Finite Group Databasebeta

← C127×#16814 #44580487

Subgroups — C127×#16814 #44580487

Order 32512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399×C127C212yesmaximal, complemented
2C2×#393×C127C212yesmaximal, complemented
3C2×#391×C127C211yesmaximal, characteristic, complemented
4C2×#390×C127C211yesmaximal, characteristic
5C127×#1444C212yesmaximal, complemented
6C127×#1439C214yesmaximal, complemented
7C127×#1438C212yesmaximal
8C127×#1434C211yesmaximal, characteristic, complemented
9C23×Q8×C127C2211yescharacteristic
10C23×D4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2214yes—
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2214yes—
16C22×C22⋊C4×C127C2214yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C42×C127C2211yescharacteristic
19#399×C127—28no—
20#393×C127—28no—
21#391×C127—24no—
22#390×C127—24no—
23C2×#95×C127C2214yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C25×C127C2312yes—
30C22×Q8×C127—22no—
31C22×Q8×C127—24no—
32C22×Q8×C127—24no—
33C22×Q8×C127—42no—
34C22×D4×C127—22no—
35C22×D4×C127—22no—
36C22×D4×C127—42no—
37C22×D4×C127—42no—
38C22×D4×C127Q814yescomplemented
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2311yescharacteristic
42C23×C4×C127C2314yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2311yescharacteristic
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127—22no—
48C2×C4⋊C4×C127—28no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—42no—
51C2×C4⋊C4×C127—44no—
52C2×C4⋊C4×C127—44no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127—28no—
56C2×C22⋊C4×C127—24no—
57C2×C22⋊C4×C127—28no—
58C2×C22⋊C4×C127—24no—
59C2×C22⋊C4×C127—28no—
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—42no—
71#95×C127D418yescomplemented
72C24×C127—22no—
73C24×C127—22no—
74C24×C127—22no—
75C24×C127—24no—
76C24×C127—22no—
77C24×C127—24no—
78C24×C127—24no—
79C24×C127—24no—
80C24×C127—22no—
81C24×C127—22no—
82C24×C127C2×Q814yes—
83C24×C127C2411yescharacteristic
84C2×Q8×C127—28no—
85C2×Q8×C127—416no—
86C2×Q8×C127—44no—
87C2×Q8×C127—44no—
88C2×D4×C127—28no—
89C2×D4×C127—48no—
90C2×D4×C127—44no—
91C2×D4×C127—44no—
92C2×D4×C127—48no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—24no—
99C22×C4×C127—24no—
100C22×C4×C127—24no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—24no—
104C22×C4×C127—22no—
105C22×C4×C127—24no—
106C22×C4×C127—24no—
107C22×C4×C127—24no—
108C22×C4×C127—22no—
109C22×C4×C127—24no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—24no—
115C22×C4×C127—22no—
116C22×C4×C127—24no—
117C22×C4×C127—24no—
118C22×C4×C127—24no—
119C22×C4×C127—22no—
120C22×C4×C127C2×D412yes—
121C22×C4×C127C2×D418yes—
122C22×C4×C127C2×D412yes—
123C22×C4×C127C2×D414yes—
124C22×C4×C127C2×D412yes—
125C22×C4×C127C2×Q812yes—
126C22×C4×C127#4114yes—
127C22×C4×C127#4112yes—
128C4⋊C4×C127—44no—
129C4⋊C4×C127—416no—
130C4⋊C4×C127—44no—
131C22⋊C4×C127—416no—
132C22⋊C4×C127—48no—
133C22⋊C4×C127—416no—
134C22⋊C4×C127—48no—
135C42×C127—44no—
136C23×C127—24no—
137C23×C127—28no—
138C23×C127—24no—
139C23×C127—24no—
140C23×C127—28no—
141C23×C127—42no—
142C23×C127—42no—
143C23×C127—42no—
144C23×C127—42no—
145C23×C127—44no—
146C23×C127—42no—
147C23×C127—44no—
148C23×C127—42no—
149C23×C127—44no—
150C23×C127—44no—
151C23×C127—44no—
152C23×C127—44no—
