Finite Group Databasebeta

← C127×#20392 #44584065

Subgroups — C127×#20392 #44584065

Order 32512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518×C127C211yesmaximal, characteristic
2C22×#99×C127C212yesmaximal
3C127×#2273C212yesmaximal, complemented
4C4×#99×C127C218yesmaximal, complemented
5C2×#342×C127C212yesmaximal, complemented
6C23×D4×C127C2211yescharacteristic
7#518×C127—28no—
8C2×C4×D4×C127C414yescomplemented
9C2×C4×D4×C127C2214yes—
10C2×#117×C127C412yescomplemented
11C22×C22⋊C4×C127C412yescomplemented
12C22×C22⋊C4×C127C2211yescharacteristic
13C22×C22⋊C4×C127C2214yes—
14C22×C22⋊C4×C127C2211yescharacteristic
15C2×#99×C127C4116yescomplemented
16C2×#99×C127C2218yes—
17C4×C22⋊C4×C127—28no—
18C4×C22⋊C4×C127C2218yescomplemented
19#342×C127C2218yescomplemented
20C25×C127C2×C411yescharacteristic
21C25×C127C2311yescharacteristic
22C22×D4×C127—24no—
23C22×D4×C127—24no—
24C22×D4×C127C2×C418yes—
25C22×D4×C127C2×C412yes—
26C22×D4×C127C2312yes—
27C23×C4×C127—24no—
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127C2×C412yes—
30C23×C4×C127C2×C414yescomplemented
31C23×C4×C127C2312yes—
32C4×D4×C127—216no—
33C4×D4×C127—216no—
34#117×C127—28no—
35C2×C4⋊C4×C127C2×C414yescomplemented
36C2×C22⋊C4×C127—24no—
37C2×C22⋊C4×C127—24no—
38C2×C22⋊C4×C127—28no—
39C2×C22⋊C4×C127—28no—
40C2×C22⋊C4×C127C2×C412yes—
41C2×C22⋊C4×C127C2×C412yes—
42C2×C22⋊C4×C127C2×C412yes—
43C2×C22⋊C4×C127C2×C414yescomplemented
44C2×C22⋊C4×C127C2×C4116yescomplemented
45C2×C22⋊C4×C127D418yes—
46C2×C22⋊C4×C127C2312yes—
47C2×C22⋊C4×C127C2318yes—
48C2×C42×C127—28no—
49C2×C42×C127C2×C414yescomplemented
50#99×C127C2×C4132yescomplemented
51#95×C127—28no—
52C24×C127—22no—
53C24×C127—22no—
54C24×C127—82no—
55C24×C127—82no—
56C24×C127C4214yescomplemented
57C24×C127C22×C412yes—
58C24×C127C22×C411yescharacteristic
59C24×C127C22×C411yescharacteristic
60C24×C127C22×C414yes—
61C24×C127C2×D414yes—
62C24×C127#4114yes—
63C24×C127C2411yescharacteristic
64C2×D4×C127—24no—
65C2×D4×C127—24no—
66C2×D4×C127—216no—
67C2×D4×C127—216no—
68C2×D4×C127—24no—
69C2×D4×C127—24no—
70C2×D4×C127C4218yescomplemented
71C2×D4×C127C22×C418yes—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—28no—
75C22×C4×C127—24no—
76C22×C4×C127—216no—
77C22×C4×C127—28no—
78C22×C4×C127—82no—
79C22×C4×C127—88no—
80C22×C4×C127—82no—
81C22×C4×C127—84no—
82C22×C4×C127C4214yescomplemented
83C22×C4×C127C22×C412yes—
84C22×C4×C127C22×C414yes—
85C22×C4×C127C22×C414yes—
86C22×C4×C127C2×D418yes—
87C4⋊C4×C127—28no—
88C22⋊C4×C127—24no—
89C22⋊C4×C127—216no—
90C22⋊C4×C127—24no—
91C22⋊C4×C127—28no—
92C22⋊C4×C127—28no—
93C22⋊C4×C127C22⋊C418yes—
94C22⋊C4×C127C22⋊C418yes—
95C22⋊C4×C127C22×C4116yes—
96C42×C127—28no—
97C42×C127—88no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—28no—
101C23×C127—42no—
102C23×C127—48no—
103C23×C127—42no—
104C23×C127—42no—
105C23×C127—44no—
106C23×C127—84no—
107C23×C127—84no—
108C23×C127—84no—
109C23×C127—84no—
110C23×C127—84no—
111C23×C127C2×C4214yes—
112C23×C127C2×C4212yes—
113C23×C127C2×C4212yes—
114C23×C127C2×C22⋊C412yes—
115C23×C127C2×C22⋊C412yes—
116C23×C127#11712yes—
117C23×C127#11712yes—
118C23×C127C4×D418yes—
119C23×C127C4×D418yes—
120C23×C127C23×C411yescharacteristic
