Finite Group Databasebeta

← C127×#20725 #44584398

Subgroups — C127×#20725 #44584398

Order 32512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520×C127C211yesmaximal, characteristic
2C127×#2313C212yesmaximal, complemented
3C127×#1820C212yesmaximal, complemented
4C127×#1799C214yesmaximal, complemented
5C127×#1782C212yesmaximal
6C4×#99×C127C214yesmaximal, complemented
7C22×#41×C127C2211yescharacteristic
8#520×C127—28no—
9C2×C4×Q8×C127C412yescomplemented
10C2×C4×D4×C127C412yescomplemented
11C2×C4×D4×C127C2212yes—
12C2×C4×D4×C127C2212yes—
13C2×#117×C127C412yescomplemented
14C2×#117×C127C2211yescharacteristic
15C22×C4⋊C4×C127C412yescomplemented
16C22×C4⋊C4×C127C2211yescharacteristic
17C2×#99×C127C2218yes—
18#387×C127—28no—
19#381×C127C2214yes—
20#380×C127C2214yes—
21C4×C22⋊C4×C127C2214yes—
22#342×C127C2218yes—
23C2×#41×C127—28no—
24C2×#41×C127—24no—
25C22×Q8×C127C2×C411yescharacteristic
26C22×D4×C127C2×C411yescharacteristic
27C22×D4×C127C2312yes—
28C23×C4×C127C2×C412yes—
29C23×C4×C127C2×C412yes—
30C23×C4×C127C2×C412yes—
31C23×C4×C127C2311yescharacteristic
32C23×C4×C127C2311yescharacteristic
33C23×C4×C127C2311yescharacteristic
34C4×Q8×C127—28no—
35C4×D4×C127—28no—
36C4×D4×C127—28no—
37C4×D4×C127—28no—
38#117×C127—24no—
39#117×C127—28no—
40C2×C4⋊C4×C127—28no—
41C2×C4⋊C4×C127—24no—
42C2×C4⋊C4×C127C2×C411yescharacteristic
43C2×C4⋊C4×C127C2×C412yes—
44C2×C4⋊C4×C127C2×C412yes—
45C2×C4⋊C4×C127C2×C411yescharacteristic
46C2×C4⋊C4×C127C2×C412yes—
47C2×C4⋊C4×C127C2×C412yes—
48C2×C4⋊C4×C127D418yes—
49C2×C22⋊C4×C127C2×C412yes—
50C2×C22⋊C4×C127C2×C412yes—
51C2×C22⋊C4×C127C2318yes—
52C2×C22⋊C4×C127C2312yes—
53C2×C42×C127—24no—
54C2×C42×C127C2×C412yes—
55C2×C42×C127C2×C412yes—
56C2×C42×C127C2×C412yes—
57#99×C127—216no—
58#95×C127—24no—
59#95×C127—24no—
60#95×C127—28no—
61C24×C127C22×C412yes—
62C24×C127C2411yescharacteristic
63#41×C127—232no—
64C2×Q8×C127—22no—
65C2×Q8×C127—24no—
66C2×D4×C127—24no—
67C2×D4×C127—24no—
68C2×D4×C127—24no—
69C2×D4×C127—22no—
70C2×D4×C127#4118yes—
71C22×C4×C127—24no—
72C22×C4×C127—24no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—28no—
78C22×C4×C127—81no—
79C22×C4×C127—81no—
80C22×C4×C127—82no—
81C22×C4×C127—82no—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C22×C411yescharacteristic
84C22×C4×C127C22×C412yes—
85C22×C4×C127C22×C412yes—
86C22×C4×C127C22×C412yes—
87C22×C4×C127C22×C412yes—
88C22×C4×C127C22×C412yes—
89C22×C4×C127C2×D414yes—
90C22×C4×C127C2×D414yes—
91C22×C4×C127C2×D414yes—
92C4⋊C4×C127—22no—
93C4⋊C4×C127—24no—
94C4⋊C4×C127—24no—
95C4⋊C4×C127—22no—
96C4⋊C4×C127—84no—
97C4⋊C4×C127C22⋊C418yes—
98C4⋊C4×C127C22⋊C418yes—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—24no—
101C22⋊C4×C127—24no—
102C22⋊C4×C127—216no—
103C42×C127—24no—
104C42×C127—24no—
105C42×C127—24no—
106C42×C127—84no—
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127—42no—
112C23×C127—42no—
113C23×C127#11714yes—
114C23×C127C23×C411yescharacteristic
115C23×C127C22×D412yes—
116C23×C127C2×#4114yes—
117Q8×C127—28no—
118D4×C127—28no—
119D4×C127—216no—
120C2×C4×C127—216no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—28no—
126C2×C4×C127—22no—
127C2×C4×C127—22no—
128C2×C4×C127—44no—
129C2×C4×C127—44no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—48no—
135C2×C4×C127—48no—
136C2×C4×C127—48no—
137C2×C4×C127—84no—
138C2×C4×C127—84no—
139C2×C4×C127C2×C22⋊C412yes—
140C2×C4×C127C2×C22⋊C414yes—
141C2×C4×C127C2×C22⋊C414yes—
142C2×C4×C127C2×C22⋊C412yes—
143C2×C4×C127#11714yes—
144C22×C127—22no—
145C22×C127—21no—
146C22×C127—22no—
147C22×C127—21no—
148C22×C127—24no—
149C22×C127—44no—
150C22×C127—44no—
151C22×C127—44no—
152C22×C127—42no—
153C22×C127—42no—
154C22×C127C22×C22⋊C411yescharacteristic
155C22×C127C2×#11712yes—
156C22×C127#52514yes—
157C4×C127—24no—
158C4×C127—24no—
159C4×C127—22no—
160C4×C127—22no—
161C4×C127—24no—
162C4×C127—42no—
163C4×C127—42no—
164C4×C127—88no—
165#20725C12711yesmaximal, characteristic, direct product factor, Sylow
166C2×C127—24no—
167C2×C127—22no—
168C2×C127—44no—
169C2×C127#228011yescharacteristic
170C2×C127#288311yescharacteristic
