Finite Group Databasebeta

← C127×#17112 #44580785

Subgroups — C127×#17112 #44580785

Order 32512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C127C212yesmaximal
2C2×#395×C127C212yesmaximal
3C127×#1446C211yesmaximal, characteristic
4C127×#1445C212yesmaximal
5C127×#1444C212yesmaximal
6C127×#1444C211yesmaximal, characteristic
7C127×#1443C212yesmaximal
8C127×#1442C211yesmaximal, characteristic
9C127×#1442C212yesmaximal
10C23×Q8×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16#400×C127—28no—
17#395×C127—28no—
18C2×#95×C127C2212yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2212yes—
32C2×#95×C127C2211yescharacteristic
33C22×Q8×C127—24no—
34C22×Q8×C127—22no—
35C22×Q8×C127—41no—
36C22×Q8×C127—42no—
37C22×Q8×C127—41no—
38C23×C4×C127C2312yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2312yes—
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2311yescharacteristic
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—24no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—24no—
53C2×C4⋊C4×C127—41no—
54C2×C4⋊C4×C127—42no—
55C2×C4⋊C4×C127—42no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127—42no—
58C2×C4⋊C4×C127—42no—
59C2×C4⋊C4×C127—41no—
60#95×C127—24no—
61#95×C127—24no—
62#95×C127—24no—
63#95×C127—24no—
64#95×C127—24no—
65#95×C127—24no—
66#95×C127—24no—
67#95×C127—22no—
68#95×C127—41no—
69#95×C127—42no—
70#95×C127—42no—
71#95×C127—42no—
72#95×C127—41no—
73#95×C127D418yes—
74#95×C127Q814yescomplemented
75C24×C127C2411yescharacteristic
76C2×Q8×C127—44no—
77C2×Q8×C127—48no—
78C2×Q8×C127—48no—
79C2×Q8×C127—44no—
80C2×Q8×C127—44no—
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—21no—
93C22×C4×C127—22no—
94C22×C4×C127—21no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
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—21no—
110C22×C4×C127—22no—
111C22×C4×C127—21no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
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×C127C2×D414yes—
132C22×C4×C127C2×D414yes—
133C22×C4×C127C2×D414yes—
134C22×C4×C127C2×Q812yes—
135C22×C4×C127C2×Q814yes—
136C22×C4×C127#4112yes—
137C22×C4×C127#4112yes—
138C4⋊C4×C127—44no—
139C4⋊C4×C127—48no—
140C4⋊C4×C127—48no—
141C4⋊C4×C127—48no—
142C4⋊C4×C127—44no—
143C23×C127C22×D412yes—
144C23×C127C22×Q811yescharacteristic
145C23×C127C2×#4111yescharacteristic
146C23×C127C2×#4111yescharacteristic
147C23×C127H(5,𝔽2)11yescharacteristic
148C23×C127H(5,𝔽2)12yes—
149C23×C127H(5,𝔽2)11yescharacteristic
150C23×C1272−1+411yescharacteristic
151C23×C1272−1+411yescharacteristic
152C23×C1272−1+412yes—
153C23×C1272−1+412yes—
154Q8×C127—416no—
155C2×C4×C127—24no—
156C2×C4×C127—24no—
157C2×C4×C127—24no—
158C2×C4×C127—22no—
159C2×C4×C127—22no—
160C2×C4×C127—24no—
161C2×C4×C127—22no—
162C2×C4×C127—24no—
163C2×C4×C127—24no—
164C2×C4×C127—24no—
165C2×C4×C127—24no—
166C2×C4×C127—24no—
167C2×C4×C127—24no—
168C2×C4×C127—24no—
169C2×C4×C127—24no—
170C2×C4×C127—24no—
171C2×C4×C127—24no—
172C2×C4×C127—24no—
173C2×C4×C127—42no—
174C2×C4×C127—42no—
175C2×C4×C127—42no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—42no—
182C2×C4×C127—42no—
183C2×C4×C127—41no—
184C2×C4×C127—41no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—42no—
189C2×C4×C127—41no—
190C2×C4×C127—42no—
191C2×C4×C127—41no—
192C2×C4×C127—42no—
193C2×C4×C127—41no—
194C2×C4×C127—41no—
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—42no—
209C2×C4×C127—42no—
210C2×C4×C127—42no—
211C2×C4×C127—42no—
212C22×C127#53512yes—
213C22×C127#53612yes—
214C22×C127#53612yes—
215C22×C127#53811yescharacteristic
216C22×C127#53911yescharacteristic
217C22×C127#53912yes—
218C22×C127#54112yes—
219C22×C127#54411yescharacteristic
220C22×C127#54412yes—
221C22×C127D4211yescharacteristic
222C22×C127#54612yes—
223C22×C127#54712yes—
224C22×C127D4×Q812yes—
225C22×C127#55411yescharacteristic
226C22×C127#55611yescharacteristic
227C22×C127#55711yescharacteristic
228C22×C127H(3,𝔽4)11yescharacteristic
229C22×C127#56311yescharacteristic
230C22×C127#56312yes—
231C22×C127#56312yes—
232C22×C127#56312yes—
233C22×C127#56311yescharacteristic
234C22×C127#56311yescharacteristic
235C4×C127—42no—
236C4×C127—44no—
237C4×C127—42no—
238C4×C127—42no—
239C4×C127—44no—
240C4×C127—44no—
241C4×C127—44no—
242C4×C127—82no—
243C4×C127—82no—
