Finite Group Databasebeta

← C127×#20724 #44584397

Subgroups — C127×#20724 #44584397

Order 32512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518×C127C211yesmaximal, characteristic
2C127×#2312C212yesmaximal, complemented
3C127×#1819C212yesmaximal, complemented
4C127×#1799C214yesmaximal, complemented
5C127×#1781C212yesmaximal
6C4×#99×C127C214yesmaximal, complemented
7C23×D4×C127C2211yescharacteristic
8#518×C127—28no—
9C2×C4×D4×C127C412yescomplemented
10C2×C4×D4×C127C412yescomplemented
11C2×C4×D4×C127C2212yes—
12C2×C4×D4×C127C2212yes—
13C2×#117×C127C412yescomplemented
14C22×C22⋊C4×C127C412yescomplemented
15C22×C22⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C2211yescharacteristic
17C2×#99×C127C2218yes—
18#388×C127—28no—
19#380×C127C2214yescomplemented
20#379×C127C2214yes—
21C4×C22⋊C4×C127C2214yescomplemented
22#342×C127C2218yescomplemented
23C25×C127C2×C411yescharacteristic
24C25×C127C2311yescharacteristic
25C22×D4×C127—24no—
26C22×D4×C127—28no—
27C22×D4×C127C2×C412yes—
28C22×D4×C127C2312yes—
29C23×C4×C127C2×C411yescharacteristic
30C23×C4×C127C2×C412yes—
31C23×C4×C127C2×C412yes—
32C23×C4×C127C2×C411yescharacteristic
33C23×C4×C127C2×C411yescharacteristic
34C23×C4×C127C2311yescharacteristic
35C23×C4×C127C2311yescharacteristic
36C4×D4×C127—28no—
37C4×D4×C127—28no—
38C4×D4×C127—28no—
39C4×D4×C127—28no—
40#117×C127—28no—
41C2×C4⋊C4×C127—24no—
42C2×C4⋊C4×C127C2×C412yes—
43C2×C4⋊C4×C127C2×C412yes—
44C2×C22⋊C4×C127—24no—
45C2×C22⋊C4×C127—28no—
46C2×C22⋊C4×C127—24no—
47C2×C22⋊C4×C127—24no—
48C2×C22⋊C4×C127C2×C412yes—
49C2×C22⋊C4×C127C2×C412yes—
50C2×C22⋊C4×C127C2×C412yes—
51C2×C22⋊C4×C127C2×C412yes—
52C2×C22⋊C4×C127C2×C412yes—
53C2×C22⋊C4×C127D418yes—
54C2×C22⋊C4×C127C2318yes—
55C2×C22⋊C4×C127C2312yes—
56C2×C42×C127—24no—
57C2×C42×C127C2×C412yes—
58C2×C42×C127C2×C412yes—
59#99×C127—216no—
60#95×C127—24no—
61#95×C127—24no—
62C24×C127—22no—
63C24×C127—24no—
64C24×C127—24no—
65C24×C127—81no—
66C24×C127—81no—
67C24×C127—81no—
68C24×C127—81no—
69C24×C127C22×C411yescharacteristic
70C24×C127C22×C411yescharacteristic
71C24×C127C22×C412yes—
72C24×C127C2×D414yes—
73C24×C127C2411yescharacteristic
74C2×D4×C127—232no—
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—24no—
78C2×D4×C127—24no—
79C2×D4×C127—24no—
80C2×D4×C127#4118yes—
81C22×C4×C127—22no—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—24no—
86C22×C4×C127—24no—
87C22×C4×C127—28no—
88C22×C4×C127—81no—
89C22×C4×C127—81no—
90C22×C4×C127—82no—
91C22×C4×C127—82no—
92C22×C4×C127—81no—
93C22×C4×C127—81no—
94C22×C4×C127C22×C412yes—
95C22×C4×C127C22×C412yes—
96C22×C4×C127C22×C412yes—
97C22×C4×C127C22×C412yes—
98C22×C4×C127C22×C412yes—
99C22×C4×C127C2×D414yes—
100C22×C4×C127C2×D414yes—
101C4⋊C4×C127—24no—
102C4⋊C4×C127—24no—
103C4⋊C4×C127—84no—
104C22⋊C4×C127—24no—
105C22⋊C4×C127—24no—
106C22⋊C4×C127—24no—
107C22⋊C4×C127—24no—
108C22⋊C4×C127—216no—
109C22⋊C4×C127—84no—
110C22⋊C4×C127—84no—
111C22⋊C4×C127C22⋊C418yes—
112C22⋊C4×C127C22⋊C418yes—
113C42×C127—24no—
114C42×C127—24no—
115C42×C127—84no—
116C23×C127—28no—
117C23×C127—22no—
