C2×C127#250912yes—
272C2×C127#254311yescharacteristic
273C2×C127#254611yescharacteristic
274C2×C127#258311yescharacteristic
275C2×C127#258512yes—
276C2×C127#260411yescharacteristic
277C2×C127#266311yescharacteristic
278C2×C127#285911yescharacteristic
279C2×#389C2×C12711yescharacteristic
280C2×#389C2×C12712yes—
281C2×#387C2×C12711yescharacteristic
282#1444C2×C12711yescharacteristic
283#1444C2×C12711yescharacteristic
284#1442C2×C12712yes—
285#1442C2×C12711yescharacteristic
286#1442C2×C12711yescharacteristic
287#1441C2×C12712yes—
288#1436C2×C12712yes—
289C4×#95C2×C12711yescharacteristic
290C127#1501211yescharacteristic, direct product factor, Sylow
291C22×C4⋊C4C22×C12712yes—
292C22×C4⋊C4C22×C12712yes—
293C22×C4⋊C4C22×C12711yescharacteristic
294C22×C4⋊C4C22×C12712yes—
295C22×C4⋊C4C22×C12711yescharacteristic
296C22×C4⋊C4C22×C12711yescharacteristic
297C22×C4⋊C4C22×C12711yescharacteristic
298C22×C42C22×C12712yes—
299C22×C42C22×C12711yescharacteristic
300#389—28no—
301#389—24no—
302#387—24no—
303C2×#95C22×C12712yes—
304C2×#95C22×C12711yescharacteristic
305C2×#95C22×C12712yes—
306C2×#95C22×C12712yes—
307C2×#95C22×C12712yes—
308C2×#95C22×C12712yes—
309C2×#95C22×C12712yes—
310C2×#95C22×C12711yescharacteristic
311C2×#95C22×C12712yes—
312C2×#95C22×C12711yescharacteristic
313C2×#95C22×C12711yescharacteristic
314C2×#95C22×C12712yes—
315C2×#95C22×C12711yescharacteristic
316C2×#95C22×C12711yescharacteristic
317C23×C4C23×C12712yes—
318C23×C4C23×C12711yescharacteristic
319C23×C4C23×C12711yescharacteristic
320C23×C4C23×C12711yescharacteristic
321C23×C4C23×C12711yescharacteristic
322C23×C4C23×C12711yescharacteristic
323C23×C4C23×C12712yes—
324C23×C4C23×C12712yes—
325C23×C4C23×C12711yescharacteristic
326C23×C4C23×C12711yescharacteristic
327C23×C4C23×C12712yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4D4×C12714yes—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42D4×C12714yes—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—42no—
363#95Q8×C12714yes—
364#95Q8×C12714yes—
365C24C24×C12711yescharacteristic
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4C2×D4×C12712yes—
416C22×C4C2×D4×C12712yes—
417C22×C4C2×D4×C12712yes—
418C22×C4C2×D4×C12712yes—
419C22×C4C2×D4×C12712yes—
420C22×C4C2×D4×C12712yes—
421C22×C4C2×Q8×C12714yes—
422C22×C4C2×Q8×C12714yes—
423C22×C4C2×Q8×C12712yes—
424C22×C4C2×Q8×C12712yes—
425C22×C4#41×C12714yes—
426C22×C4#41×C12712yes—
427C22×C4#41×C12714yes—
428C22×C4#41×C12714yes—
429C4⋊C4—28no—
430C4⋊C4—216no—
431C4⋊C4—28no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C42—44no—
437C42—48no—
438C23C22×D4×C12711yescharacteristic
439C23C22×D4×C12711yescharacteristic
440C23C22×Q8×C12711yescharacteristic
441C23C22×Q8×C12711yescharacteristic
442C23C2×#41×C12711yescharacteristic
443C23C2×#41×C12712yes—
444C23C2×#41×C12712yes—
445C23C2×#41×C12712yes—
446C23H(5,𝔽2)×C12712yes—
447C232−1+4×C12711yescharacteristic
448C232−1+4×C12711yescharacteristic
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4#121×C12714yes—
510C2×C4#125×C12718yes—
511C2×C4C4⋊Q8×C12714yes—
512C22C2×#121×C12711yescharacteristic
513C22#525×C12711yescharacteristic
514C22#525×C12712yes—
515C22C2×#125×C12712yes—
516C22#529×C12712yes—
517C22C2×C4⋊Q8×C12711yescharacteristic
518C22#532×C12711yescharacteristic
519C22#532×C12711yescharacteristic
520C22#533×C12712yes—
521C22#534×C12711yescharacteristic
522C22#536×C12711yescharacteristic
523C22#539×C12712yes—
524C22#542×C12712yes—
525C22#546×C12712yes—
526C22D4×Q8×C12711yescharacteristic
527C22D4×Q8×C12711yescharacteristic
528C22#552×C12712yes—
529C22#553×C12712yes—
530C22#554×C12712yes—
531C22#554×C12712yes—
532C22#556×C12712yes—
533C22#557×C12711yescharacteristic
534C22#557×C12711yescharacteristic
535C4—24no—
536C4—28no—
537C4—24no—
538C4—42no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—81no—
544C4—81no—
545C4—82no—
546C2C127×#239211yescharacteristic
547C2C127×#246512yes—
548C2C127×#247312yes—
549C2C127×#250912yes—
550C2C127×#254311yescharacteristic
551C2C127×#254611yescharacteristic
552C2C127×#258311yescharacteristic
553C2C127×#258512yes—
554C2C127×#260411yescharacteristic
555C2C127×#266311yescharacteristic
556C2C127×#285911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
267. #15012C127direct (N × H)
290. C127#15012direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#389×C1272
C2×#387×C1272
C127×#14442
C127×#14422
C127×#14412
C127×#14362
C4×#95×C1272
#15012127