Finite Group Databasebeta

← C127×#14885 #44578558

Subgroups — C127×#14885 #44578558

Order 32512.

4766 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C127C211yesmaximal, characteristic, complemented
2C2×#393×C127C211yesmaximal, characteristic, complemented
3C2×#392×C127C213yesmaximal, complemented
4C2×#386×C127C213yesmaximal, complemented
5C127×#1446C211yesmaximal, characteristic, complemented
6C127×#1438C213yesmaximal, complemented
7C127×C22⋊#95C213yesmaximal, complemented
8C22×#41×C127—24no—
9C22×#41×C127—212no—
10C22×#41×C127—212no—
11C23×Q8×C127C2211yescharacteristic
12C23×D4×C127C2213yes—
13C23×D4×C127C2213yes—
14C24×C4×C127C2213yes—
15C22×C22⋊C4×C127C2213yes—
16C22×C22⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2216yes—
18C22×C22⋊C4×C127C2213yes—
19#393×C127—24no—
20#392×C127—212no—
21#386×C127—212no—
22C2×#95×C127C2213yes—
23C2×#95×C127C2213yes—
24C2×#95×C127C2213yes—
25C2×#95×C127C2213yes—
26C25×C127C2311yescharacteristic
27C25×C127C2313yes—
28C2×#41×C127—296no—
29C2×#41×C127—296no—
30C2×#41×C127—232no—
31C22×Q8×C127—26no—
32C22×Q8×C127—26no—
33C22×Q8×C127D414yescomplemented
34C22×D4×C127—212no—
35C22×D4×C127—212no—
36C22×D4×C127—26no—
37C22×D4×C127—26no—
38C22×D4×C127—212no—
39C22×D4×C127—26no—
40C22×D4×C127—26no—
41C22×D4×C127—26no—
42C22×D4×C127—46no—
43C22×D4×C127D4112yescomplemented
44C23×C4×C127—212no—
45C23×C4×C127—26no—
46C23×C4×C127—26no—
47C23×C4×C127—212no—
48C23×C4×C127D4112yescomplemented
49C23×C4×C127C2313yes—
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2313yes—
52C23×C4×C127C2311yescharacteristic
53C23×C4×C127C2313yes—
54C2×C22⋊C4×C127—26no—
55C2×C22⋊C4×C127—212no—
56C2×C22⋊C4×C127—26no—
57C2×C22⋊C4×C127—26no—
58C2×C22⋊C4×C127—26no—
59C2×C22⋊C4×C127—26no—
60C2×C22⋊C4×C127—212no—
61C2×C22⋊C4×C127—46no—
62C2×C22⋊C4×C127—46no—
63#95×C127—26no—
64#95×C127—26no—
65#95×C127—26no—
66#95×C127—26no—
67C24×C127—26no—
68C24×C127—23no—
69C24×C127—23no—
70C24×C127—26no—
71C24×C127—23no—
72C24×C127—23no—
73C24×C127—23no—
74C24×C127—26no—
75C24×C127—23no—
76C24×C127—21no—
77C24×C127—23no—
78C24×C127—23no—
79C24×C127—23no—
80C24×C127—23no—
81C24×C127—21no—
82C24×C127C2×D4112yes—
83C24×C127C2×D416yes—
84C24×C127#4112yescomplemented
85C24×C127C2411yescharacteristic
86#41×C127—2256no—
87C2×Q8×C127—28no—
88C2×Q8×C127—224no—
89C2×Q8×C127—224no—
90C2×D4×C127—224no—
91C2×D4×C127—224no—
92C2×D4×C127—248no—
93C2×D4×C127—248no—
94C2×D4×C127—224no—
95C2×D4×C127—224no—
96C2×D4×C127—424no—
97C2×D4×C127—424no—
98C2×D4×C127—412no—
99C2×D4×C127—412no—
100C22×C4×C127—248no—
101C22×C4×C127—248no—
102C22×C4×C127—224no—
103C22×C4×C127—23no—
104C22×C4×C127—21no—
105C22×C4×C127—26no—
106C22×C4×C127—26no—
107C22×C4×C127—224no—
108C22×C4×C127—224no—
109C22×C4×C127—23no—
110C22×C4×C127—26no—
111C22×C4×C127—23no—