118C23×C127—22no—
119C23×C127—22no—
120C23×C127—22no—
121C23×C127—28no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—44no—
129C23×C127—44no—
130C23×C127—42no—
131C23×C127—42no—
132C23×C127—82no—
133C23×C127—82no—
134C23×C127—88no—
135C23×C127—88no—
136C23×C127C2×C22⋊C412yes—
137C23×C127C2×C22⋊C412yes—
138C23×C127#11714yes—
139C23×C127C23×C411yescharacteristic
140C23×C127C22×D412yes—
141C23×C127C2×#4114yes—
142D4×C127—216no—
143D4×C127—216no—
144C2×C4×C127—28no—
145C2×C4×C127—22no—
146C2×C4×C127—22no—
147C2×C4×C127—22no—
148C2×C4×C127—22no—
149C2×C4×C127—44no—
150C2×C4×C127—44no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—42no—
154C2×C4×C127—42no—
155C2×C4×C127—48no—
156C2×C4×C127—48no—
157C2×C4×C127—48no—
158C2×C4×C127—84no—
159C2×C4×C127—84no—
160C2×C4×C127—82no—
161C2×C4×C127—82no—
162C2×C4×C127C2×C22⋊C414yes—
163C2×C4×C127C2×C22⋊C414yes—
164C2×C4×C127#11714yes—
165C22×C127—22no—
166C22×C127—22no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—22no—
170C22×C127—22no—
171C22×C127—22no—
172C22×C127—22no—
173C22×C127—22no—
174C22×C127—22no—
175C22×C127—24no—
176C22×C127—44no—
177C22×C127—44no—
178C22×C127—44no—
179C22×C127—44no—
180C22×C127—44no—
181C22×C127—42no—
182C22×C127—42no—
183C22×C127—88no—
184C22×C127—88no—
185C22×C127C22×C22⋊C411yescharacteristic
186C22×C127C2×#11712yes—
187C22×C127#52514yes—
188C4×C127—24no—
189C4×C127—24no—
190C4×C127—24no—
191C4×C127—42no—
192C4×C127—42no—
193C4×C127—88no—
194#20724C12711yesmaximal, characteristic, direct product factor, Sylow
195C2×C127—24no—
196C2×C127—22no—
197C2×C127—42no—
198C2×C127—42no—
199C2×C127—44no—
200C2×C127#228011yescharacteristic
201C2×C127#288311yescharacteristic
202C2×C127#288311yescharacteristic
203C2×#518C2×C12711yescharacteristic
204#2312C2×C12712yescomplemented
205#1819C2×C12712yescomplemented
206#1799C2×C12714yescomplemented
207#1781C2×C12712yes—
208C4×#99C2×C12714yescomplemented
209C127#2072411yescharacteristic, direct product factor, Sylow
210C23×D4C22×C12711yescharacteristic
211#518—28no—
212C2×C4×D4C4×C12712yescomplemented
213C2×C4×D4C4×C12712yescomplemented
214C2×C4×D4C22×C12712yes—
215C2×C4×D4C22×C12712yes—
216C2×#117C4×C12712yescomplemented
217C22×C22⋊C4C4×C12712yescomplemented
218C22×C22⋊C4C22×C12711yescharacteristic
219C22×C22⋊C4C22×C12711yescharacteristic
220C2×#99C22×C12718yes—
221#388—28no—
222#380C22×C12714yescomplemented
223#379C22×C12714yes—
224C4×C22⋊C4C22×C12714yescomplemented
225#342C22×C12718yescomplemented
226C25C2×C4×C12711yescharacteristic
227C25C23×C12711yescharacteristic
228C22×D4—24no—
229C22×D4—28no—
230C22×D4C2×C4×C12712yes—
231C22×D4C23×C12712yes—
232C23×C4C2×C4×C12711yescharacteristic
233C23×C4C2×C4×C12712yes—
234C23×C4C2×C4×C12712yes—
235C23×C4C2×C4×C12711yescharacteristic
236C23×C4C2×C4×C12711yescharacteristic
237C23×C4C23×C12711yescharacteristic
238C23×C4C23×C12711yescharacteristic
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—28no—
243#117—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4C2×C4×C12712yes—
246C2×C4⋊C4C2×C4×C12712yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4C2×C4×C12712yes—
