Finite Group Databasebeta

← C127×#17025 #44580698

Subgroups — C127×#17025 #44580698

Order 32512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#391×C127C211yesmaximal, characteristic
2C2×#387×C127C211yesmaximal, characteristic
3C127×#1446C211yesmaximal, characteristic
4C127×#1446C211yesmaximal, characteristic
5C127×#1445C212yesmaximal
6C127×#1444C211yesmaximal, characteristic
7C127×#1443C212yesmaximal
8C127×#1442C212yesmaximal
9C127×#1442C211yesmaximal, characteristic
10C127×#1441C212yesmaximal
11C127×#1434C211yesmaximal, characteristic
12C23×Q8×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2211yescharacteristic
17C22×C42×C127C2211yescharacteristic
18#391×C127—24no—
19#387×C127—24no—
20C2×#95×C127C2212yes—
21C2×#95×C127C2211yescharacteristic
22C2×#95×C127C2211yescharacteristic
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2211yescharacteristic
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2212yes—
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C2×#95×C127C2212yes—
36C2×#95×C127C2212yes—
37C22×Q8×C127—22no—
38C22×Q8×C127—22no—
39C22×Q8×C127—42no—
40C22×Q8×C127—42no—
41C22×Q8×C127Q814yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2311yescharacteristic
48C23×C4×C127C2312yes—
49C23×C4×C127C2311yescharacteristic
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2312yes—
52C23×C4×C127C2311yescharacteristic
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127—22no—
57C2×C4⋊C4×C127—22no—
58C2×C4⋊C4×C127—24no—
59C2×C4⋊C4×C127—24no—
60C2×C4⋊C4×C127—42no—
61C2×C4⋊C4×C127—42no—
62C2×C4⋊C4×C127—42no—
63C2×C42×C127—42no—
64C2×C42×C127D414yes—
65#95×C127—22no—
66#95×C127—22no—
67#95×C127—22no—
68#95×C127—22no—
69#95×C127—24no—
70#95×C127—24no—
71#95×C127—22no—
72#95×C127—24no—
73#95×C127—24no—
74#95×C127—24no—
75#95×C127—22no—
76#95×C127—42no—
77#95×C127—42no—
78#95×C127—42no—
79#95×C127—42no—
80#95×C127—42no—
81#95×C127—42no—
82C24×C127C2411yescharacteristic
83C2×Q8×C127—28no—
84C2×Q8×C127—48no—
85C2×Q8×C127—44no—
86C2×Q8×C127—44no—
87C2×Q8×C127—48no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—21no—
91C22×C4×C127—22no—
92C22×C4×C127—21no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—21no—
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—22no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—21no—
113C22×C4×C127—22no—
114C22×C4×C127—21no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—21no—
119C22×C4×C127—22no—
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—22no—
133C22×C4×C127—22no—
134C22×C4×C127—22no—
135C22×C4×C127—21no—
136C22×C4×C127—21no—
137C22×C4×C127—21no—
138C22×C4×C127—22no—
139C22×C4×C127—21no—
140C22×C4×C127—22no—
141C22×C4×C127—22no—
142C22×C4×C127C2×D412yes—
143C22×C4×C127C2×D412yes—
144C22×C4×C127C2×D412yes—
145C22×C4×C127C2×Q814yes—
146C22×C4×C127C2×Q812yes—
147C22×C4×C127#4112yes—
148C22×C4×C127#4114yes—
149C22×C4×C127#4112yes—
150C22×C4×C127#4112yes—
151C4⋊C4×C127—28no—
152C4⋊C4×C127—48no—
153C4⋊C4×C127—44no—
154C4⋊C4×C127—44no—
155C42×C127—44no—
156C23×C127C22×D411yescharacteristic
157C23×C127C22×Q811yescharacteristic
158C23×C127C2×#4111yescharacteristic
159C23×C127C2×#4111yescharacteristic
160C23×C127C2×#4111yescharacteristic
161C23×C127C2×#4112yes—
162C23×C127H(5,𝔽2)12yes—