112C22×C4×C127—26no—
113C22×C4×C127—23no—
114C22×C4×C127—23no—
115C22×C4×C127—23no—
116C22×C4×C127—23no—
117C22×C4×C127—26no—
118C22×C4×C127—23no—
119C22×C4×C127—26no—
120C22×C4×C127—21no—
121C22×C4×C127C2×D416yes—
122C22×C4×C127C2×D416yes—
123C22×C4×C127C2×D416yes—
124C22×C4×C127C2×D416yes—
125C22×C4×C127#4116yescomplemented
126C22⋊C4×C127—424no—
127C22⋊C4×C127—412no—
128C22⋊C4×C127—44no—
129C22⋊C4×C127—412no—
130C23×C127—26no—
131C23×C127—212no—
132C23×C127—212no—
133C23×C127—26no—
134C23×C127—26no—
135C23×C127—26no—
136C23×C127—26no—
137C23×C127—26no—
138C23×C127—212no—
139C23×C127—212no—
140C23×C127—212no—
141C23×C127—212no—
142C23×C127—26no—
143C23×C127—26no—
144C23×C127—22no—
145C23×C127—46no—
146C23×C127—46no—
147C23×C127—46no—
148C23×C127—46no—
149C23×C127—43no—
150C23×C127—46no—
151C23×C127—43no—
152C23×C127—43no—
153C23×C127—46no—
154C23×C127—43no—
155C23×C127—41no—
156C23×C127—43no—
157C23×C127—43no—
158C23×C127—43no—
159C23×C127—43no—
160C23×C127—43no—
161C23×C127—46no—
162C23×C127—43no—
163C23×C127—43no—
164C23×C127#120112yes—
165C23×C127C22×D413yes—
166C23×C127C22×D413yes—
167C23×C127C22×D411yescharacteristic
168C23×C127C2×#4111yescharacteristic
169C23×C127C2×#4113yes—
170C23×C127H(5,𝔽2)13yes—
171C23×C1272−1+411yescharacteristic
172Q8×C127—232no—
173D4×C127—296no—
174D4×C127—448no—
175C2×C4×C127—296no—
176C2×C4×C127—26no—
177C2×C4×C127—26no—
178C2×C4×C127—26no—
179C2×C4×C127—212no—
180C2×C4×C127—26no—
181C2×C4×C127—212no—
182C2×C4×C127—26no—
183C2×C4×C127—212no—
184C2×C4×C127—43no—
185C2×C4×C127—43no—
186C2×C4×C127—46no—
187C2×C4×C127—41no—
188C2×C4×C127—43no—
189C2×C4×C127—43no—
190C2×C4×C127—43no—
191C2×C4×C127—43no—
192C2×C4×C127—43no—
193C2×C4×C127—46no—
194C2×C4×C127—46no—
195C2×C4×C127#12014yes—
196C2×C4×C127#120112yes—
197C22×C127—212no—
198C22×C127—212no—
199C22×C127—212no—
200C22×C127—46no—
201C22×C127—46no—
202C22×C127—412no—
203C22×C127—46no—
204C22×C127—412no—
205C22×C127—412no—
206C22×C127—46no—
207C22×C127—412no—
208C22×C127—42no—
209C22×C127—46no—
210C22×C127—46no—
211C22×C127—81no—
212C22×C127—81no—
213C22×C127—83no—
214C22×C127—83no—
215C22×C127C2×#12011yescharacteristic
216C22×C127C2×#12013yes—
217C22×C127C2×#12013yes—
218C22×C127#52513yes—
219C22×C127#53213yes—
220C22×C127#53513yes—
221C22×C127#53713yes—
222C22×C127#54613yes—
223C22×C127#54616yes—
224C22×C127#54713yes—
225C22×C127#54811yescharacteristic
226C22×C127#55013yes—
227C4×C127—24no—
228C4×C127—212no—
229C4×C127—46no—
230C4×C127—81no—
231C4×C127—83no—
232#14885C12711yesmaximal, characteristic, direct product factor, Sylow
233C2×C127—412no—
234C2×C127—82no—
235C2×C127#238811yescharacteristic
236C2×C127#240413yes—
237C2×C127#240713yes—
238C2×C127#243013yes—
239C2×C127#243113yes—
240C2×C127#261511yescharacteristic
241C2×C127#284611yescharacteristic