244#17112C12711yesmaximal, characteristic, direct product factor, Sylow
245C2×C127#267011yescharacteristic
246C2×C127#267711yescharacteristic
247C2×C127#268011yescharacteristic
248C2×C127#268912yes—
249C2×C127#271312yes—
250C2×C127#272012yes—
251C2×C127#280511yescharacteristic
252C2×C127#282112yes—
253C2×C127#283311yescharacteristic
254C2×C127#283511yescharacteristic
255C2×C127#284411yescharacteristic
256C2×#400C2×C12712yes—
257C2×#395C2×C12712yes—
258#1446C2×C12711yescharacteristic
259#1445C2×C12712yes—
260#1444C2×C12712yes—
261#1444C2×C12711yescharacteristic
262#1443C2×C12712yes—
263#1442C2×C12711yescharacteristic
264#1442C2×C12712yes—
265C127#1711211yescharacteristic, direct product factor, Sylow
266C23×Q8C22×C12711yescharacteristic
267C22×C4⋊C4C22×C12712yes—
268C22×C4⋊C4C22×C12712yes—
269C22×C4⋊C4C22×C12711yescharacteristic
270C22×C4⋊C4C22×C12712yes—
271C22×C4⋊C4C22×C12711yescharacteristic
272#400—28no—
273#395—28no—
274C2×#95C22×C12712yes—
275C2×#95C22×C12712yes—
276C2×#95C22×C12712yes—
277C2×#95C22×C12712yes—
278C2×#95C22×C12712yes—
279C2×#95C22×C12712yes—
280C2×#95C22×C12712yes—
281C2×#95C22×C12711yescharacteristic
282C2×#95C22×C12712yes—
283C2×#95C22×C12712yes—
284C2×#95C22×C12711yescharacteristic
285C2×#95C22×C12712yes—
286C2×#95C22×C12711yescharacteristic
287C2×#95C22×C12712yes—
288C2×#95C22×C12711yescharacteristic
289C22×Q8—24no—
290C22×Q8—22no—
291C22×Q8—41no—
292C22×Q8—42no—
293C22×Q8—41no—
294C23×C4C23×C12712yes—
295C23×C4C23×C12712yes—
296C23×C4C23×C12712yes—
297C23×C4C23×C12712yes—
298C23×C4C23×C12712yes—
299C23×C4C23×C12711yescharacteristic
300C23×C4C23×C12712yes—
301C23×C4C23×C12711yescharacteristic
302C23×C4C23×C12711yescharacteristic
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—41no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—41no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—41no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—41no—
329#95D4×C12718yes—
330#95Q8×C12714yescomplemented
331C24C24×C12711yescharacteristic
332C2×Q8—44no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—44no—
336C2×Q8—44no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
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×C4C2×D4×C12714yes—
388C22×C4C2×D4×C12714yes—
389C22×C4C2×D4×C12714yes—
390C22×C4C2×Q8×C12712yes—
391C22×C4C2×Q8×C12714yes—
392C22×C4#41×C12712yes—
393C22×C4#41×C12712yes—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C23C22×D4×C12712yes—
400C23C22×Q8×C12711yescharacteristic
401C23C2×#41×C12711yescharacteristic
402C23C2×#41×C12711yescharacteristic
403C23H(5,𝔽2)×C12711yescharacteristic
404C23H(5,𝔽2)×C12712yes—
405C23H(5,𝔽2)×C12711yescharacteristic
406C232−1+4×C12711yescharacteristic
407C232−1+4×C12711yescharacteristic
408C232−1+4×C12712yes—
409C232−1+4×C12712yes—
410Q8—416no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C22#535×C12712yes—
469C22#536×C12712yes—
470C22#536×C12712yes—
471C22#538×C12711yescharacteristic
472C22#539×C12711yescharacteristic
473C22#539×C12712yes—
474C22#541×C12712yes—
475C22#544×C12711yescharacteristic
476C22#544×C12712yes—
477C22D42×C12711yescharacteristic
478C22#546×C12712yes—
479C22#547×C12712yes—
480C22D4×Q8×C12712yes—
481C22#554×C12711yescharacteristic
482C22#556×C12711yescharacteristic
483C22#557×C12711yescharacteristic
484C22H(3,𝔽4)×C12711yescharacteristic
485C22#563×C12711yescharacteristic
486C22#563×C12711yescharacteristic
487C22#563×C12712yes—
488C22#563×C12712yes—
489C22#563×C12712yes—
490C22#563×C12711yescharacteristic
491C4—42no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—44no—
496C4—44no—
497C4—44no—
498C4—82no—
499C4—82no—
500C2C127×#267011yescharacteristic
501C2C127×#267711yescharacteristic
502C2C127×#268011yescharacteristic
503C2C127×#268912yes—
504C2C127×#271312yes—
505C2C127×#272012yes—
506C2C127×#280511yescharacteristic
507C2C127×#282112yes—
508C2C127×#283311yescharacteristic
509C2C127×#283511yescharacteristic
510C2C127×#284411yescharacteristic
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
74. #95×C127Q8semidirect
244. #17112C127direct (N × H)
265. C127#17112direct (N × H)
330. #95Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#400×C1272
C2×#395×C1272
C127×#14462
C127×#14452
C127×#14442
C127×#14432
C127×#14422
#17112127