Finite Group Databasebeta

← C127×#10729 #44574402

Subgroups — C127×#10729 #44574402

Order 32512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C127C211yesmaximal, characteristic
2C2×#407×C127C211yesmaximal, characteristic
3C127×#2280C211yesmaximal, characteristic
4C2×#344×C127C214yesmaximal, complemented
5C24×C4×C127C2211yescharacteristic
6C22×M16×C127—22no—
7C22×M16×C127—22no—
8C22×M16×C127—28no—
9C22×M16×C127C412yescomplemented
10C22×M16×C127C2212yes—
11C22×M16×C127C2212yes—
12C23×C8×C127C412yescomplemented
13C2×#117×C127C2212yes—
14#407×C127—22no—
15#407×C127C412yescomplemented
16#407×C127C412yescomplemented
17C2×C22⋊C8×C127—22no—
18#381×C127C412yes—
19#380×C127—22no—
20#379×C127C412yes—
21C4×C22⋊C4×C127—22no—
22#344×C127—28no—
23C25×C127C2×C411yescharacteristic
24C23×C4×C127—24no—
25C23×C4×C127—24no—
26C23×C4×C127C2×C411yescharacteristic
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127C2×C411yescharacteristic
30C23×C4×C127C2×C411yescharacteristic
31C23×C4×C127D412yes—
32C23×C4×C127D412yes—
33C23×C4×C127D412yes—
34C23×C4×C127Q812yes—
35C23×C4×C127C2311yescharacteristic
36C2×M16×C127—216no—
37C2×M16×C127—216no—
38C2×M16×C127—24no—
39C2×M16×C127—24no—
40C2×M16×C127—22no—
41C2×M16×C127—22no—
42C2×M16×C127—22no—
43C2×M16×C127—28no—
44C2×M16×C127—24no—
45C2×M16×C127—24no—
46C2×M16×C127C2×C414yes—
47C2×M16×C127D418yes—
48C22×C8×C127—24no—
49C22×C8×C127—28no—
50C22×C8×C127—22no—
51C22×C8×C127C2×C414yes—
52#117×C127—24no—
53#117×C127—24no—
54C2×C4⋊C4×C127—22no—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127C2×C414yes—
57C2×C42×C127—22no—
58C22⋊C8×C127—24no—
59C22⋊C8×C127—24no—
60#95×C127—24no—
61C24×C127—24no—
62C24×C127—44no—
63C24×C127C4211yescharacteristic
64C24×C127C4211yescharacteristic
65C24×C127C22⋊C412yes—
66C24×C127C4⋊C412yes—
67C24×C127C22×C411yescharacteristic
68C22×C4×C127—24no—
69C22×C4×C127—22no—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—28no—
78C22×C4×C127—28no—
79C22×C4×C127—24no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127C4211yescharacteristic
83C22×C4×C127C4211yescharacteristic
84C22×C4×C127C22⋊C412yes—
85C22×C4×C127C22⋊C412yes—
86C22×C4×C127C22⋊C412yes—
87C22×C4×C127C22⋊C412yes—
88C22×C4×C127C22⋊C412yes—
89C22×C4×C127C4⋊C412yes—
90C22×C4×C127C4⋊C412yes—
91C22×C4×C127C4⋊C412yes—
92C22×C4×C127C4⋊C412yes—
93C22×C4×C127C4⋊C412yes—
94C22×C4×C127C22×C411yescharacteristic
95C22×C4×C127C22×C411yescharacteristic
96C22×C4×C127C2×D411yescharacteristic
97C22×C4×C127C2×D411yescharacteristic
98C22×C4×C127C2×D414yes—
99C22×C4×C127C2×D411yescharacteristic
100C22×C4×C127C2×Q811yescharacteristic
101C22×C4×C127#4114yes—
102M16×C127—216no—
103M16×C127—28no—
104M16×C127—28no—
105M16×C127—28no—
106M16×C127—44no—
107C2×C8×C127—216no—
108C2×C8×C127—24no—
109C2×C8×C127—24no—
110C2×C8×C127—24no—
111C2×C8×C127—42no—
112C2×C8×C127—42no—
113C2×C8×C127C22⋊C418yescomplemented
114C4⋊C4×C127—44no—
115C22⋊C4×C127—28no—
116C22⋊C4×C127—48no—
117C42×C127—42no—
118C42×C127—42no—
119C23×C127—24no—
120C23×C127—22no—
121C23×C127—24no—
122C23×C127—24no—
123C23×C127—22no—
124C23×C127—24no—
125C23×C127—24no—
126C23×C127—42no—
127C23×C127—42no—
128C23×C127—42no—
129C23×C127—42no—
130C23×C127—416no—
131C23×C127#9512yes—
132C23×C127#9512yes—