242C23×#41C2×C12711yescharacteristic, complemented
243C2×#393C2×C12711yescharacteristic, complemented
244C2×#392C2×C12713yescomplemented
245C2×#386C2×C12713yescomplemented
246#1446C2×C12711yescharacteristic, complemented
247#1438C2×C12713yescomplemented
248C22⋊#95C2×C12713yescomplemented
249C127#1488511yescharacteristic, direct product factor, Sylow
250C22×#41—24no—
251C22×#41—212no—
252C22×#41—212no—
253C23×Q8C22×C12711yescharacteristic
254C23×D4C22×C12713yes—
255C23×D4C22×C12713yes—
256C24×C4C22×C12713yes—
257C22×C22⋊C4C22×C12713yes—
258C22×C22⋊C4C22×C12711yescharacteristic
259C22×C22⋊C4C22×C12716yes—
260C22×C22⋊C4C22×C12713yes—
261#393—24no—
262#392—212no—
263#386—212no—
264C2×#95C22×C12713yes—
265C2×#95C22×C12713yes—
266C2×#95C22×C12713yes—
267C2×#95C22×C12713yes—
268C25C23×C12711yescharacteristic
269C25C23×C12713yes—
270C2×#41—296no—
271C2×#41—296no—
272C2×#41—232no—
273C22×Q8—26no—
274C22×Q8—26no—
275C22×Q8D4×C12714yescomplemented
276C22×D4—212no—
277C22×D4—212no—
278C22×D4—26no—
279C22×D4—26no—
280C22×D4—212no—
281C22×D4—26no—
282C22×D4—26no—
283C22×D4—26no—
284C22×D4—46no—
285C22×D4D4×C127112yescomplemented
286C23×C4—212no—
287C23×C4—26no—
288C23×C4—26no—
289C23×C4—212no—
290C23×C4D4×C127112yescomplemented
291C23×C4C23×C12713yes—
292C23×C4C23×C12711yescharacteristic
293C23×C4C23×C12713yes—
294C23×C4C23×C12711yescharacteristic
295C23×C4C23×C12713yes—
296C2×C22⋊C4—26no—
297C2×C22⋊C4—212no—
298C2×C22⋊C4—26no—
299C2×C22⋊C4—26no—
300C2×C22⋊C4—26no—
301C2×C22⋊C4—26no—
302C2×C22⋊C4—212no—
303C2×C22⋊C4—46no—
304C2×C22⋊C4—46no—
305#95—26no—
306#95—26no—
307#95—26no—
308#95—26no—
309C24—26no—
310C24—23no—
311C24—23no—
312C24—26no—
313C24—23no—
314C24—23no—
315C24—23no—
316C24—26no—
317C24—23no—
318C24—21no—
319C24—23no—
320C24—23no—
321C24—23no—
322C24—23no—
323C24—21no—
324C24C2×D4×C127112yes—
325C24C2×D4×C12716yes—
326C24#41×C12712yescomplemented
327C24C24×C12711yescharacteristic
328#41—2256no—
329C2×Q8—28no—
330C2×Q8—224no—
331C2×Q8—224no—
332C2×D4—224no—
333C2×D4—224no—
334C2×D4—248no—
335C2×D4—248no—
336C2×D4—224no—
337C2×D4—224no—
338C2×D4—424no—
339C2×D4—424no—
340C2×D4—412no—
341C2×D4—412no—
342C22×C4—248no—
343C22×C4—248no—
344C22×C4—224no—
345C22×C4—23no—
346C22×C4—21no—
347C22×C4—26no—
348C22×C4—26no—
349C22×C4—224no—
350C22×C4—224no—
351C22×C4—23no—
352C22×C4—26no—
353C22×C4—23no—
354C22×C4—26no—
355C22×C4—23no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—23no—
359C22×C4—26no—
360C22×C4—23no—
361C22×C4—26no—
362C22×C4—21no—
363C22×C4C2×D4×C12716yes—
364C22×C4C2×D4×C12716yes—
365C22×C4C2×D4×C12716yes—
366C22×C4C2×D4×C12716yes—
367C22×C4#41×C12716yescomplemented
368C22⋊C4—424no—
369C22⋊C4—412no—
370C22⋊C4—44no—
371C22⋊C4—412no—
372C23—26no—
373C23—212no—
374C23—212no—
375C23—26no—
376C23—26no—
