Finite Group Databasebeta

← C127×#17077 #44580750

Subgroups — C127×#17077 #44580750

Order 32512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393×C127C212yesmaximal, complemented
2C2×#391×C127C211yesmaximal, characteristic, complemented
3C2×#390×C127C211yesmaximal, characteristic
4C127×#1442C212yesmaximal, complemented
5C127×#1439C212yesmaximal, complemented
6C127×#1439C214yesmaximal, complemented
7C127×#1438C212yesmaximal
8C127×C4⋊#95C211yesmaximal, characteristic, complemented
9C23×Q8×C127C2211yescharacteristic
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2214yes—
17C22×C22⋊C4×C127C2214yes—
18C22×C42×C127C2211yescharacteristic
19#393×C127—28no—
20#391×C127—24no—
21#390×C127—24no—
22C2×#95×C127C2212yes—
23C2×#95×C127C2214yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2214yes—
28C25×C127C2312yes—
29C22×Q8×C127—22no—
30C22×Q8×C127—22no—
31C22×Q8×C127—42no—
32C22×Q8×C127—42no—
33C22×Q8×C127D414yescomplemented
34C22×D4×C127—22no—
35C22×D4×C127—22no—
36C22×D4×C127—42no—
37C22×D4×C127—42no—
38C22×D4×C127Q814yescomplemented
39C23×C4×C127C2312yes—
40C23×C4×C127C2314yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2311yescharacteristic
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127—22no—
48C2×C4⋊C4×C127—24no—
49C2×C4⋊C4×C127—42no—
50C2×C4⋊C4×C127—42no—
51C2×C4⋊C4×C127—42no—
52C2×C4⋊C4×C127—42no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127—28no—
56C2×C22⋊C4×C127—24no—
57C2×C22⋊C4×C127—24no—
58C2×C22⋊C4×C127—28no—
59C2×C22⋊C4×C127—44no—
60C2×C22⋊C4×C127—44no—
61C2×C22⋊C4×C127—44no—
62C2×C22⋊C4×C127—44no—
63C2×C22⋊C4×C127—44no—
64C2×C42×C127—42no—
65C2×C42×C127D414yes—
66#95×C127—24no—
67#95×C127—24no—
68#95×C127—28no—
69#95×C127—24no—
70#95×C127—24no—
71#95×C127—44no—
72C24×C127—22no—
73C24×C127—22no—
74C24×C127—22no—
75C24×C127—24no—
76C24×C127—22no—
77C24×C127—24no—
78C24×C127—22no—
79C24×C127—24no—
80C24×C127—24no—
81C24×C127—22no—
82C24×C127C2×Q814yes—
83C24×C127C2411yescharacteristic
84C2×Q8×C127—28no—
85C2×Q8×C127—48no—
86C2×Q8×C127—44no—
87C2×Q8×C127—44no—
88C2×Q8×C127—48no—
89C2×D4×C127—28no—
90C2×D4×C127—48no—
91C2×D4×C127—44no—
92C2×D4×C127—44no—
93C2×D4×C127—48no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—21no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—21no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—24no—
104C22×C4×C127—24no—
105C22×C4×C127—22no—
106C22×C4×C127—24no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—24no—
110C22×C4×C127—22no—
111C22×C4×C127—24no—
112C22×C4×C127—24no—
113C22×C4×C127—24no—
114C22×C4×C127—24no—
115C22×C4×C127—24no—
116C22×C4×C127—24no—
117C22×C4×C127—22no—
118C22×C4×C127—24no—
119C22×C4×C127—24no—
120C22×C4×C127—24no—
121C22×C4×C127—22no—
122C22×C4×C127C2×D414yes—
123C22×C4×C127C2×D412yes—
124C22×C4×C127C2×D412yes—
125C22×C4×C127C2×D412yes—
126C22×C4×C127C2×D412yes—
127C22×C4×C127C2×Q812yes—
128C22×C4×C127#4112yes—
129C22×C4×C127#4112yes—
130C22×C4×C127#4114yes—
131C4⋊C4×C127—44no—
132C4⋊C4×C127—44no—
133C4⋊C4×C127—48no—
134C22⋊C4×C127—416no—
135C22⋊C4×C127—416no—
136C22⋊C4×C127—48no—
137C22⋊C4×C127—48no—
138C42×C127—44no—
139C23×C127—24no—
140C23×C127—28no—
141C23×C127—24no—
142C23×C127—28no—
143C23×C127—24no—
144C23×C127—42no—
145C23×C127—42no—
146C23×C127—44no—
147C23×C127—42no—
148C23×C127—42no—
149C23×C127—42no—
150C23×C127—42no—
151C23×C127—44no—
152C23×C127—44no—
153C23×C127—44no—
154C23×C127—44no—
155C23×C127—44no—
156C23×C127—42no—
157C23×C127—44no—
158C23×C127—42no—
159C23×C127—44no—
160C23×C127—44no—