171C2×C127#288411yescharacteristic
172C2×#520C2×C12711yescharacteristic
173#2313C2×C12712yescomplemented
174#1820C2×C12712yescomplemented
175#1799C2×C12714yescomplemented
176#1782C2×C12712yes—
177C4×#99C2×C12714yescomplemented
178C127#2072511yescharacteristic, direct product factor, Sylow
179C22×#41C22×C12711yescharacteristic
180#520—28no—
181C2×C4×Q8C4×C12712yescomplemented
182C2×C4×D4C4×C12712yescomplemented
183C2×C4×D4C22×C12712yes—
184C2×C4×D4C22×C12712yes—
185C2×#117C4×C12712yescomplemented
186C2×#117C22×C12711yescharacteristic
187C22×C4⋊C4C4×C12712yescomplemented
188C22×C4⋊C4C22×C12711yescharacteristic
189C2×#99C22×C12718yes—
190#387—28no—
191#381C22×C12714yes—
192#380C22×C12714yes—
193C4×C22⋊C4C22×C12714yes—
194#342C22×C12718yes—
195C2×#41—28no—
196C2×#41—24no—
197C22×Q8C2×C4×C12711yescharacteristic
198C22×D4C2×C4×C12711yescharacteristic
199C22×D4C23×C12712yes—
200C23×C4C2×C4×C12712yes—
201C23×C4C2×C4×C12712yes—
202C23×C4C2×C4×C12712yes—
203C23×C4C23×C12711yescharacteristic
204C23×C4C23×C12711yescharacteristic
205C23×C4C23×C12711yescharacteristic
206C4×Q8—28no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4—28no—
210#117—24no—
211#117—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4C2×C4×C12711yescharacteristic
215C2×C4⋊C4C2×C4×C12712yes—
216C2×C4⋊C4C2×C4×C12712yes—
217C2×C4⋊C4C2×C4×C12711yescharacteristic
218C2×C4⋊C4C2×C4×C12712yes—
219C2×C4⋊C4C2×C4×C12712yes—
220C2×C4⋊C4D4×C12718yes—
221C2×C22⋊C4C2×C4×C12712yes—
222C2×C22⋊C4C2×C4×C12712yes—
223C2×C22⋊C4C23×C12718yes—
224C2×C22⋊C4C23×C12712yes—
225C2×C42—24no—
226C2×C42C2×C4×C12712yes—
227C2×C42C2×C4×C12712yes—
228C2×C42C2×C4×C12712yes—
229#99—216no—
230#95—24no—
231#95—24no—
232#95—28no—
233C24C22×C4×C12712yes—
234C24C24×C12711yescharacteristic
235#41—232no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4#41×C12718yes—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C22×C4×C12711yescharacteristic
255C22×C4C22×C4×C12711yescharacteristic
256C22×C4C22×C4×C12712yes—
257C22×C4C22×C4×C12712yes—
258C22×C4C22×C4×C12712yes—
259C22×C4C22×C4×C12712yes—
260C22×C4C22×C4×C12712yes—
261C22×C4C2×D4×C12714yes—
262C22×C4C2×D4×C12714yes—
263C22×C4C2×D4×C12714yes—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—84no—
269C4⋊C4C22⋊C4×C12718yes—
270C4⋊C4C22⋊C4×C12718yes—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—216no—
275C42—24no—
276C42—24no—
277C42—24no—
278C42—84no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—42no—
284C23—42no—
285C23#117×C12714yes—
286C23C23×C4×C12711yescharacteristic
287C23C22×D4×C12712yes—
288C23C2×#41×C12714yes—
289Q8—28no—
290D4—28no—
291D4—216no—
292C2×C4—216no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—28no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4C2×C22⋊C4×C12712yes—
312C2×C4C2×C22⋊C4×C12714yes—
313C2×C4C2×C22⋊C4×C12714yes—
314C2×C4C2×C22⋊C4×C12712yes—
315C2×C4#117×C12714yes—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—24no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22C22×C22⋊C4×C12711yescharacteristic
327C22C2×#117×C12712yes—
328C22#525×C12714yes—
329C4—24no—
330C4—24no—
331C4—22no—
332C4—22no—
333C4—24no—
334C4—42no—
335C4—42no—
336C4—88no—
337C2—24no—
338C2—22no—
339C2—44no—
340C2C127×#228011yescharacteristic
341C2C127×#288311yescharacteristic
342C2C127×#288411yescharacteristic
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC22×#41×C127C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2313C2semidirect
3. C127×#1820C2semidirect
4. C127×#1799C2semidirect
6. C4×#99×C127C2semidirect
9. C2×C4×Q8×C127C4semidirect
10. C2×C4×D4×C127C4semidirect
13. C2×#117×C127C4semidirect
15. C22×C4⋊C4×C127C4semidirect
165. #20725C127direct (N × H)
173. #2313C2×C127semidirect
174. #1820C2×C127semidirect
175. #1799C2×C127semidirect
177. C4×#99C2×C127semidirect
178. C127#20725direct (N × H)
181. C2×C4×Q8C4×C127semidirect
182. C2×C4×D4C4×C127semidirect
185. C2×#117C4×C127semidirect
187. C22×C4⋊C4C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#520×C1272
C127×#23132
C127×#18202
C127×#17992
C127×#17822
C4×#99×C1272
#20725127