Finite Group Databasebeta

← C22×#339×C127 #44584014

Subgroups — C22×#339×C127 #44584014

Order 32512.

2830 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C127C211yesmaximal, characteristic
2C22×#117×C127C211yesmaximal, characteristic
3C23×C42×C127C211yesmaximal, characteristic
4C2×#339×C127C2112yesmaximal, direct product factor
5C24×C4×C127—21no—
6C24×C4×C127C2211yescharacteristic
7C22×M16×C127—24no—
8C22×M16×C127—23no—
9C22×M16×C127C418yescomplemented
10C22×M16×C127C2216yes—
11C23×C8×C127—21no—
12C2×#117×C127—23no—
13C2×#117×C127—23no—
14C2×#117×C127C416yescomplemented
15C2×#117×C127C2216yes—
16C22×C4⋊C4×C127C412yescomplemented
17C22×C22⋊C4×C127—21no—
18C22×C42×C127—23no—
19C22×C42×C127—24no—
20C22×C42×C127—21no—
21C22×C42×C127C418yescomplemented
22C22×C42×C127C2216yes—
23#339×C127C22116yesdirect product factor
24C25×C127D411yescharacteristic
25C23×C4×C127—216no—
26C23×C4×C127—28no—
27C23×C4×C127—26no—
28C23×C4×C127—22no—
29C23×C4×C127C2×C414yes—
30C23×C4×C127C2×C414yes—
31C23×C4×C127C2×C411yescharacteristic
32C23×C4×C127C2×C413yes—
33C23×C4×C127D413yes—
34C23×C4×C127D414yes—
35C23×C4×C127D413yes—
36C23×C4×C127D411yescharacteristic
37C23×C4×C127Q814yes—
38C23×C4×C127C2313yes—
39C2×M16×C127—224no—
40C2×M16×C127—212no—
41C2×M16×C127—26no—
42C2×M16×C127C2×C4124yescomplemented
43C2×M16×C127C2314yes—
44C22×C8×C127—28no—
45C22×C8×C127—26no—
46#117×C127—212no—
47#117×C127—26no—
48#117×C127—26no—
49#117×C127C2×C4112yescomplemented
50#117×C127C2314yes—
51C2×C4⋊C4×C127—26no—
52C2×C4⋊C4×C127C2×C4112yescomplemented
53C2×C22⋊C4×C127—26no—
54C2×C22⋊C4×C127—26no—
55C2×C42×C127—224no—
56C2×C42×C127—26no—
57C2×C42×C127—26no—
58C2×C42×C127—212no—
59C2×C42×C127—26no—
60C2×C42×C127C2×C4124yescomplemented
61C2×C42×C127C2314yes—
62C24×C127—28no—
63C24×C127C22⋊C413yes—
64C24×C127C22⋊C414yes—
65C24×C127C22⋊C411yescharacteristic
66C24×C127C4⋊C414yes—
67C24×C127C2×D413yes—
68C22×C4×C127—264no—
69C22×C4×C127—248no—
70C22×C4×C127—224no—
71C22×C4×C127—24no—
72C22×C4×C127—212no—
73C22×C4×C127—48no—
74C22×C4×C127C42112yes—
75C22×C4×C127C4214yes—
76C22×C4×C127C22⋊C4112yes—
77C22×C4×C127C22⋊C413yes—
78C22×C4×C127C22⋊C413yes—
79C22×C4×C127C22⋊C416yes—
80C22×C4×C127C22⋊C414yes—
81C22×C4×C127C22⋊C4112yes—
82C22×C4×C127C4⋊C4112yes—
83C22×C4×C127C4⋊C4112yes—
84C22×C4×C127C4⋊C4112yes—
85C22×C4×C127C4⋊C414yes—
86C22×C4×C127C4⋊C414yes—
87C22×C4×C127C22×C416yes—
88C22×C4×C127C22×C416yes—
89C22×C4×C127C22×C413yes—
90C22×C4×C127C22×C413yes—
91C22×C4×C127C2×D413yes—
92C22×C4×C127C2×D413yes—
93C22×C4×C127C2×D416yes—
94C22×C4×C127C2×D413yes—
95C22×C4×C127C2×Q816yes—
96C22×C4×C127C2411yescharacteristic