153C23×C127—44no—
154C23×C127—42no—
155C23×C127—42no—
156C23×C127—44no—
157C23×C127—44no—
158C23×C127—44no—
159C23×C127C22×D411yescharacteristic
160C23×C127C22×D412yes—
161C23×C127C22×Q811yescharacteristic
162C23×C127C2×#4111yescharacteristic
163C23×C127C2×#4112yes—
164C23×C127H(5,𝔽2)12yes—
165C23×C127H(5,𝔽2)12yes—
166C23×C127H(5,𝔽2)12yes—
167C23×C1272−1+412yes—
168Q8×C127—416no—
169D4×C127—416no—
170C2×C4×C127—22no—
171C2×C4×C127—22no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—24no—
175C2×C4×C127—24no—
176C2×C4×C127—28no—
177C2×C4×C127—24no—
178C2×C4×C127—22no—
179C2×C4×C127—22no—
180C2×C4×C127—28no—
181C2×C4×C127—22no—
182C2×C4×C127—28no—
183C2×C4×C127—28no—
184C2×C4×C127—28no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—44no—
188C2×C4×C127—44no—
189C2×C4×C127—44no—
190C2×C4×C127—44no—
191C2×C4×C127—44no—
192C2×C4×C127—44no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—42no—
196C2×C4×C127—44no—
197C2×C4×C127—42no—
198C2×C4×C127—44no—
199C2×C4×C127—44no—
200C2×C4×C127—42no—
201C2×C4×C127—42no—
202C2×C4×C127—44no—
203C2×C4×C127C22⋊Q814yes—
204C2×C4×C127#12714yes—
205C22×C127—44no—
206C22×C127—44no—
207C22×C127—48no—
208C22×C127—48no—
209C22×C127—44no—
210C22×C127—82no—
211C22×C127—82no—
212C22×C127—82no—
213C22×C127—84no—
214C22×C127—84no—
215C22×C127—82no—
216C22×C127C2×C22⋊Q811yescharacteristic
217C22×C127#52511yescharacteristic
218C22×C127C2×#12711yescharacteristic
219C22×C127#53411yescharacteristic
220C22×C127#53511yescharacteristic
221C22×C127#53511yescharacteristic
222C22×C127#53514yes—
223C22×C127#53611yescharacteristic
224C22×C127#53812yes—
225C22×C127#54214yes—
226C22×C127#54312yes—
227C22×C127#54614yes—
228C22×C127#54712yes—
229C22×C127D4×Q812yes—
230C22×C127#55412yes—
231C22×C127#55912yes—
232C22×C127#56214yes—
233C4×C127—24no—
234C4×C127—24no—
235C4×C127—42no—
236C4×C127—48no—
237C4×C127—44no—
238C4×C127—82no—
239C4×C127—82no—
240#16814C12711yesmaximal, characteristic, direct product factor, Sylow
241C2×C127—84no—
242C2×C127#253911yescharacteristic
243C2×C127#254211yescharacteristic
244C2×C127#258011yescharacteristic
245C2×C127#262611yescharacteristic
246C2×C127#263211yescharacteristic
247C2×C127#265711yescharacteristic
248C2×C127#269214yes—
249C2×C127#271014yes—
250C2×C127#281611yescharacteristic
251C2×#399C2×C12712yescomplemented
252C2×#393C2×C12712yescomplemented
253C2×#391C2×C12711yescharacteristic, complemented
254C2×#390C2×C12711yescharacteristic
255#1444C2×C12712yescomplemented
256#1439C2×C12714yescomplemented
257#1438C2×C12712yes—
258#1434C2×C12711yescharacteristic, complemented
259C127#1681411yescharacteristic, direct product factor, Sylow
260C23×Q8C22×C12711yescharacteristic
261C23×D4C22×C12711yescharacteristic
262C22×C4⋊C4C22×C12711yescharacteristic
263C22×C4⋊C4C22×C12714yes—
264C22×C4⋊C4C22×C12711yescharacteristic
265C22×C22⋊C4C22×C12712yes—
266C22×C22⋊C4C22×C12714yes—
267C22×C22⋊C4C22×C12714yes—
268C22×C22⋊C4C22×C12712yes—
269C22×C42C22×C12711yescharacteristic
270#399—28no—
271#393—28no—
272#391—24no—
273#390—24no—
274C2×#95C22×C12714yes—
275C2×#95C22×C12712yes—
276C2×#95C22×C12712yes—
277C2×#95C22×C12712yes—
278C2×#95C22×C12712yes—
279C2×#95C22×C12712yes—
280C25C23×C12712yes—
281C22×Q8—22no—
282C22×Q8—24no—
283C22×Q8—24no—
284C22×Q8—42no—
