Finite Group Databasebeta

← C127×#20608 #44584281

Subgroups — C127×#20608 #44584281

Order 32512.

3798 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585×C127C211yesmaximal, characteristic
2C2×#410×C127C213yesmaximal, complemented
3C127×#2297C213yesmaximal, complemented
4C127×#1790C218yesmaximal, complemented
5#585×C127—212no—
6#585×C127C418yescomplemented
7C2×2−1+4×C127—23no—
8C2×H(5,𝔽2)×C127—23no—
9C2×H(5,𝔽2)×C127C2214yes—
10C22×#41×C127—26no—
11C22×#41×C127C2213yes—
12C2×#117×C127C2213yescomplemented
13C22×C42×C127C2211yescharacteristic, complemented
14#410×C127—212no—
15C2×#99×C127C22112yescomplemented
16#391×C127—23no—
17#390×C127—23no—
18#386×C127C22112yescomplemented
19C4×C22⋊C4×C127—26no—
202−1+4×C127—224no—
212−1+4×C127—224no—
22H(5,𝔽2)×C127—224no—
23H(5,𝔽2)×C127—224no—
24H(5,𝔽2)×C127—224no—
25H(5,𝔽2)×C127C2×C4116yes—
26C2×#41×C127—24no—
27C2×#41×C127—212no—
28C2×#41×C127—248no—
29C2×#41×C127—248no—
30C2×#41×C127—26no—
31C2×#41×C127—212no—
32C2×#41×C127—26no—
33C2×#41×C127—212no—
34C2×#41×C127C2×C4112yes—
35C2×#41×C127D4112yes—
36C22×Q8×C127—26no—
37C22×Q8×C127D413yes—
38C22×D4×C127—26no—
39C22×D4×C127—212no—
40C22×D4×C127D413yes—
41C22×D4×C127C2316yes—
42C23×C4×C127—24no—
43C23×C4×C127D416yes—
44C23×C4×C127C2312yes—
45C23×C4×C127C2311yescharacteristic
46#117×C127—212no—
47C2×C4⋊C4×C127—23no—
48C2×C22⋊C4×C127—212no—
49C2×C22⋊C4×C127C2316yes—
50C2×C42×C127—23no—
51C2×C42×C127—23no—
52C2×C42×C127—23no—
53#99×C127—224no—
54#95×C127—212no—
55C24×C127—24no—
56C24×C127C2×D416yes—
57C24×C127C2411yescharacteristic
58#41×C127—232no—
59#41×C127—248no—
60#41×C127—296no—
61#41×C127—224no—
62#41×C127—248no—
63#41×C127—224no—
64#41×C127—224no—
65#41×C127—224no—
66C2×Q8×C127—24no—
67C2×Q8×C127—248no—
68C2×Q8×C127—26no—
69C2×Q8×C127—26no—
70C2×Q8×C127C22⋊C4112yescomplemented
71C2×D4×C127—212no—
72C2×D4×C127—248no—
73C2×D4×C127—248no—
74C2×D4×C127—248no—
75C2×D4×C127—212no—
76C2×D4×C127—212no—
77C2×D4×C127—26no—
78C2×D4×C127—26no—
79C2×D4×C127C22⋊C4112yescomplemented
80C2×D4×C127C22×C4112yes—
81C2×D4×C127C2×D4112yes—
82C22×C4×C127—212no—
83C22×C4×C127—232no—
84C22×C4×C127—26no—
85C22×C4×C127—212no—
86C22×C4×C127—26no—
87C22×C4×C127—26no—
88C22×C4×C127—26no—
89C22×C4×C127—44no—
90C22×C4×C127C22⋊C4124yescomplemented
91C22×C4×C127C22×C412yes—
92C22×C4×C127C2×D416yes—
93C22×C4×C127C2×D413yes—
94C22×C4×C127C2×D416yes—
95C22×C4×C127C2×D413yes—
96C4⋊C4×C127—46no—
97C22⋊C4×C127—212no—
98C22⋊C4×C127—424no—
99C42×C127—43no—
100C42×C127—46no—
101C42×C127—41no—
102C23×C127—216no—
103C23×C127—26no—
104C23×C127—26no—
105C23×C127—42no—
106C23×C127—412no—
107C23×C127—88no—
108C23×C127C2×C22⋊C4112yes—