97M16×C127—224no—
98M16×C127—24no—
99M16×C127C22×C418yescomplemented
100C2×C8×C127—224no—
101C2×C8×C127—24no—
102C4⋊C4×C127—212no—
103C4⋊C4×C127C22×C418yescomplemented
104C22⋊C4×C127—212no—
105C22⋊C4×C127—24no—
106C42×C127—224no—
107C42×C127—212no—
108C42×C127—24no—
109C42×C127—24no—
110C42×C127C22×C418yescomplemented
111C23×C127—216no—
112C23×C127—216no—
113C23×C127—212no—
114C23×C127—212no—
115C23×C127#95112yes—
116C23×C127#9514yes—
117C23×C127#10418yes—
118C23×C127C2×C22⋊C413yes—
119C23×C127C2×C22⋊C416yes—
120C23×C127C2×C22⋊C413yes—
121C23×C127C2×C4⋊C416yes—
122C23×C127C22×D411yescharacteristic
123C2×C4×C127—216no—
124C2×C4×C127—296no—
125C2×C4×C127—28no—
126C2×C4×C127—28no—
127C2×C4×C127—424no—
128C2×C4×C127#95124yes—
129C2×C4×C127#95112yes—
130C2×C4×C127#95112yes—
131C2×C4×C127C2×C4216yes—
132C2×C4×C127C2×C4216yes—
133C2×C4×C127C2×C22⋊C416yes—
134C2×C4×C127C2×C22⋊C416yes—
135C2×C4×C127C2×C22⋊C416yes—
136C2×C4×C127C2×C22⋊C413yes—
137C2×C4×C127C2×C22⋊C413yes—
138C2×C4×C127C2×C4⋊C416yes—
139C2×C4×C127C2×C4⋊C416yes—
140C2×C4×C127C2×C4⋊C416yes—
141C2×C4×C127C2×C4⋊C416yes—
142C2×C4×C127C2×C4⋊C416yes—
143C2×C4×C127C23×C411yescharacteristic
144C2×C4×C127C23×C411yescharacteristic
145C2×C4×C127C23×C411yescharacteristic
146C2×C4×C127C22×D411yescharacteristic
147C2×C4×C127C22×D411yescharacteristic
148C2×C4×C127C22×Q811yescharacteristic
149C8×C127—28no—
150C22×C127—22no—
151C22×C127—26no—
152C22×C127—212no—
153C22×C127—224no—
154C22×C127—212no—
155C22×C127#339116yesdirect product factor
156C22×C127C2×#9516yes—
157C22×C127C2×#9516yes—
158C22×C127C2×#104112yes—
159C22×C127C22×C22⋊C411yescharacteristic
160C22×C127C22×C22⋊C411yescharacteristic
161C22×C127C22×C4⋊C411yescharacteristic
162C4×C127—216no—
163C4×C127—48no—
164C4×C127C2×#9516yes—
165C4×C127C2×#9516yes—
166C4×C127C22×C4211yescharacteristic
167C4×C127C22×C22⋊C411yescharacteristic
168C4×C127C22×C4⋊C411yescharacteristic
169C4×C127C22×C4⋊C411yescharacteristic
170C22×#339C12711yesmaximal, characteristic, direct product factor, Sylow
171C2×C127—21no—
172C2×C127—21no—
173C2×C127—26no—
174C2×C127C2×#339112yesdirect product factor
175C2×C127C22×#9511yescharacteristic
176C2×C127C22×#10412yes—
177C23×M16C2×C12711yescharacteristic
178C22×#117C2×C12711yescharacteristic
179C23×C42C2×C12711yescharacteristic
180C2×#339C2×C127112yesdirect product factor
181C127C22×#33911yescharacteristic, direct product factor, Sylow
182C24×C4—21no—
183C24×C4C22×C12711yescharacteristic
184C22×M16—24no—
185C22×M16—23no—
186C22×M16C4×C12718yescomplemented
187C22×M16C22×C12716yes—
188C23×C8—21no—
189C2×#117—23no—
190C2×#117—23no—