121C23×C127C23×C412yes—
122C23×C127C22×D412yes—
123C23×C127C2×#4112yes—
124D4×C127—216no—
125D4×C127—216no—
126C2×C4×C127—22no—
127C2×C4×C127—22no—
128C2×C4×C127—28no—
129C2×C4×C127—24no—
130C2×C4×C127—216no—
131C2×C4×C127—48no—
132C2×C4×C127—44no—
133C2×C4×C127—416no—
134C2×C4×C127—44no—
135C2×C4×C127—88no—
136C2×C4×C127—84no—
137C2×C4×C127—816no—
138C2×C4×C127C2×C4214yes—
139C2×C4×C127C2×C22⋊C418yes—
140C22×C127—24no—
141C22×C127—22no—
142C22×C127—24no—
143C22×C127—24no—
144C22×C127—24no—
145C22×C127—48no—
146C22×C127—42no—
147C22×C127—42no—
148C22×C127—42no—
149C22×C127—42no—
150C22×C127—48no—
151C22×C127—42no—
152C22×C127—42no—
153C22×C127—44no—
154C22×C127—44no—
155C22×C127—44no—
156C22×C127—88no—
157C22×C127C4×C22⋊C414yes—
158C22×C127C4×C22⋊C414yes—
159C22×C127C22×C4211yescharacteristic
160C22×C127C22×C22⋊C411yescharacteristic
161C22×C127C2×#11711yescharacteristic
162C22×C127C2×C4×D414yes—
163C4×C127—28no—
164C4×C127—24no—
165C4×C127—416no—
166C4×C127—44no—
167#20392C12711yesmaximal, characteristic, direct product factor, Sylow
168C2×C127—24no—
169C2×C127—21no—
170C2×C127—21no—
171C2×C127—24no—
172C2×C127—24no—
173C2×C127—44no—
174C2×C127C2×C4×C22⋊C411yescharacteristic
175C2×C127#288312yes—
176C2×#518C2×C12711yescharacteristic
177C22×#99C2×C12712yes—
178#2273C2×C12712yescomplemented
179C4×#99C2×C12718yescomplemented
180C2×#342C2×C12712yescomplemented
181C127#2039211yescharacteristic, direct product factor, Sylow
182C23×D4C22×C12711yescharacteristic
183#518—28no—
184C2×C4×D4C4×C12714yescomplemented
185C2×C4×D4C22×C12714yes—
186C2×#117C4×C12712yescomplemented
187C22×C22⋊C4C4×C12712yescomplemented
188C22×C22⋊C4C22×C12711yescharacteristic
189C22×C22⋊C4C22×C12714yes—
190C22×C22⋊C4C22×C12711yescharacteristic
191C2×#99C4×C127116yescomplemented
192C2×#99C22×C12718yes—
193C4×C22⋊C4—28no—
194C4×C22⋊C4C22×C12718yescomplemented
195#342C22×C12718yescomplemented
196C25C2×C4×C12711yescharacteristic
197C25C23×C12711yescharacteristic
198C22×D4—24no—
199C22×D4—24no—
200C22×D4C2×C4×C12718yes—
201C22×D4C2×C4×C12712yes—
202C22×D4C23×C12712yes—
203C23×C4—24no—
204C23×C4C2×C4×C12711yescharacteristic
205C23×C4C2×C4×C12712yes—
206C23×C4C2×C4×C12714yescomplemented
207C23×C4C23×C12712yes—
208C4×D4—216no—
209C4×D4—216no—
210#117—28no—
211C2×C4⋊C4C2×C4×C12714yescomplemented
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4C2×C4×C12712yes—
217C2×C22⋊C4C2×C4×C12712yes—
218C2×C22⋊C4C2×C4×C12712yes—
219C2×C22⋊C4C2×C4×C12714yescomplemented
220C2×C22⋊C4C2×C4×C127116yescomplemented
221C2×C22⋊C4D4×C12718yes—
222C2×C22⋊C4C23×C12712yes—
223C2×C22⋊C4C23×C12718yes—
224C2×C42—28no—
225C2×C42C2×C4×C12714yescomplemented
226#99C2×C4×C127132yescomplemented
227#95—28no—
228C24—22no—
229C24—22no—
230C24—82no—
231C24—82no—
232C24C42×C12714yescomplemented
233C24C22×C4×C12712yes—
234C24C22×C4×C12711yescharacteristic
235C24C22×C4×C12711yescharacteristic
236C24C22×C4×C12714yes—
237C24C2×D4×C12714yes—
238C24#41×C12714yes—
239C24C24×C12711yescharacteristic
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—216no—
243C2×D4—216no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4C42×C12718yescomplemented
247C2×D4C22×C4×C12718yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—216no—