377C23—26no—
378C23—26no—
379C23—26no—
380C23—212no—
381C23—212no—
382C23—212no—
383C23—212no—
384C23—26no—
385C23—26no—
386C23—22no—
387C23—46no—
388C23—46no—
389C23—46no—
390C23—46no—
391C23—43no—
392C23—46no—
393C23—43no—
394C23—43no—
395C23—46no—
396C23—43no—
397C23—41no—
398C23—43no—
399C23—43no—
400C23—43no—
401C23—43no—
402C23—43no—
403C23—46no—
404C23—43no—
405C23—43no—
406C23#120×C127112yes—
407C23C22×D4×C12713yes—
408C23C22×D4×C12713yes—
409C23C22×D4×C12711yescharacteristic
410C23C2×#41×C12711yescharacteristic
411C23C2×#41×C12713yes—
412C23H(5,𝔽2)×C12713yes—
413C232−1+4×C12711yescharacteristic
414Q8—232no—
415D4—296no—
416D4—448no—
417C2×C4—296no—
418C2×C4—26no—
419C2×C4—26no—
420C2×C4—26no—
421C2×C4—212no—
422C2×C4—26no—
423C2×C4—212no—
424C2×C4—26no—
425C2×C4—212no—
426C2×C4—43no—
427C2×C4—43no—
428C2×C4—46no—
429C2×C4—41no—
430C2×C4—43no—
431C2×C4—43no—
432C2×C4—43no—
433C2×C4—43no—
434C2×C4—43no—
435C2×C4—46no—
436C2×C4—46no—
437C2×C4#120×C12714yes—
438C2×C4#120×C127112yes—
439C22—212no—
440C22—212no—
441C22—212no—
442C22—46no—
443C22—46no—
444C22—412no—
445C22—46no—
446C22—412no—
447C22—412no—
448C22—46no—
449C22—412no—
450C22—42no—
451C22—46no—
452C22—46no—
453C22—81no—
454C22—81no—
455C22—83no—
456C22—83no—
457C22C2×#120×C12711yescharacteristic
458C22C2×#120×C12713yes—
459C22C2×#120×C12713yes—
460C22#525×C12713yes—
461C22#532×C12713yes—
462C22#535×C12713yes—
463C22#537×C12713yes—
464C22#546×C12713yes—
465C22#546×C12716yes—
466C22#547×C12713yes—
467C22#548×C12711yescharacteristic
468C22#550×C12713yes—
469C4—24no—
470C4—212no—
471C4—46no—
472C4—81no—
473C4—83no—
474C2—412no—
475C2—82no—
476C2C127×#238811yescharacteristic
477C2C127×#240413yes—
478C2C127×#240713yes—
479C2C127×#243013yes—
480C2C127×#243113yes—
481C2C127×#261511yescharacteristic
482C2C127×#284611yescharacteristic
subgroupG/H
center, socleC24×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
1. C23×#41×C127C2semidirect
2. C2×#393×C127C2semidirect
3. C2×#392×C127C2semidirect
4. C2×#386×C127C2semidirect
5. C127×#1446C2semidirect
6. C127×#1438C2semidirect
7. C127×C22⋊#95C2semidirect
33. C22×Q8×C127D4semidirect
43. C22×D4×C127D4semidirect
48. C23×C4×C127D4semidirect
84. C24×C127#41semidirect
125. C22×C4×C127#41semidirect
232. #14885C127direct (N × H)
242. C23×#41C2×C127semidirect
243. C2×#393C2×C127semidirect
244. C2×#392C2×C127semidirect
245. C2×#386C2×C127semidirect
246. #1446C2×C127semidirect
247. #1438C2×C127semidirect
248. C22⋊#95C2×C127semidirect
249. C127#14885direct (N × H)
275. C22×Q8D4×C127semidirect
285. C22×D4D4×C127semidirect
290. C23×C4D4×C127semidirect
326. C24#41×C127semidirect
367. C22×C4#41×C127semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C2×#393×C1272
C2×#392×C1272
C2×#386×C1272
C127×#14462
C127×#14382
C127×C22⋊#952
#14885127