191C2×#117C4×C12716yescomplemented
192C2×#117C22×C12716yes—
193C22×C4⋊C4C4×C12712yescomplemented
194C22×C22⋊C4—21no—
195C22×C42—23no—
196C22×C42—24no—
197C22×C42—21no—
198C22×C42C4×C12718yescomplemented
199C22×C42C22×C12716yes—
200#339C22×C127116yesdirect product factor
201C25D4×C12711yescharacteristic
202C23×C4—216no—
203C23×C4—28no—
204C23×C4—26no—
205C23×C4—22no—
206C23×C4C2×C4×C12714yes—
207C23×C4C2×C4×C12711yescharacteristic
208C23×C4C2×C4×C12713yes—
209C23×C4C2×C4×C12714yes—
210C23×C4D4×C12713yes—
211C23×C4D4×C12713yes—
212C23×C4D4×C12711yescharacteristic
213C23×C4D4×C12714yes—
214C23×C4Q8×C12714yes—
215C23×C4C23×C12713yes—
216C2×M16—224no—
217C2×M16—212no—
218C2×M16—26no—
219C2×M16C2×C4×C127124yescomplemented
220C2×M16C23×C12714yes—
221C22×C8—28no—
222C22×C8—26no—
223#117—212no—
224#117—26no—
225#117—26no—
226#117C2×C4×C127112yescomplemented
227#117C23×C12714yes—
228C2×C4⋊C4—26no—
229C2×C4⋊C4C2×C4×C127112yescomplemented
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C42—26no—
233C2×C42—224no—
234C2×C42—26no—
235C2×C42—26no—
236C2×C42—212no—
237C2×C42C2×C4×C127124yescomplemented
238C2×C42C23×C12714yes—
239C24—28no—
240C24C22⋊C4×C12713yes—
241C24C22⋊C4×C12714yes—
242C24C22⋊C4×C12711yescharacteristic
243C24C4⋊C4×C12714yes—
244C24C2×D4×C12713yes—
245C22×C4—264no—
246C22×C4—248no—
247C22×C4—224no—
248C22×C4—24no—
249C22×C4—212no—
250C22×C4—48no—
251C22×C4C42×C127112yes—
252C22×C4C42×C12714yes—
253C22×C4C22⋊C4×C127112yes—
254C22×C4C22⋊C4×C12713yes—
255C22×C4C22⋊C4×C12713yes—
256C22×C4C22⋊C4×C12716yes—
257C22×C4C22⋊C4×C12714yes—
258C22×C4C22⋊C4×C127112yes—
259C22×C4C4⋊C4×C127112yes—
260C22×C4C4⋊C4×C127112yes—
261C22×C4C4⋊C4×C127112yes—
262C22×C4C4⋊C4×C12714yes—
263C22×C4C4⋊C4×C12714yes—
264C22×C4C22×C4×C12716yes—
265C22×C4C22×C4×C12716yes—
266C22×C4C22×C4×C12713yes—
267C22×C4C22×C4×C12713yes—
268C22×C4C2×D4×C12713yes—
269C22×C4C2×D4×C12713yes—
270C22×C4C2×D4×C12716yes—
271C22×C4C2×D4×C12713yes—
272C22×C4C2×Q8×C12716yes—
273C22×C4C24×C12711yescharacteristic
274M16—224no—
275M16—24no—
276M16C22×C4×C12718yescomplemented
277C2×C8—224no—
278C2×C8—24no—
279C4⋊C4—212no—
280C4⋊C4C22×C4×C12718yescomplemented
281C22⋊C4—212no—
282C22⋊C4—24no—
283C42—224no—
284C42—212no—
285C42—24no—
286C42—24no—
287C42C22×C4×C12718yescomplemented
288C23—216no—
289C23—216no—
290C23—212no—
291C23—212no—
292C23#95×C127112yes—
293C23#95×C12714yes—
294C23#104×C12718yes—
295C23C2×C22⋊C4×C12713yes—
296C23C2×C22⋊C4×C12716yes—
297C23C2×C22⋊C4×C12713yes—
298C23C2×C4⋊C4×C12716yes—
299C23C22×D4×C12711yescharacteristic
300C2×C4—216no—
301C2×C4—296no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—424no—