109C23×C127C23×C411yescharacteristic
110C23×C127C22×D413yes—
111C23×C127C22×D413yes—
112Q8×C127—216no—
113Q8×C127—224no—
114D4×C127—248no—
115D4×C127—224no—
116D4×C127—224no—
117D4×C127—224no—
118C2×C4×C127—248no—
119C2×C4×C127—26no—
120C2×C4×C127—26no—
121C2×C4×C127—224no—
122C2×C4×C127—224no—
123C2×C4×C127—212no—
124C2×C4×C127—46no—
125C2×C4×C127—42no—
126C2×C4×C127—412no—
127C2×C4×C127—46no—
128C2×C4×C127C2×C22⋊C4112yes—
129C2×C4×C127C2×C22⋊C416yes—
130C2×C4×C127C2×C22⋊C416yes—
131C2×C4×C127#12013yes—
132C2×C4×C127#12016yes—
133C2×C4×C127#12016yes—
134C2×C4×C127#12011yescharacteristic
135C22×C127—26no—
136C22×C127—212no—
137C22×C127—41no—
138C22×C127—43no—
139C22×C127—412no—
140C22×C127—412no—
141C22×C127—86no—
142C22×C127—816no—
143C22×C127C22×C22⋊C413yes—
144C22×C127C2×#12013yes—
145C22×C127C2×#12011yescharacteristic
146C4×C127—212no—
147C4×C127—44no—
148C4×C127—86no—
149C4×C127#37912yes—
150C4×C127#37916yes—
151#20608C12711yesmaximal, characteristic, direct product factor, Sylow
152C2×C127—23no—
153C2×C127—23no—
154C2×C127—412no—
155C2×C127C2×#37911yescharacteristic
156C2×C127#302512yes—
157C2×#585C2×C12711yescharacteristic
158C2×#410C2×C12713yescomplemented
159#2297C2×C12713yescomplemented
160#1790C2×C12718yescomplemented
161C127#2060811yescharacteristic, direct product factor, Sylow
162#585—212no—
163#585C4×C12718yescomplemented
164C2×2−1+4—23no—
165C2×H(5,𝔽2)—23no—
166C2×H(5,𝔽2)C22×C12714yes—
167C22×#41—26no—
168C22×#41C22×C12713yes—
169C2×#117C22×C12713yescomplemented
170C22×C42C22×C12711yescharacteristic, complemented
171#410—212no—
172C2×#99C22×C127112yescomplemented
173#391—23no—
174#390—23no—
175#386C22×C127112yescomplemented
176C4×C22⋊C4—26no—
1772−1+4—224no—
1782−1+4—224no—
179H(5,𝔽2)—224no—
180H(5,𝔽2)—224no—
181H(5,𝔽2)—224no—
182H(5,𝔽2)C2×C4×C127116yes—
183C2×#41—24no—
184C2×#41—212no—
185C2×#41—248no—
186C2×#41—248no—
187C2×#41—26no—
188C2×#41—212no—
189C2×#41—26no—
190C2×#41—212no—
191C2×#41C2×C4×C127112yes—
192C2×#41D4×C127112yes—
193C22×Q8—26no—
194C22×Q8D4×C12713yes—
195C22×D4—26no—
196C22×D4—212no—
197C22×D4D4×C12713yes—
198C22×D4C23×C12716yes—
199C23×C4—24no—
200C23×C4D4×C12716yes—
201C23×C4C23×C12712yes—
202C23×C4C23×C12711yescharacteristic
203#117—212no—
204C2×C4⋊C4—23no—
205C2×C22⋊C4—212no—
206C2×C22⋊C4C23×C12716yes—
207C2×C42—23no—
208C2×C42—23no—
209C2×C42—23no—
210#99—224no—
211#95—212no—
212C24—24no—
213C24C2×D4×C12716yes—
214C24C24×C12711yescharacteristic
215#41—232no—
216#41—248no—
217#41—296no—
218#41—224no—
219#41—248no—
220#41—224no—
221#41—224no—
222#41—224no—
223C2×Q8—24no—
224C2×Q8—248no—
225C2×Q8—26no—
226C2×Q8—26no—
