Finite Group Databasebeta

← C127×#17064 #44580737

Subgroups — C127×#17064 #44580737

Order 32512.

1466 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
1C2×#400×C127C212yesmaximal, complemented
2C2×#399×C127C212yesmaximal
3C2×#388×C127C211yesmaximal, characteristic
4C2×#384×C127C211yesmaximal, characteristic, complemented
5C127×#1444C212yesmaximal, complemented
6C127×#1444C212yesmaximal, complemented
7C127×#1439C212yesmaximal
8C127×#1439C212yesmaximal
9C127×#1434C211yesmaximal, characteristic, complemented
10C22×C4⋊C4×C127C2212yes—
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2214yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C22⋊C4×C127C2214yes—
19C22×C42×C127C2211yescharacteristic
20#400×C127—28no—
21#399×C127—28no—
22#388×C127—24no—
23#384×C127—24no—
24C2×#95×C127C2214yes—
25C2×#95×C127C2214yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2211yescharacteristic
30C2×#95×C127C2212yes—
31C25×C127C2311yescharacteristic
32C23×C4×C127C2311yescharacteristic
33C23×C4×C127C2314yes—
34C23×C4×C127C2311yescharacteristic
35C23×C4×C127C2314yes—
36C23×C4×C127C2311yescharacteristic
37C23×C4×C127C2311yescharacteristic
38C23×C4×C127C2311yescharacteristic
39C23×C4×C127C2311yescharacteristic
40C2×C4⋊C4×C127—24no—
41C2×C4⋊C4×C127—24no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—28no—
45C2×C4⋊C4×C127—24no—
46C2×C4⋊C4×C127—24no—
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—44no—
49C2×C4⋊C4×C127—44no—
50C2×C4⋊C4×C127—44no—
51C2×C4⋊C4×C127—44no—
52C2×C22⋊C4×C127—22no—
53C2×C22⋊C4×C127—22no—
54C2×C22⋊C4×C127—22no—
55C2×C22⋊C4×C127—22no—
56C2×C22⋊C4×C127—28no—
57C2×C22⋊C4×C127—24no—
58C2×C22⋊C4×C127—24no—
59C2×C22⋊C4×C127—44no—
60C2×C22⋊C4×C127—44no—
61C2×C22⋊C4×C127Q814yes—
62C2×C42×C127—24no—
63C2×C42×C127D414yes—
64#95×C127—22no—
65#95×C127—22no—
66#95×C127—28no—
67#95×C127—28no—
68#95×C127—24no—
69#95×C127—24no—
70#95×C127D418yes—
71C24×C127—21no—
72C24×C127—22no—
73C24×C127—21no—
74C24×C127—22no—
75C24×C127—22no—
76C24×C127—22no—
77C24×C127—22no—
78C24×C127—22no—
79C24×C127C2×Q812yes—
80C24×C127C2411yescharacteristic
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—21no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—21no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—22no—
97C22×C4×C127—21no—
98C22×C4×C127—24no—
99C22×C4×C127—22no—
100C22×C4×C127—21no—
101C22×C4×C127—22no—
102C22×C4×C127—24no—
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×C127C2×D418yes—
113C22×C4×C127C2×D412yes—
114C22×C4×C127C2×D412yes—
115C22×C4×C127C2×D414yes—
116C22×C4×C127C2×D412yes—
117C22×C4×C127C2×Q812yes—
118C22×C4×C127C2×Q812yes—
119C22×C4×C127#4112yes—
120C22×C4×C127#4114yes—
121C4⋊C4×C127—48no—
122C4⋊C4×C127—48no—
123C4⋊C4×C127—416no—
124C4⋊C4×C127—44no—
125C4⋊C4×C127—48no—
126C22⋊C4×C127—28no—
127C22⋊C4×C127—416no—
128C22⋊C4×C127—44no—
129C22⋊C4×C127—44no—
130C42×C127—28no—
131C23×C127—22no—
132C23×C127—24no—
133C23×C127—24no—
134C23×C127—24no—
135C23×C127—41no—
136C23×C127—41no—
137C23×C127—44no—
138C23×C127—41no—
139C23×C127—41no—
140C23×C127—41no—
141C23×C127—41no—
142C23×C127—44no—
143C23×C127—42no—
144C23×C127—44no—
145C23×C127—42no—
146C23×C127—42no—
147C23×C127—44no—
148C23×C127C22×D411yescharacteristic