285C22×D4—22no—
286C22×D4—22no—
287C22×D4—42no—
288C22×D4—42no—
289C22×D4Q8×C12714yescomplemented
290C23×C4C23×C12712yes—
291C23×C4C23×C12712yes—
292C23×C4C23×C12714yes—
293C23×C4C23×C12711yescharacteristic
294C23×C4C23×C12712yes—
295C23×C4C23×C12711yescharacteristic
296C23×C4C23×C12711yescharacteristic
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C42—42no—
316C2×C42D4×C12714yes—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—42no—
322#95D4×C12718yescomplemented
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24C2×Q8×C12714yes—
334C24C24×C12711yescharacteristic
335C2×Q8—28no—
336C2×Q8—416no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×D4—28no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4C2×D4×C12712yes—
372C22×C4C2×D4×C12718yes—
373C22×C4C2×D4×C12712yes—
374C22×C4C2×D4×C12714yes—
375C22×C4C2×D4×C12712yes—
376C22×C4C2×Q8×C12712yes—
377C22×C4#41×C12714yes—
378C22×C4#41×C12712yes—
379C4⋊C4—44no—
380C4⋊C4—416no—
381C4⋊C4—44no—
382C22⋊C4—416no—
383C22⋊C4—48no—
384C22⋊C4—416no—
385C22⋊C4—48no—
386C42—44no—
387C23—24no—
388C23—28no—
389C23—24no—
390C23—24no—
391C23—28no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23C22×D4×C12711yescharacteristic
411C23C22×D4×C12712yes—
412C23C22×Q8×C12711yescharacteristic
413C23C2×#41×C12711yescharacteristic
414C23C2×#41×C12712yes—
415C23H(5,𝔽2)×C12712yes—
416C23H(5,𝔽2)×C12712yes—
417C23H(5,𝔽2)×C12712yes—
418C232−1+4×C12712yes—
419Q8—416no—
420D4—416no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4C22⋊Q8×C12714yes—
455C2×C4#127×C12714yes—
456C22—44no—
457C22—44no—
458C22—48no—
459C22—48no—
460C22—44no—
461C22—82no—
462C22—82no—
463C22—84no—
464C22—84no—
465C22—82no—
466C22—82no—
467C22C2×C22⋊Q8×C12711yescharacteristic
468C22#525×C12711yescharacteristic
469C22C2×#127×C12711yescharacteristic
470C22#534×C12711yescharacteristic
471C22#535×C12711yescharacteristic
472C22#535×C12711yescharacteristic
473C22#535×C12714yes—
474C22#536×C12711yescharacteristic
475C22#538×C12712yes—
476C22#542×C12714yes—
477C22#543×C12712yes—
478C22#546×C12714yes—
479C22#547×C12712yes—
480C22D4×Q8×C12712yes—
481C22#554×C12712yes—
482C22#559×C12712yes—
483C22#562×C12714yes—
484C4—24no—
485C4—24no—
486C4—48no—
487C4—44no—
488C4—42no—
489C4—82no—
490C4—82no—
491C2—84no—
492C2C127×#253911yescharacteristic
493C2C127×#254211yescharacteristic
494C2C127×#258011yescharacteristic
495C2C127×#262611yescharacteristic
496C2C127×#263211yescharacteristic
497C2C127×#265711yescharacteristic
498C2C127×#269214yes—
499C2C127×#271014yes—
500C2C127×#281611yescharacteristic
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×#399×C127C2semidirect
2. C2×#393×C127C2semidirect
3. C2×#391×C127C2semidirect
5. C127×#1444C2semidirect
6. C127×#1439C2semidirect
8. C127×#1434C2semidirect
38. C22×D4×C127Q8semidirect
71. #95×C127D4semidirect
240. #16814C127direct (N × H)
251. C2×#399C2×C127semidirect
252. C2×#393C2×C127semidirect
253. C2×#391C2×C127semidirect
255. #1444C2×C127semidirect
256. #1439C2×C127semidirect
258. #1434C2×C127semidirect
259. C127#16814direct (N × H)
289. C22×D4Q8×C127semidirect
322. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#399×C1272
C2×#393×C1272
C2×#391×C1272
C2×#390×C1272
C127×#14442
C127×#14392
C127×#14382
C127×#14342
#16814127