227C2×Q8C22⋊C4×C127112yescomplemented
228C2×D4—212no—
229C2×D4—248no—
230C2×D4—248no—
231C2×D4—248no—
232C2×D4—212no—
233C2×D4—212no—
234C2×D4—26no—
235C2×D4—26no—
236C2×D4C22⋊C4×C127112yescomplemented
237C2×D4C22×C4×C127112yes—
238C2×D4C2×D4×C127112yes—
239C22×C4—212no—
240C22×C4—232no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—212no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—44no—
247C22×C4C22⋊C4×C127124yescomplemented
248C22×C4C22×C4×C12712yes—
249C22×C4C2×D4×C12716yes—
250C22×C4C2×D4×C12713yes—
251C22×C4C2×D4×C12713yes—
252C22×C4C2×D4×C12716yes—
253C4⋊C4—46no—
254C22⋊C4—212no—
255C22⋊C4—424no—
256C42—41no—
257C42—43no—
258C42—46no—
259C23—216no—
260C23—26no—
261C23—26no—
262C23—42no—
263C23—412no—
264C23—88no—
265C23C2×C22⋊C4×C127112yes—
266C23C23×C4×C12711yescharacteristic
267C23C22×D4×C12713yes—
268C23C22×D4×C12713yes—
269Q8—216no—
270Q8—224no—
271D4—248no—
272D4—224no—
273D4—224no—
274D4—224no—
275C2×C4—248no—
276C2×C4—26no—
277C2×C4—26no—
278C2×C4—224no—
279C2×C4—224no—
280C2×C4—212no—
281C2×C4—46no—
282C2×C4—42no—
283C2×C4—412no—
284C2×C4—46no—
285C2×C4C2×C22⋊C4×C127112yes—
286C2×C4C2×C22⋊C4×C12716yes—
287C2×C4C2×C22⋊C4×C12716yes—
288C2×C4#120×C12711yescharacteristic
289C2×C4#120×C12713yes—
290C2×C4#120×C12716yes—
291C2×C4#120×C12716yes—
292C22—26no—
293C22—212no—
294C22—41no—
295C22—43no—
296C22—412no—
297C22—412no—
298C22—86no—
299C22—816no—
300C22C22×C22⋊C4×C12713yes—
301C22C2×#120×C12713yes—
302C22C2×#120×C12711yescharacteristic
303C4—212no—
304C4—44no—
305C4—86no—
306C4#379×C12712yes—
307C4#379×C12716yes—
308C2—23no—
309C2—23no—
310C2—412no—
311C2C2×#379×C12711yescharacteristic
312C2C127×#302512yes—
subgroupG/H
centerC2×C4×C127#120
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
socleC22×C127C2×#120
prime radicalC2×#585×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#410×C127C2semidirect
3. C127×#2297C2semidirect
4. C127×#1790C2semidirect
6. #585×C127C4semidirect
12. C2×#117×C127C22semidirect
13. C22×C42×C127C22semidirect
15. C2×#99×C127C22semidirect
18. #386×C127C22semidirect
70. C2×Q8×C127C22⋊C4semidirect
79. C2×D4×C127C22⋊C4semidirect
90. C22×C4×C127C22⋊C4semidirect
151. #20608C127direct (N × H)
158. C2×#410C2×C127semidirect
159. #2297C2×C127semidirect
160. #1790C2×C127semidirect
161. C127#20608direct (N × H)
163. #585C4×C127semidirect
169. C2×#117C22×C127semidirect
170. C22×C42C22×C127semidirect
172. C2×#99C22×C127semidirect
175. #386C22×C127semidirect
227. C2×Q8C22⋊C4×C127semidirect
236. C2×D4C22⋊C4×C127semidirect
247. C22×C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#585×C1272
C2×#410×C1272
C127×#22972
C127×#17902
#20608127