252C2×C22⋊C4C2×C4×C12712yes—
253C2×C22⋊C4C2×C4×C12712yes—
254C2×C22⋊C4C2×C4×C12712yes—
255C2×C22⋊C4C2×C4×C12712yes—
256C2×C22⋊C4D4×C12718yes—
257C2×C22⋊C4C23×C12718yes—
258C2×C22⋊C4C23×C12712yes—
259C2×C42—24no—
260C2×C42C2×C4×C12712yes—
261C2×C42C2×C4×C12712yes—
262#99—216no—
263#95—24no—
264#95—24no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24—81no—
269C24—81no—
270C24—81no—
271C24—81no—
272C24C22×C4×C12711yescharacteristic
273C24C22×C4×C12711yescharacteristic
274C24C22×C4×C12712yes—
275C24C2×D4×C12714yes—
276C24C24×C12711yescharacteristic
277C2×D4—232no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4#41×C12718yes—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C22×C4×C12712yes—
298C22×C4C22×C4×C12712yes—
299C22×C4C22×C4×C12712yes—
300C22×C4C22×C4×C12712yes—
301C22×C4C22×C4×C12712yes—
302C22×C4C2×D4×C12714yes—
303C22×C4C2×D4×C12714yes—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—84no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—216no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C22⋊C4C22⋊C4×C12718yes—
315C22⋊C4C22⋊C4×C12718yes—
316C42—24no—
317C42—24no—
318C42—84no—
319C23—28no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—28no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—44no—
332C23—44no—
333C23—42no—
334C23—42no—
335C23—82no—
336C23—82no—
337C23—88no—
338C23—88no—
339C23C2×C22⋊C4×C12712yes—
340C23C2×C22⋊C4×C12712yes—
341C23#117×C12714yes—
342C23C23×C4×C12711yescharacteristic
343C23C22×D4×C12712yes—
344C23C2×#41×C12714yes—
345D4—216no—
346D4—216no—
347C2×C4—28no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4C2×C22⋊C4×C12714yes—
366C2×C4C2×C22⋊C4×C12714yes—
367C2×C4#117×C12714yes—
368C22—22no—
369C22—22no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—44no—
384C22—42no—
385C22—42no—
386C22—88no—
387C22—88no—
388C22C22×C22⋊C4×C12711yescharacteristic
389C22C2×#117×C12712yes—
390C22#525×C12714yes—
391C4—24no—
392C4—24no—
393C4—24no—
394C4—42no—
395C4—42no—
396C4—88no—
397C2—24no—
398C2—22no—
399C2—42no—
400C2—42no—
401C2—44no—
402C2C127×#228011yescharacteristic
403C2C127×#288311yescharacteristic
404C2C127×#288311yescharacteristic
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2312C2semidirect
3. C127×#1819C2semidirect
4. C127×#1799C2semidirect
6. C4×#99×C127C2semidirect
9. C2×C4×D4×C127C4semidirect
10. C2×C4×D4×C127C4semidirect
13. C2×#117×C127C4semidirect
14. C22×C22⋊C4×C127C4semidirect
19. #380×C127C22semidirect
21. C4×C22⋊C4×C127C22semidirect
22. #342×C127C22semidirect
194. #20724C127direct (N × H)
204. #2312C2×C127semidirect
205. #1819C2×C127semidirect
206. #1799C2×C127semidirect
208. C4×#99C2×C127semidirect
209. C127#20724direct (N × H)
212. C2×C4×D4C4×C127semidirect
213. C2×C4×D4C4×C127semidirect
216. C2×#117C4×C127semidirect
217. C22×C22⋊C4C4×C127semidirect
222. #380C22×C127semidirect
224. C4×C22⋊C4C22×C127semidirect
225. #342C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#518×C1272
C127×#23122
C127×#18192
C127×#17992
C127×#17812
C4×#99×C1272
#20724127