161C23×C127—44no—
162C23×C127C22×D411yescharacteristic
163C23×C127C22×D411yescharacteristic
164C23×C127C22×Q811yescharacteristic
165C23×C127C2×#4111yescharacteristic
166C23×C127C2×#4111yescharacteristic
167C23×C127C2×#4112yes—
168C23×C127H(5,𝔽2)12yes—
169C23×C127H(5,𝔽2)12yes—
170C23×C127H(5,𝔽2)12yes—
171C23×C127H(5,𝔽2)11yescharacteristic
172C23×C1272−1+411yescharacteristic
173Q8×C127—416no—
174D4×C127—416no—
175C2×C4×C127—22no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—24no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—22no—
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—28no—
189C2×C4×C127—42no—
190C2×C4×C127—42no—
191C2×C4×C127—44no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—44no—
195C2×C4×C127—42no—
196C2×C4×C127—42no—
197C2×C4×C127—42no—
198C2×C4×C127—42no—
199C2×C4×C127—44no—
200C2×C4×C127—44no—
201C2×C4×C127—44no—
202C2×C4×C127—44no—
203C2×C4×C127—44no—
204C2×C4×C127—44no—
205C2×C4×C127—44no—
206C2×C4×C127—44no—
207C2×C4×C127—44no—
208C2×C4×C127—44no—
209C2×C4×C127#12114yes—
210C2×C4×C127C22⋊Q814yes—
211C2×C4×C127#12414yes—
212C22×C127—44no—
213C22×C127—44no—
214C22×C127—48no—
215C22×C127—44no—
216C22×C127—48no—
217C22×C127—82no—
218C22×C127—82no—
219C22×C127—82no—
220C22×C127—84no—
221C22×C127—82no—
222C22×C127—84no—
223C22×C127C2×#12111yescharacteristic
224C22×C127C2×C22⋊Q811yescharacteristic
225C22×C127C2×#12411yescharacteristic
226C22×C127#53411yescharacteristic
227C22×C127#53412yes—
228C22×C127#53511yescharacteristic
229C22×C127#53511yescharacteristic
230C22×C127#53711yescharacteristic
231C22×C127#53812yes—
232C22×C127#53812yes—
233C22×C127#53812yes—
234C22×C127#53912yes—
235C22×C127#54312yes—
236C22×C127D4×Q811yescharacteristic
237C22×C127#55012yes—
238C22×C127#55011yescharacteristic
239C22×C127#55412yes—
240C22×C127#55411yescharacteristic
241C22×C127#55512yes—
242C22×C127#55611yescharacteristic
243C22×C127#55912yes—
244C22×C127H(3,𝔽4)12yes—
245C22×C127#56212yes—
246C4×C127—24no—
247C4×C127—24no—
248C4×C127—24no—
249C4×C127—44no—
250C4×C127—82no—
251C4×C127—84no—
252C4×C127—82no—
253#17077C12711yesmaximal, characteristic, direct product factor, Sylow
254C2×C127—84no—
255C2×C127#253811yescharacteristic
256C2×C127#254811yescharacteristic
257C2×C127#256311yescharacteristic
258C2×C127#261711yescharacteristic
259C2×C127#262611yescharacteristic
260C2×C127#263911yescharacteristic
261C2×C127#273212yes—
262C2×C127#273312yes—
263C2×C127#280412yes—
264C2×C127#280812yes—
265C2×C127#281911yescharacteristic
266C2×#393C2×C12712yescomplemented
267C2×#391C2×C12711yescharacteristic, complemented
268C2×#390C2×C12711yescharacteristic
269#1442C2×C12712yescomplemented
270#1439C2×C12712yescomplemented
271#1439C2×C12714yescomplemented
272#1438C2×C12712yes—
273C4⋊#95C2×C12711yescharacteristic, complemented
274C127#1707711yescharacteristic, direct product factor, Sylow
275C23×Q8C22×C12711yescharacteristic
276C23×D4C22×C12711yescharacteristic
277C22×C4⋊C4C22×C12712yes—
278C22×C4⋊C4C22×C12711yescharacteristic
279C22×C4⋊C4C22×C12711yescharacteristic
280C22×C22⋊C4C22×C12712yes—
281C22×C22⋊C4C22×C12712yes—
282C22×C22⋊C4C22×C12714yes—
283C22×C22⋊C4C22×C12714yes—
284C22×C42C22×C12711yescharacteristic
285#393—28no—
286#391—24no—
287#390—24no—
288C2×#95C22×C12712yes—
289C2×#95C22×C12714yes—
290C2×#95C22×C12712yes—
291C2×#95C22×C12712yes—
292C2×#95C22×C12712yes—
293C2×#95C22×C12714yes—
294C25C23×C12712yes—
295C22×Q8—22no—
296C22×Q8—22no—
297C22×Q8—42no—
298C22×Q8—42no—
299C22×Q8D4×C12714yescomplemented
300C22×D4—22no—