163C23×C127H(5,𝔽2)12yes—
164C23×C1272−1+411yescharacteristic
165C23×C1272−1+412yes—
166C23×C1272−1+411yescharacteristic
167Q8×C127—416no—
168C2×C4×C127—22no—
169C2×C4×C127—22no—
170C2×C4×C127—24no—
171C2×C4×C127—24no—
172C2×C4×C127—24no—
173C2×C4×C127—22no—
174C2×C4×C127—22no—
175C2×C4×C127—22no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—24no—
182C2×C4×C127—24no—
183C2×C4×C127—24no—
184C2×C4×C127—24no—
185C2×C4×C127—24no—
186C2×C4×C127—24no—
187C2×C4×C127—24no—
188C2×C4×C127—41no—
189C2×C4×C127—41no—
190C2×C4×C127—42no—
191C2×C4×C127—41no—
192C2×C4×C127—42no—
193C2×C4×C127—41no—
194C2×C4×C127—42no—
195C2×C4×C127—42no—
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—42no—
203C2×C4×C127—42no—
204C2×C4×C127—42no—
205C2×C4×C127—42no—
206C2×C4×C127—42no—
207C2×C4×C127—42no—
208C2×C4×C127—41no—
209C2×C4×C127—41no—
210C2×C4×C127—42no—
211C2×C4×C127—41no—
212C2×C4×C127—41no—
213C2×C4×C127—41no—
214C2×C4×C127—41no—
215C2×C4×C127—42no—
216C2×C4×C127—42no—
217C2×C4×C127—42no—
218C2×C4×C127—42no—
219C2×C4×C127—42no—
220C2×C4×C127—42no—
221C2×C4×C127—42no—
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×C127C22⋊Q814yes—
231C22×C127C2×C22⋊Q811yescharacteristic
232C22×C127#52511yescharacteristic
233C22×C127#53211yescharacteristic
234C22×C127#53411yescharacteristic
235C22×C127#53511yescharacteristic
236C22×C127#53611yescharacteristic
237C22×C127#53611yescharacteristic
238C22×C127#53812yes—
239C22×C127#53912yes—
240C22×C127#53912yes—
241C22×C127#54212yes—
242C22×C127#54212yes—
243C22×C127#54312yes—
244C22×C127#55112yes—
245C22×C127#55212yes—
246C22×C127#55512yes—
247C22×C127#55611yescharacteristic
248C22×C127#55611yescharacteristic
249C22×C127#55712yes—
250C22×C127#55711yescharacteristic
251C22×C127#55711yescharacteristic
252C22×C127#56312yes—
253C22×C127#56312yes—
254C4×C127—24no—
255C4×C127—42no—
256C4×C127—44no—
257C4×C127—42no—
258C4×C127—42no—
259C4×C127—42no—
260C4×C127—44no—
261C4×C127—81no—
262C4×C127—82no—
263C4×C127—81no—
264C4×C127—82no—
265#17025C12711yesmaximal, characteristic, direct product factor, Sylow
266C2×C127#256511yescharacteristic
267C2×C127#258111yescharacteristic
268C2×C127#261011yescharacteristic
269C2×C127#262611yescharacteristic
270C2×C127#263811yescharacteristic
271C2×C127#266311yescharacteristic
272C2×C127#274512yes—
273C2×C127#277512yes—
274C2×C127#279912yes—
275C2×C127#280912yes—
276C2×C127#282111yescharacteristic
277C2×#391C2×C12711yescharacteristic
278C2×#387C2×C12711yescharacteristic
279#1446C2×C12711yescharacteristic
280#1446C2×C12711yescharacteristic
281#1445C2×C12712yes—
282#1444C2×C12711yescharacteristic
283#1443C2×C12712yes—
284#1442C2×C12712yes—
285#1442C2×C12711yescharacteristic
286#1441C2×C12712yes—
287#1434C2×C12711yescharacteristic
288C127#1702511yescharacteristic, direct product factor, Sylow
289C23×Q8C22×C12711yescharacteristic
290C22×C4⋊C4C22×C12711yescharacteristic
291C22×C4⋊C4C22×C12712yes—
292C22×C4⋊C4C22×C12711yescharacteristic
293C22×C4⋊C4C22×C12711yescharacteristic
294C22×C42C22×C12711yescharacteristic
295#391—24no—
296#387—24no—
297C2×#95C22×C12712yes—
298C2×#95C22×C12711yescharacteristic
299C2×#95C22×C12711yescharacteristic
300C2×#95C22×C12712yes—
301C2×#95C22×C12712yes—
302C2×#95C22×C12712yes—