149C23×C127C22×D412yes—
150C23×C127C22×Q811yescharacteristic
151C23×C127C2×#4111yescharacteristic
152C23×C127C2×#4112yes—
153C23×C127H(5,𝔽2)12yes—
154C23×C127H(5,𝔽2)12yes—
155C23×C1272−1+412yes—
156C23×C1272−1+412yes—
157C2×C4×C127—22no—
158C2×C4×C127—22no—
159C2×C4×C127—22no—
160C2×C4×C127—22no—
161C2×C4×C127—22no—
162C2×C4×C127—22no—
163C2×C4×C127—28no—
164C2×C4×C127—28no—
165C2×C4×C127—24no—
166C2×C4×C127—24no—
167C2×C4×C127—22no—
168C2×C4×C127—28no—
169C2×C4×C127—28no—
170C2×C4×C127—28no—
171C2×C4×C127—24no—
172C2×C4×C127—41no—
173C2×C4×C127—41no—
174C2×C4×C127—44no—
175C2×C4×C127—44no—
176C2×C4×C127—42no—
177C2×C4×C127—44no—
178C2×C4×C127—44no—
179C2×C4×C127—41no—
180C2×C4×C127—41no—
181C2×C4×C127—44no—
182C2×C4×C127—41no—
183C2×C4×C127—44no—
184C2×C4×C127—44no—
185C2×C4×C127—44no—
186C2×C4×C127—44no—
187C2×C4×C127—44no—
188C2×C4×C127—41no—
189C2×C4×C127—41no—
190C2×C4×C127—44no—
191C2×C4×C127—44no—
192C2×C4×C127—41no—
193C2×C4×C127—44no—
194C2×C4×C127C22⋊Q814yes—
195C2×C4×C127#12314yes—
196C2×C4×C127#12714yes—
197C22×C127—42no—
198C22×C127—48no—
199C22×C127—42no—
200C22×C127—42no—
201C22×C127—81no—
202C22×C127—81no—
203C22×C127—81no—
204C22×C127—84no—
205C22×C127—81no—
206C22×C127C2×C22⋊Q811yescharacteristic
207C22×C127C2×#12311yescharacteristic
208C22×C127C2×#12711yescharacteristic
209C22×C127#53511yescharacteristic
210C22×C127#53511yescharacteristic
211C22×C127#53514yes—
212C22×C127#53611yescharacteristic
213C22×C127#53611yescharacteristic
214C22×C127#53914yes—
215C22×C127#54212yes—
216C22×C127#54412yes—
217C22×C127#54712yes—
218C22×C127#54714yes—
219C22×C127D4×Q812yes—
220C22×C127#55312yes—
221C22×C127#55412yes—
222C22×C127#56314yes—
223C4×C127—24no—
224C4×C127—24no—
225C4×C127—24no—
226C4×C127—48no—
227C4×C127—42no—
228C4×C127—81no—
229C4×C127—84no—
230C4×C127—81no—
231#17064C12711yesmaximal, characteristic, direct product factor, Sylow
232C2×C127—82no—
233C2×C127#248711yescharacteristic
234C2×C127#250111yescharacteristic
235C2×C127#254211yescharacteristic
236C2×C127#262311yescharacteristic
237C2×C127#262811yescharacteristic
238C2×C127#265711yescharacteristic
239C2×C127#271814yes—
240C2×C127#272014yes—
241C2×C127#282111yescharacteristic
242C2×#400C2×C12712yescomplemented
243C2×#399C2×C12712yes—
244C2×#388C2×C12711yescharacteristic
245C2×#384C2×C12711yescharacteristic, complemented
246#1444C2×C12712yescomplemented
247#1444C2×C12712yescomplemented
248#1439C2×C12712yes—
249#1439C2×C12712yes—
250#1434C2×C12711yescharacteristic, complemented
251C127#1706411yescharacteristic, direct product factor, Sylow
252C22×C4⋊C4C22×C12712yes—
253C22×C4⋊C4C22×C12711yescharacteristic
254C22×C4⋊C4C22×C12714yes—
255C22×C4⋊C4C22×C12712yes—
256C22×C4⋊C4C22×C12712yes—
257C22×C22⋊C4C22×C12711yescharacteristic
258C22×C22⋊C4C22×C12711yescharacteristic
259C22×C22⋊C4C22×C12711yescharacteristic
260C22×C22⋊C4C22×C12714yes—
261C22×C42C22×C12711yescharacteristic
262#400—28no—
263#399—28no—
264#388—24no—
265#384—24no—
266C2×#95C22×C12714yes—
267C2×#95C22×C12714yes—
268C2×#95C22×C12712yes—
269C2×#95C22×C12712yes—
270C2×#95C22×C12711yescharacteristic
271C2×#95C22×C12711yescharacteristic
272C2×#95C22×C12712yes—
273C25C23×C12711yescharacteristic