301C22×D4—22no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4Q8×C12714yescomplemented
305C23×C4C23×C12712yes—
306C23×C4C23×C12714yes—
307C23×C4C23×C12712yes—
308C23×C4C23×C12712yes—
309C23×C4C23×C12711yescharacteristic
310C23×C4C23×C12711yescharacteristic
311C23×C4C23×C12711yescharacteristic
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C42—42no—
331C2×C42D4×C12714yes—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—44no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24C2×Q8×C12714yes—
349C24C24×C12711yescharacteristic
350C2×Q8—28no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—48no—
355C2×D4—28no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4C2×D4×C12714yes—
389C22×C4C2×D4×C12712yes—
390C22×C4C2×D4×C12712yes—
391C22×C4C2×D4×C12712yes—
392C22×C4C2×D4×C12712yes—
393C22×C4C2×Q8×C12712yes—
394C22×C4#41×C12712yes—
395C22×C4#41×C12712yes—
396C22×C4#41×C12714yes—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C42—44no—
405C23—24no—
406C23—28no—
407C23—24no—
408C23—28no—
409C23—24no—
410C23—42no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23C22×D4×C12711yescharacteristic
429C23C22×D4×C12711yescharacteristic
430C23C22×Q8×C12711yescharacteristic
431C23C2×#41×C12711yescharacteristic
432C23C2×#41×C12711yescharacteristic
433C23C2×#41×C12712yes—
434C23H(5,𝔽2)×C12712yes—
435C23H(5,𝔽2)×C12712yes—
436C23H(5,𝔽2)×C12712yes—
437C23H(5,𝔽2)×C12711yescharacteristic
438C232−1+4×C12711yescharacteristic
439Q8—416no—
440D4—416no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4#121×C12714yes—
476C2×C4C22⋊Q8×C12714yes—
477C2×C4#124×C12714yes—
478C22—44no—
479C22—48no—
480C22—44no—
481C22—48no—
482C22—44no—
483C22—82no—
484C22—82no—
485C22—84no—
486C22—82no—
487C22—84no—
488C22—82no—
489C22C2×#121×C12711yescharacteristic
490C22C2×C22⋊Q8×C12711yescharacteristic
491C22C2×#124×C12711yescharacteristic
492C22#534×C12711yescharacteristic
493C22#534×C12712yes—
494C22#535×C12711yescharacteristic
495C22#535×C12711yescharacteristic
496C22#537×C12711yescharacteristic
497C22#538×C12712yes—
498C22#538×C12712yes—
499C22#538×C12712yes—
500C22#539×C12712yes—
501C22#543×C12712yes—
502C22D4×Q8×C12711yescharacteristic
503C22#550×C12712yes—
504C22#550×C12711yescharacteristic
505C22#554×C12712yes—
506C22#554×C12711yescharacteristic
507C22#555×C12712yes—
508C22#556×C12711yescharacteristic
509C22#559×C12712yes—
510C22H(3,𝔽4)×C12712yes—
511C22#562×C12712yes—
512C4—24no—
513C4—24no—
514C4—24no—
515C4—44no—
516C4—84no—
517C4—82no—
518C4—82no—
519C2—84no—
520C2C127×#253811yescharacteristic
521C2C127×#254811yescharacteristic
522C2C127×#256311yescharacteristic
523C2C127×#261711yescharacteristic
524C2C127×#262611yescharacteristic
525C2C127×#263911yescharacteristic
526C2C127×#273212yes—
527C2C127×#273312yes—
528C2C127×#280412yes—
529C2C127×#280812yes—
530C2C127×#281911yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#393×C127C2semidirect
2. C2×#391×C127C2semidirect
4. C127×#1442C2semidirect
5. C127×#1439C2semidirect
6. C127×#1439C2semidirect
8. C127×C4⋊#95C2semidirect
33. C22×Q8×C127D4semidirect
38. C22×D4×C127Q8semidirect
253. #17077C127direct (N × H)
266. C2×#393C2×C127semidirect
267. C2×#391C2×C127semidirect
269. #1442C2×C127semidirect
270. #1439C2×C127semidirect
271. #1439C2×C127semidirect
273. C4⋊#95C2×C127semidirect
274. C127#17077direct (N × H)
299. C22×Q8D4×C127semidirect
304. C22×D4Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#393×C1272
C2×#391×C1272
C2×#390×C1272
C127×#14422
C127×#14392
C127×#14382
C127×C4⋊#952
#17077127