133C23×C127C2×C4211yescharacteristic
134C23×C127C2×C22⋊C411yescharacteristic
135C23×C127C2×C4⋊C411yescharacteristic
136C23×C127C4×D414yes—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—24no—
140C2×C4×C127—24no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—24no—
144C2×C4×C127—24no—
145C2×C4×C127—22no—
146C2×C4×C127—22no—
147C2×C4×C127—24no—
148C2×C4×C127—24no—
149C2×C4×C127—24no—
150C2×C4×C127—24no—
151C2×C4×C127—48no—
152C2×C4×C127—44no—
153C2×C4×C127#9512yes—
154C2×C4×C127#9512yes—
155C2×C4×C127C2×C22⋊C412yes—
156C2×C4×C127C2×C22⋊C411yescharacteristic
157C2×C4×C127C2×C22⋊C411yescharacteristic
158C2×C4×C127C2×C4⋊C411yescharacteristic
159C2×C4×C127C2×C4⋊C411yescharacteristic
160C2×C4×C127#11712yes—
161C2×C4×C127C4×D414yes—
162C2×C4×C127#12012yes—
163C2×C4×C127#12012yes—
164C2×C4×C127#12114yes—
165C2×C4×C127C22⋊Q814yes—
166C2×C4×C127#12312yes—
167C2×C4×C127#12312yes—
168C8×C127—28no—
169C8×C127—44no—
170C22×C127—28no—
171C22×C127—22no—
172C22×C127—22no—
173C22×C127—22no—
174C22×C127—22no—
175C22×C127—22no—
176C22×C127—22no—
177C22×C127—22no—
178C22×C127—22no—
179C22×C127—44no—
180C22×C127—48no—
181C22×C127—48no—
182C22×C127—44no—
183C22×C127#34414yescomplemented
184C22×C127C2×#9511yescharacteristic
185C22×C127C4×C22⋊C412yes—
186C22×C127#38512yes—
187C22×C127#38612yes—
188C4×C127—22no—
189C4×C127—22no—
190C4×C127—84no—
191C4×C127#37911yescharacteristic
192C4×C127#38012yes—
193C4×C127#38111yescharacteristic
194C4×C127#38512yes—
195C4×C127#38612yes—
196#10729C12711yesmaximal, characteristic, direct product factor, Sylow
197C2×C127—22no—
198C2×C127—24no—
199C2×C127—22no—
200C2×C127—22no—
201C2×C127—42no—
202C2×C127C22⋊#9511yescharacteristic
203C2×C127C2×#34412yes—
204C23×M16C2×C12711yescharacteristic
205C2×#407C2×C12711yescharacteristic
206#2280C2×C12711yescharacteristic
207C2×#344C2×C12714yescomplemented
208C127#1072911yescharacteristic, direct product factor, Sylow
209C24×C4C22×C12711yescharacteristic
210C22×M16—22no—
211C22×M16—22no—
212C22×M16—28no—
213C22×M16C4×C12712yescomplemented
214C22×M16C22×C12712yes—
215C22×M16C22×C12712yes—
216C23×C8C4×C12712yescomplemented
217C2×#117C22×C12712yes—
218#407—22no—
219#407C4×C12712yescomplemented
220#407C4×C12712yescomplemented
221C2×C22⋊C8—22no—
222#381C4×C12712yes—
223#380—22no—
224#379C4×C12712yes—
225C4×C22⋊C4—22no—
226#344—28no—
227C25C2×C4×C12711yescharacteristic
228C23×C4—24no—
229C23×C4—24no—
230C23×C4C2×C4×C12711yescharacteristic
231C23×C4C2×C4×C12711yescharacteristic
232C23×C4C2×C4×C12711yescharacteristic
233C23×C4C2×C4×C12711yescharacteristic
234C23×C4C2×C4×C12711yescharacteristic
235C23×C4D4×C12712yes—
236C23×C4D4×C12712yes—
237C23×C4D4×C12712yes—
238C23×C4Q8×C12712yes—
239C23×C4C23×C12711yescharacteristic
240C2×M16—216no—
241C2×M16—216no—
242C2×M16—24no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—24no—
246C2×M16—22no—
247C2×M16—28no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16C2×C4×C12714yes—
251C2×M16D4×C12718yes—
252C22×C8—24no—
253C22×C8—28no—
254C22×C8—22no—
255C22×C8C2×C4×C12714yes—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4C2×C4×C12714yes—
261C2×C42—22no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264#95—24no—
265C24—24no—
266C24—44no—
