Finite Group Databasebeta

← C127×#14127 #44577800

Subgroups — C127×#14127 #44577800

Order 32512.

4046 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C4×C127C212yesmaximal
2C2×#385×C127C214yesmaximal, complemented
3C2×#379×C127C211yesmaximal, characteristic
4C127×#1439C214yesmaximal, complemented
5C127×C22⋊#95C214yesmaximal, complemented
6C26×C127C2211yescharacteristic
7C24×C4×C127C2212yes—
8C24×C4×C127C2212yes—
9C22×C4⋊C4×C127C2214yescomplemented
10C22×C22⋊C4×C127—28no—
11C22×C22⋊C4×C127—216no—
12C22×C22⋊C4×C127C2214yes—
13C22×C22⋊C4×C127C2214yes—
14C22×C22⋊C4×C127C2214yes—
15C22×C22⋊C4×C127C2212yes—
16#385×C127—216no—
17#379×C127C418yescomplemented
18C2×#95×C127C2218yescomplemented
19C2×#95×C127C2214yescomplemented
20C25×C127—28no—
21C25×C127—28no—
22C25×C127—42no—
23C25×C127—42no—
24C25×C127C2×C414yes—
25C25×C127D414yes—
26C25×C127Q814yes—
27C25×C127C2311yescharacteristic
28C25×C127C2312yes—
29C23×C4×C127—24no—
30C23×C4×C127—24no—
31C23×C4×C127—28no—
32C23×C4×C127—28no—
33C23×C4×C127—28no—
34C23×C4×C127—28no—
35C23×C4×C127—44no—
36C23×C4×C127—44no—
37C23×C4×C127C2314yes—
38C23×C4×C127C2314yes—
39C23×C4×C127C2314yes—
40C2×C4⋊C4×C127—216no—
41C2×C4⋊C4×C127—44no—
42C2×C22⋊C4×C127—264no—
43C2×C22⋊C4×C127—28no—
44C2×C22⋊C4×C127—28no—
45C2×C22⋊C4×C127—28no—
46C2×C22⋊C4×C127—28no—
47C2×C22⋊C4×C127—28no—
48C2×C22⋊C4×C127—28no—
49C2×C22⋊C4×C127—28no—
50C2×C22⋊C4×C127—42no—
51C2×C22⋊C4×C127—42no—
52C2×C22⋊C4×C127C2×C4116yescomplemented
53C2×C22⋊C4×C127C2×C418yescomplemented
54C2×C22⋊C4×C127D4116yes—
55#95×C127—216no—
56#95×C127—44no—
57C24×C127—24no—
58C24×C127—24no—
59C24×C127—24no—
60C24×C127—24no—
61C24×C127—232no—
62C24×C127—216no—
63C24×C127—232no—
64C24×C127—216no—
65C24×C127—24no—
66C24×C127—24no—
67C24×C127—22no—
68C24×C127—22no—
69C24×C127—48no—
70C24×C127—44no—
71C24×C127—44no—
72C24×C127—48no—
73C24×C127—416no—
74C24×C127—432no—
75C24×C127—416no—
76C24×C127C4⋊C4116yescomplemented
77C24×C127C22×C412yes—
78C24×C127C22×C414yes—
79C24×C127C2×D412yes—
80C24×C127C2×D414yes—
81C24×C127C2×D418yes—
82C24×C127C2×Q812yes—
83C24×C127C2×Q814yes—
84C24×C127#4118yes—
85C24×C127C2411yescharacteristic
86C22×C4×C127—216no—
87C22×C4×C127—216no—
88C22×C4×C127—28no—
89C22×C4×C127—28no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—28no—
95C22×C4×C127—28no—
96C22×C4×C127—24no—
97C22×C4×C127—28no—
98C22×C4×C127—24no—
99C22×C4×C127—28no—
100C22×C4×C127—432no—
101C22×C4×C127—432no—
102C22×C4×C127—48no—
103C22×C4×C127—48no—
104C22×C4×C127—48no—
105C22×C4×C127—48no—
106C22×C4×C127C22×C418yes—
107C22×C4×C127C2×D418yes—
108C22×C4×C127C2×D418yes—
109C4⋊C4×C127—416no—
110C22⋊C4×C127—232no—
111C22⋊C4×C127—232no—
112C22⋊C4×C127—232no—
113C22⋊C4×C127—48no—
114C23×C127—28no—
115C23×C127—28no—
116C23×C127—24no—
117C23×C127—28no—
118C23×C127—24no—
119C23×C127—24no—
120C23×C127—24no—
121C23×C127—28no—
122C23×C127—216no—
123C23×C127—216no—
124C23×C127—216no—
125C23×C127—216no—
126C23×C127—28no—
127C23×C127—24no—
128C23×C127—24no—
129C23×C127—28no—
130C23×C127—24no—
131C23×C127—22no—
132C23×C127—22no—
133C23×C127—24no—
134C23×C127—264no—
135C23×C127—416no—
136C23×C127—464no—
137C23×C127—416no—
138C23×C127—432no—
139C23×C127—464no—
140C23×C127—416no—
141C23×C127—44no—
142C23×C127—48no—