303C2×#95C22×C12712yes—
304C2×#95C22×C12711yescharacteristic
305C2×#95C22×C12711yescharacteristic
306C2×#95C22×C12711yescharacteristic
307C2×#95C22×C12711yescharacteristic
308C2×#95C22×C12712yes—
309C2×#95C22×C12712yes—
310C2×#95C22×C12712yes—
311C2×#95C22×C12712yes—
312C2×#95C22×C12712yes—
313C2×#95C22×C12712yes—
314C22×Q8—22no—
315C22×Q8—22no—
316C22×Q8—42no—
317C22×Q8—42no—
318C22×Q8Q8×C12714yes—
319C23×C4C23×C12712yes—
320C23×C4C23×C12711yescharacteristic
321C23×C4C23×C12711yescharacteristic
322C23×C4C23×C12712yes—
323C23×C4C23×C12711yescharacteristic
324C23×C4C23×C12711yescharacteristic
325C23×C4C23×C12712yes—
326C23×C4C23×C12711yescharacteristic
327C23×C4C23×C12711yescharacteristic
328C23×C4C23×C12712yes—
329C23×C4C23×C12711yescharacteristic
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C42—42no—
341C2×C42D4×C12714yes—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359C24C24×C12711yescharacteristic
360C2×Q8—28no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
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—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4C2×D4×C12712yes—
420C22×C4C2×D4×C12712yes—
421C22×C4C2×D4×C12712yes—
422C22×C4C2×Q8×C12714yes—
423C22×C4C2×Q8×C12712yes—
424C22×C4#41×C12712yes—
425C22×C4#41×C12714yes—
426C22×C4#41×C12712yes—
427C22×C4#41×C12712yes—
428C4⋊C4—28no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C42—44no—
433C23C22×D4×C12711yescharacteristic
434C23C22×Q8×C12711yescharacteristic
435C23C2×#41×C12711yescharacteristic
436C23C2×#41×C12711yescharacteristic
437C23C2×#41×C12711yescharacteristic
438C23C2×#41×C12712yes—
439C23H(5,𝔽2)×C12712yes—
440C23H(5,𝔽2)×C12712yes—
441C232−1+4×C12711yescharacteristic
442C232−1+4×C12712yes—
443C232−1+4×C12711yescharacteristic
444Q8—416no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4C22⋊Q8×C12714yes—
508C22C2×C22⋊Q8×C12711yescharacteristic
509C22#525×C12711yescharacteristic
510C22#532×C12711yescharacteristic
511C22#534×C12711yescharacteristic
512C22#535×C12711yescharacteristic
513C22#536×C12711yescharacteristic
514C22#536×C12711yescharacteristic
515C22#538×C12712yes—
516C22#539×C12712yes—
517C22#539×C12712yes—
518C22#542×C12712yes—
519C22#542×C12712yes—
520C22#543×C12712yes—
521C22#551×C12712yes—
522C22#552×C12712yes—
523C22#555×C12712yes—
524C22#556×C12711yescharacteristic
525C22#556×C12711yescharacteristic
526C22#557×C12711yescharacteristic
527C22#557×C12711yescharacteristic
528C22#557×C12712yes—
529C22#563×C12712yes—
530C22#563×C12712yes—
531C4—24no—
532C4—42no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—82no—
539C4—81no—
540C4—82no—
541C4—81no—
542C2C127×#256511yescharacteristic
543C2C127×#258111yescharacteristic
544C2C127×#261011yescharacteristic
545C2C127×#262611yescharacteristic
546C2C127×#263811yescharacteristic
547C2C127×#266311yescharacteristic
548C2C127×#274512yes—
549C2C127×#277512yes—
550C2C127×#279912yes—
551C2C127×#280912yes—
552C2C127×#282111yescharacteristic
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
265. #17025C127direct (N × H)
288. C127#17025direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#391×C1272
C2×#387×C1272
C127×#14462
C127×#14452
C127×#14442
C127×#14432
C127×#14422
C127×#14412
C127×#14342
#17025127