253C22×C4—28no—
254C22×C4—82no—
255C22×C4—88no—
256C22×C4—82no—
257C22×C4—84no—
258C22×C4C42×C12714yescomplemented
259C22×C4C22×C4×C12712yes—
260C22×C4C22×C4×C12714yes—
261C22×C4C22×C4×C12714yes—
262C22×C4C2×D4×C12718yes—
263C4⋊C4—28no—
264C22⋊C4—24no—
265C22⋊C4—216no—
266C22⋊C4—24no—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4C22⋊C4×C12718yes—
270C22⋊C4C22⋊C4×C12718yes—
271C22⋊C4C22×C4×C127116yes—
272C42—28no—
273C42—88no—
274C23—22no—
275C23—22no—
276C23—28no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—44no—
282C23—84no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23C2×C42×C12714yes—
288C23C2×C42×C12712yes—
289C23C2×C42×C12712yes—
290C23C2×C22⋊C4×C12712yes—
291C23C2×C22⋊C4×C12712yes—
292C23#117×C12712yes—
293C23#117×C12712yes—
294C23C4×D4×C12718yes—
295C23C4×D4×C12718yes—
296C23C23×C4×C12711yescharacteristic
297C23C23×C4×C12712yes—
298C23C22×D4×C12712yes—
299C23C2×#41×C12712yes—
300D4—216no—
301D4—216no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—216no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4—416no—
310C2×C4—44no—
311C2×C4—88no—
312C2×C4—84no—
313C2×C4—816no—
314C2×C4C2×C42×C12714yes—
315C2×C4C2×C22⋊C4×C12718yes—
316C22—24no—
317C22—22no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—48no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—48no—
327C22—42no—
328C22—42no—
329C22—44no—
330C22—44no—
331C22—44no—
332C22—88no—
333C22C4×C22⋊C4×C12714yes—
334C22C4×C22⋊C4×C12714yes—
335C22C22×C42×C12711yescharacteristic
336C22C22×C22⋊C4×C12711yescharacteristic
337C22C2×#117×C12711yescharacteristic
338C22C2×C4×D4×C12714yes—
339C4—28no—
340C4—24no—
341C4—416no—
342C4—44no—
343C2—24no—
344C2—21no—
345C2—21no—
346C2—24no—
347C2—24no—
348C2—44no—
349C2C2×C4×C22⋊C4×C12711yescharacteristic
350C2C127×#288312yes—
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC22C22×C42×C127
Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#2273C2semidirect
4. C4×#99×C127C2semidirect
5. C2×#342×C127C2semidirect
8. C2×C4×D4×C127C4semidirect
10. C2×#117×C127C4semidirect
11. C22×C22⋊C4×C127C4semidirect
15. C2×#99×C127C4semidirect
18. C4×C22⋊C4×C127C22semidirect
19. #342×C127C22semidirect
30. C23×C4×C127C2×C4semidirect
35. C2×C4⋊C4×C127C2×C4semidirect
43. C2×C22⋊C4×C127C2×C4semidirect
44. C2×C22⋊C4×C127C2×C4semidirect
49. C2×C42×C127C2×C4semidirect
50. #99×C127C2×C4semidirect
56. C24×C127C42semidirect
70. C2×D4×C127C42semidirect
82. C22×C4×C127C42semidirect
167. #20392C127direct (N × H)
178. #2273C2×C127semidirect
179. C4×#99C2×C127semidirect
180. C2×#342C2×C127semidirect
181. C127#20392direct (N × H)
184. C2×C4×D4C4×C127semidirect
186. C2×#117C4×C127semidirect
187. C22×C22⋊C4C4×C127semidirect
191. C2×#99C4×C127semidirect
194. C4×C22⋊C4C22×C127semidirect
195. #342C22×C127semidirect
206. C23×C4C2×C4×C127semidirect
211. C2×C4⋊C4C2×C4×C127semidirect
219. C2×C22⋊C4C2×C4×C127semidirect
220. C2×C22⋊C4C2×C4×C127semidirect
225. C2×C42C2×C4×C127semidirect
226. #99C2×C4×C127semidirect
232. C24C42×C127semidirect
246. C2×D4C42×C127semidirect
258. C22×C4C42×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#518×C1272
C22×#99×C1272
C127×#22732
C4×#99×C1272
C2×#342×C1272
#20392127