143C23×C127—44no—
144C23×C127C2×C4⋊C414yes—
145C23×C127C2×C4⋊C418yes—
146C23×C127C4×D4116yes—
147C23×C127#12018yes—
148C23×C127C22⋊Q8116yes—
149C23×C127#12318yes—
150C23×C127C23×C411yescharacteristic
151C23×C127C22×D411yescharacteristic
152C23×C127C22×D414yes—
153C23×C127C22×Q811yescharacteristic
154C23×C127C2×#4114yes—
155C23×C127H(5,𝔽2)12yes—
156C23×C127H(5,𝔽2)12yes—
157C2×C4×C127—216no—
158C2×C4×C127—28no—
159C2×C4×C127—28no—
160C2×C4×C127—28no—
161C2×C4×C127—216no—
162C2×C4×C127—216no—
163C2×C4×C127—432no—
164C2×C4×C127—432no—
165C2×C4×C127—44no—
166C2×C4×C127—44no—
167C2×C4×C127—44no—
168C2×C4×C127—44no—
169C2×C4×C127—44no—
170C2×C4×C127—48no—
171C2×C4×C127—48no—
172C2×C4×C127—48no—
173C2×C4×C127—44no—
174C22×C127—28no—
175C22×C127—232no—
176C22×C127—28no—
177C22×C127—28no—
178C22×C127—28no—
179C22×C127—232no—
180C22×C127—28no—
181C22×C127—28no—
182C22×C127—24no—
183C22×C127—28no—
184C22×C127—464no—
185C22×C127—48no—
186C22×C127—42no—
187C22×C127—44no—
188C22×C127—48no—
189C22×C127—42no—
190C22×C127#385116yescomplemented
191C22×C127C22×C4⋊C411yescharacteristic
192C22×C127C2×C4×D414yes—
193C22×C127#51812yes—
194C22×C127C2×#12012yes—
195C22×C127C2×C22⋊Q814yes—
196C22×C127C2×#12312yes—
197C22×C127#53411yescharacteristic
198C22×C127#53411yescharacteristic
199C22×C127#54312yes—
200C22×C127D4214yes—
201C22×C127#54618yes—
202C22×C127#55114yes—
203C4×C127—48no—
204C4×C127—48no—
205C4×C127—48no—
206#14127C12711yesmaximal, characteristic, direct product factor, Sylow
207C2×C127—216no—
208C2×C127—44no—
209C2×C127C2×#38512yes—
210C2×C127#231511yescharacteristic
211C2×C127#235814yes—
212C2×C127#240311yescharacteristic
213C2×C127#240512yes—
214C2×C127#243714yes—
215C2×C127#248311yescharacteristic
216C23×C22⋊C4C2×C12712yes—
217C2×#385C2×C12714yescomplemented
218C2×#379C2×C12711yescharacteristic
219#1439C2×C12714yescomplemented
220C22⋊#95C2×C12714yescomplemented
221C127#1412711yescharacteristic, direct product factor, Sylow
222C26C22×C12711yescharacteristic
223C24×C4C22×C12712yes—
224C24×C4C22×C12712yes—
225C22×C4⋊C4C22×C12714yescomplemented
226C22×C22⋊C4—28no—
227C22×C22⋊C4—216no—
228C22×C22⋊C4C22×C12714yes—
229C22×C22⋊C4C22×C12714yes—
230C22×C22⋊C4C22×C12714yes—
231C22×C22⋊C4C22×C12712yes—
232#385—216no—
233#379C4×C12718yescomplemented
234C2×#95C22×C12718yescomplemented
235C2×#95C22×C12714yescomplemented
236C25—28no—
237C25—28no—
238C25—42no—
239C25—42no—
240C25C2×C4×C12714yes—
241C25D4×C12714yes—
242C25Q8×C12714yes—
243C25C23×C12711yescharacteristic
244C25C23×C12712yes—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—28no—
248C23×C4—28no—
249C23×C4—28no—
250C23×C4—28no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C23×C12714yes—
254C23×C4C23×C12714yes—
255C23×C4C23×C12714yes—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—44no—
258C2×C22⋊C4—264no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4C2×C4×C127116yescomplemented
269C2×C22⋊C4C2×C4×C12718yescomplemented
270C2×C22⋊C4D4×C127116yes—
271#95—216no—
272#95—44no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—232no—
278C24—216no—
279C24—232no—
280C24—216no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—48no—
286C24—44no—
287C24—44no—
288C24—48no—
289C24—416no—
290C24—432no—
291C24—416no—