305C2×C4#95×C127124yes—
306C2×C4#95×C127112yes—
307C2×C4#95×C127112yes—
308C2×C4C2×C42×C12716yes—
309C2×C4C2×C42×C12716yes—
310C2×C4C2×C22⋊C4×C12716yes—
311C2×C4C2×C22⋊C4×C12716yes—
312C2×C4C2×C22⋊C4×C12716yes—
313C2×C4C2×C22⋊C4×C12713yes—
314C2×C4C2×C22⋊C4×C12713yes—
315C2×C4C2×C4⋊C4×C12716yes—
316C2×C4C2×C4⋊C4×C12716yes—
317C2×C4C2×C4⋊C4×C12716yes—
318C2×C4C2×C4⋊C4×C12716yes—
319C2×C4C2×C4⋊C4×C12716yes—
320C2×C4C23×C4×C12711yescharacteristic
321C2×C4C23×C4×C12711yescharacteristic
322C2×C4C23×C4×C12711yescharacteristic
323C2×C4C22×D4×C12711yescharacteristic
324C2×C4C22×D4×C12711yescharacteristic
325C2×C4C22×Q8×C12711yescharacteristic
326C8—28no—
327C22—22no—
328C22—26no—
329C22—212no—
330C22—224no—
331C22—212no—
332C22#339×C127116yesdirect product factor
333C22C2×#95×C12716yes—
334C22C2×#95×C12716yes—
335C22C2×#104×C127112yes—
336C22C22×C22⋊C4×C12711yescharacteristic
337C22C22×C22⋊C4×C12711yescharacteristic
338C22C22×C4⋊C4×C12711yescharacteristic
339C4—216no—
340C4—48no—
341C4C2×#95×C12716yes—
342C4C2×#95×C12716yes—
343C4C22×C42×C12711yescharacteristic
344C4C22×C22⋊C4×C12711yescharacteristic
345C4C22×C4⋊C4×C12711yescharacteristic
346C4C22×C4⋊C4×C12711yescharacteristic
347C2—21no—
348C2—21no—
349C2—26no—
350C2C2×#339×C127112yesdirect product factor
351C2C22×#95×C12711yescharacteristic
352C2C22×#104×C12712yes—
subgroupG/H
centerC23×C4×C127D4
commutatorC4C22×C42×C127
Frattini, cosocleC22×C4C24×C127
socleC24×C127C22⋊C4
prime radicalC25×C127D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#339×C127C2direct (N × H)
9. C22×M16×C127C4semidirect
14. C2×#117×C127C4semidirect
16. C22×C4⋊C4×C127C4semidirect
21. C22×C42×C127C4semidirect
23. #339×C127C22direct (N × H)
42. C2×M16×C127C2×C4semidirect
49. #117×C127C2×C4semidirect
52. C2×C4⋊C4×C127C2×C4semidirect
60. C2×C42×C127C2×C4semidirect
99. M16×C127C22×C4semidirect
103. C4⋊C4×C127C22×C4semidirect
110. C42×C127C22×C4semidirect
155. C22×C127#339direct (N × H)
170. C22×#339C127direct (N × H)
174. C2×C127C2×#339direct (N × H)
180. C2×#339C2×C127direct (N × H)
181. C127C22×#339direct (N × H)
186. C22×M16C4×C127semidirect
191. C2×#117C4×C127semidirect
193. C22×C4⋊C4C4×C127semidirect
198. C22×C42C4×C127semidirect
200. #339C22×C127direct (N × H)
219. C2×M16C2×C4×C127semidirect
226. #117C2×C4×C127semidirect
229. C2×C4⋊C4C2×C4×C127semidirect
237. C2×C42C2×C4×C127semidirect
276. M16C22×C4×C127semidirect
280. C4⋊C4C22×C4×C127semidirect
287. C42C22×C4×C127semidirect
332. C22#339×C127direct (N × H)
350. C2C2×#339×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C23×M16×C1272
C22×#117×C1272
C23×C42×C1272
C2×#339×C1272
C22×#339127