274C23×C4C23×C12711yescharacteristic
275C23×C4C23×C12714yes—
276C23×C4C23×C12711yescharacteristic
277C23×C4C23×C12714yes—
278C23×C4C23×C12711yescharacteristic
279C23×C4C23×C12711yescharacteristic
280C23×C4C23×C12711yescharacteristic
281C23×C4C23×C12711yescharacteristic
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4Q8×C12714yes—
304C2×C42—24no—
305C2×C42D4×C12714yes—
306#95—22no—
307#95—22no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95D4×C12718yes—
313C24—21no—
314C24—22no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C2×Q8×C12712yes—
322C24C24×C12711yescharacteristic
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4C2×D4×C12718yes—
355C22×C4C2×D4×C12712yes—
356C22×C4C2×D4×C12712yes—
357C22×C4C2×D4×C12714yes—
358C22×C4C2×D4×C12712yes—
359C22×C4C2×Q8×C12712yes—
360C22×C4C2×Q8×C12712yes—
361C22×C4#41×C12712yes—
362C22×C4#41×C12714yes—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—416no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C22⋊C4—28no—
369C22⋊C4—416no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C42—28no—
373C23—22no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—41no—
378C23—41no—
379C23—44no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—44no—
385C23—42no—
386C23—44no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23C22×D4×C12711yescharacteristic
391C23C22×D4×C12712yes—
392C23C22×Q8×C12711yescharacteristic
393C23C2×#41×C12711yescharacteristic
394C23C2×#41×C12712yes—
395C23H(5,𝔽2)×C12712yes—
396C23H(5,𝔽2)×C12712yes—
397C232−1+4×C12712yes—
398C232−1+4×C12712yes—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4C22⋊Q8×C12714yes—
437C2×C4#123×C12714yes—
438C2×C4#127×C12714yes—
439C22—42no—
440C22—48no—
441C22—42no—
442C22—42no—
443C22—81no—
444C22—81no—
445C22—84no—
446C22—81no—
447C22—81no—
448C22C2×C22⋊Q8×C12711yescharacteristic
449C22C2×#123×C12711yescharacteristic
450C22C2×#127×C12711yescharacteristic
451C22#535×C12711yescharacteristic
452C22#535×C12714yes—
453C22#535×C12711yescharacteristic
454C22#536×C12711yescharacteristic
455C22#536×C12711yescharacteristic
456C22#539×C12714yes—
457C22#542×C12712yes—
458C22#544×C12712yes—
459C22#547×C12712yes—
460C22#547×C12714yes—
461C22D4×Q8×C12712yes—
462C22#553×C12712yes—
463C22#554×C12712yes—
464C22#563×C12714yes—
465C4—24no—
466C4—24no—
467C4—24no—
468C4—48no—
469C4—42no—
470C4—81no—
471C4—84no—
472C4—81no—
473C2—82no—
474C2C127×#248711yescharacteristic
475C2C127×#250111yescharacteristic
476C2C127×#254211yescharacteristic
477C2C127×#262311yescharacteristic
478C2C127×#262811yescharacteristic
479C2C127×#265711yescharacteristic
480C2C127×#271814yes—
481C2C127×#272014yes—
482C2C127×#282111yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#400×C127C2semidirect
4. C2×#384×C127C2semidirect
5. C127×#1444C2semidirect
6. C127×#1444C2semidirect
9. C127×#1434C2semidirect
231. #17064C127direct (N × H)
242. C2×#400C2×C127semidirect
245. C2×#384C2×C127semidirect
246. #1444C2×C127semidirect
247. #1444C2×C127semidirect
250. #1434C2×C127semidirect
251. C127#17064direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C1272
C2×#399×C1272
C2×#388×C1272
C2×#384×C1272
C127×#14442
C127×#14392
C127×#14342
#17064127