292C24C4⋊C4×C127116yescomplemented
293C24C22×C4×C12712yes—
294C24C22×C4×C12714yes—
295C24C2×D4×C12712yes—
296C24C2×D4×C12714yes—
297C24C2×D4×C12718yes—
298C24C2×Q8×C12712yes—
299C24C2×Q8×C12714yes—
300C24#41×C12718yes—
301C24C24×C12711yescharacteristic
302C22×C4—216no—
303C22×C4—216no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—432no—
317C22×C4—432no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4C22×C4×C12718yes—
323C22×C4C2×D4×C12718yes—
324C22×C4C2×D4×C12718yes—
325C4⋊C4—416no—
326C22⋊C4—232no—
327C22⋊C4—232no—
328C22⋊C4—232no—
329C22⋊C4—48no—
330C23—28no—
331C23—28no—
332C23—24no—
333C23—28no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—28no—
338C23—216no—
339C23—216no—
340C23—216no—
341C23—216no—
342C23—28no—
343C23—24no—
344C23—24no—
345C23—28no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—264no—
351C23—416no—
352C23—464no—
353C23—416no—
354C23—432no—
355C23—464no—
356C23—416no—
357C23—44no—
358C23—48no—
359C23—44no—
360C23C2×C4⋊C4×C12714yes—
361C23C2×C4⋊C4×C12718yes—
362C23C4×D4×C127116yes—
363C23#120×C12718yes—
364C23C22⋊Q8×C127116yes—
365C23#123×C12718yes—
366C23C23×C4×C12711yescharacteristic
367C23C22×D4×C12711yescharacteristic
368C23C22×D4×C12714yes—
369C23C22×Q8×C12711yescharacteristic
370C23C2×#41×C12714yes—
371C23H(5,𝔽2)×C12712yes—
372C23H(5,𝔽2)×C12712yes—
373C2×C4—216no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—216no—
378C2×C4—216no—
379C2×C4—432no—
380C2×C4—432no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C22—28no—
391C22—232no—
392C22—28no—
393C22—28no—
394C22—28no—
395C22—232no—
396C22—28no—
397C22—28no—
398C22—24no—
399C22—28no—
400C22—464no—
401C22—48no—
402C22—42no—
403C22—44no—
404C22—48no—
405C22—42no—
406C22#385×C127116yescomplemented
407C22C22×C4⋊C4×C12711yescharacteristic
408C22C2×C4×D4×C12714yes—
409C22#518×C12712yes—
410C22C2×#120×C12712yes—
411C22C2×C22⋊Q8×C12714yes—
412C22C2×#123×C12712yes—
413C22#534×C12711yescharacteristic
414C22#534×C12711yescharacteristic
415C22#543×C12712yes—
416C22D42×C12714yes—
417C22#546×C12718yes—
418C22#551×C12714yes—
419C4—48no—
420C4—48no—
421C4—48no—
422C2—216no—
423C2—44no—
424C2C2×#385×C12712yes—
425C2C127×#231511yescharacteristic
426C2C127×#235814yes—
427C2C127×#240311yescharacteristic
428C2C127×#240512yes—
429C2C127×#243714yes—
430C2C127×#248311yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC26×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#385×C127C2semidirect
4. C127×#1439C2semidirect
5. C127×C22⋊#95C2semidirect
9. C22×C4⋊C4×C127C22semidirect
17. #379×C127C4semidirect
18. C2×#95×C127C22semidirect
19. C2×#95×C127C22semidirect
52. C2×C22⋊C4×C127C2×C4semidirect
53. C2×C22⋊C4×C127C2×C4semidirect
76. C24×C127C4⋊C4semidirect
190. C22×C127#385semidirect
206. #14127C127direct (N × H)
217. C2×#385C2×C127semidirect
219. #1439C2×C127semidirect
220. C22⋊#95C2×C127semidirect
221. C127#14127direct (N × H)
225. C22×C4⋊C4C22×C127semidirect
233. #379C4×C127semidirect
234. C2×#95C22×C127semidirect
235. C2×#95C22×C127semidirect
268. C2×C22⋊C4C2×C4×C127semidirect
269. C2×C22⋊C4C2×C4×C127semidirect
292. C24C4⋊C4×C127semidirect
406. C22#385×C127semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C1272
C2×#385×C1272
C2×#379×C1272
C127×#14392
C127×C22⋊#952
#14127127