267C24C42×C12711yescharacteristic
268C24C42×C12711yescharacteristic
269C24C22⋊C4×C12712yes—
270C24C4⋊C4×C12712yes—
271C24C22×C4×C12711yescharacteristic
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4C42×C12711yescharacteristic
287C22×C4C42×C12711yescharacteristic
288C22×C4C22⋊C4×C12712yes—
289C22×C4C22⋊C4×C12712yes—
290C22×C4C22⋊C4×C12712yes—
291C22×C4C22⋊C4×C12712yes—
292C22×C4C22⋊C4×C12712yes—
293C22×C4C4⋊C4×C12712yes—
294C22×C4C4⋊C4×C12712yes—
295C22×C4C4⋊C4×C12712yes—
296C22×C4C4⋊C4×C12712yes—
297C22×C4C4⋊C4×C12712yes—
298C22×C4C22×C4×C12711yescharacteristic
299C22×C4C22×C4×C12711yescharacteristic
300C22×C4C2×D4×C12711yescharacteristic
301C22×C4C2×D4×C12711yescharacteristic
302C22×C4C2×D4×C12711yescharacteristic
303C22×C4C2×D4×C12714yes—
304C22×C4C2×Q8×C12711yescharacteristic
305C22×C4#41×C12714yes—
306M16—216no—
307M16—28no—
308M16—28no—
309M16—28no—
310M16—44no—
311C2×C8—216no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8C22⋊C4×C12718yescomplemented
318C4⋊C4—44no—
319C22⋊C4—28no—
320C22⋊C4—48no—
321C42—42no—
322C42—42no—
323C23—24no—
324C23—22no—
325C23—24no—
326C23—24no—
327C23—22no—
328C23—24no—
329C23—24no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—416no—
335C23#95×C12712yes—
336C23#95×C12712yes—
337C23C2×C42×C12711yescharacteristic
338C23C2×C22⋊C4×C12711yescharacteristic
339C23C2×C4⋊C4×C12711yescharacteristic
340C23C4×D4×C12714yes—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4#95×C12712yes—
358C2×C4#95×C12712yes—
359C2×C4C2×C22⋊C4×C12712yes—
360C2×C4C2×C22⋊C4×C12711yescharacteristic
361C2×C4C2×C22⋊C4×C12711yescharacteristic
362C2×C4C2×C4⋊C4×C12711yescharacteristic
363C2×C4C2×C4⋊C4×C12711yescharacteristic
364C2×C4#117×C12712yes—
365C2×C4C4×D4×C12714yes—
366C2×C4#120×C12712yes—
367C2×C4#120×C12712yes—
368C2×C4#121×C12714yes—
369C2×C4C22⋊Q8×C12714yes—
370C2×C4#123×C12712yes—
371C2×C4#123×C12712yes—
372C8—28no—
373C8—44no—
374C22—28no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—44no—
384C22—48no—
385C22—48no—
386C22—44no—
387C22#344×C12714yescomplemented
388C22C2×#95×C12711yescharacteristic
389C22C4×C22⋊C4×C12712yes—
390C22#385×C12712yes—
391C22#386×C12712yes—
392C4—22no—
393C4—22no—
394C4—84no—
395C4#379×C12711yescharacteristic
396C4#380×C12712yes—
397C4#381×C12711yescharacteristic
398C4#385×C12712yes—
399C4#386×C12712yes—
400C2—22no—
401C2—24no—
402C2—22no—
403C2—22no—
404C2—42no—
405C2C127×C22⋊#9511yescharacteristic
406C2C2×#344×C12712yes—
subgroupG/H
centerC2×C4×C127C2×C22⋊C4
commutatorC23C2×C42×C127
Frattini, cosocleC23×C4C23×C127
socleC22×C127C2×#95
prime radicalC25×C127C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#344×C127C2semidirect
9. C22×M16×C127C4semidirect
12. C23×C8×C127C4semidirect
15. #407×C127C4semidirect
16. #407×C127C4semidirect
113. C2×C8×C127C22⋊C4semidirect
183. C22×C127#344semidirect
196. #10729C127direct (N × H)
207. C2×#344C2×C127semidirect
208. C127#10729direct (N × H)
213. C22×M16C4×C127semidirect
216. C23×C8C4×C127semidirect
219. #407C4×C127semidirect
220. #407C4×C127semidirect
317. C2×C8C22⋊C4×C127semidirect
387. C22#344×C127semidirect

Maximal subgroups

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

subgroupindex
C23×M16×C1272
C2×#407×C1272
C127×#22802
C2×#344×C1272
#10729127