Finite Group Databasebeta

← C127×#20610 #44584283

Subgroups — C127×#20610 #44584283

Order 32512.

3754 conjugacy classes of subgroups.

The subgroup lattice has 326 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×#2311C213yesmaximal, 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×C4⋊C4×C127C2211yescharacteristic, complemented
14#410×C127—212no—
15C2×#99×C127C22112yescomplemented
16#391×C127—23no—
17#390×C127—23no—
18#386×C127C22112yescomplemented
19#380×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×C4⋊C4×C127—23no—
49C2×C4⋊C4×C127—23no—
50C2×C22⋊C4×C127—212no—
51C2×C22⋊C4×C127C2316yes—
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—26no—
86C22×C4×C127—212no—
87C22×C4×C127—26no—
88C22×C4×C127—26no—
89C22×C4×C127—82no—
90C22×C4×C127C22⋊C4124yescomplemented
91C22×C4×C127C22×C412yes—
92C22×C4×C127C2×D416yes—
93C22×C4×C127C2×D413yes—
94C22×C4×C127C2×D413yes—
95C22×C4×C127C2×D416yes—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—43no—
98C4⋊C4×C127—46no—
99C22⋊C4×C127—212no—
100C22⋊C4×C127—812no—
101C42×C127—46no—
102C23×C127—216no—
103C23×C127—26no—
104C23×C127—26no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—46no—
108C23×C127—46no—
109C23×C127—84no—
110C23×C127—84no—
111C23×C127C2×C22⋊C4112yes—
112C23×C127C23×C411yescharacteristic
113C23×C127C22×D413yes—
114C23×C127C22×D413yes—
115Q8×C127—216no—
116Q8×C127—224no—
117D4×C127—248no—
118D4×C127—224no—
119D4×C127—224no—
120D4×C127—224no—
121C2×C4×C127—248no—
122C2×C4×C127—26no—
123C2×C4×C127—26no—
124C2×C4×C127—224no—
125C2×C4×C127—224no—
126C2×C4×C127—46no—
127C2×C4×C127—412no—
128C2×C4×C127—46no—
129C2×C4×C127—46no—
130C2×C4×C127—42no—
131C2×C4×C127C2×C22⋊C4112yes—
132C2×C4×C127C2×C22⋊C416yes—
133C2×C4×C127C2×C22⋊C416yes—
134C2×C4×C127#12016yes—
135C2×C4×C127#12016yes—
136C2×C4×C127#12011yescharacteristic
137C2×C4×C127#12013yes—
138C22×C127—23no—
139C22×C127—23no—
140C22×C127—212no—
141C22×C127—41no—
142C22×C127—43no—
143C22×C127—412no—
144C22×C127—46no—
145C22×C127—46no—
146C22×C127—86no—
147C22×C127—816no—
148C22×C127C22×C22⋊C413yes—
149C22×C127C2×#12013yes—
150C22×C127C2×#12011yescharacteristic
151C4×C127—212no—
152C4×C127—86no—
153C4×C127—82no—
154C4×C127#37912yes—
155C4×C127#37916yes—
156#20610C12711yesmaximal, characteristic, direct product factor, Sylow
157C2×C127—23no—
158C2×C127—23no—
159C2×C127—412no—
160C2×C127C2×#37911yescharacteristic
161C2×C127#302711yescharacteristic
162C2×C127#302811yescharacteristic
163C2×#585C2×C12711yescharacteristic
164C2×#410C2×C12713yescomplemented
165#2311C2×C12713yescomplemented
166#1790C2×C12718yescomplemented
167C127#2061011yescharacteristic, direct product factor, Sylow
168#585—212no—
169#585C4×C12718yescomplemented
170C2×2−1+4—23no—
171C2×H(5,𝔽2)—23no—
172C2×H(5,𝔽2)C22×C12714yes—
173C22×#41—26no—
174C22×#41C22×C12713yes—
175C2×#117C22×C12713yescomplemented
176C22×C4⋊C4C22×C12711yescharacteristic, complemented
177#410—212no—
178C2×#99C22×C127112yescomplemented
179#391—23no—
180#390—23no—
181#386C22×C127112yescomplemented
182#380—26no—
1832−1+4—224no—
1842−1+4—224no—
185H(5,𝔽2)—224no—
186H(5,𝔽2)—224no—
187H(5,𝔽2)—224no—
188H(5,𝔽2)C2×C4×C127116yes—
189C2×#41—24no—
190C2×#41—212no—
191C2×#41—248no—
192C2×#41—248no—
193C2×#41—26no—
194C2×#41—212no—
195C2×#41—26no—
196C2×#41—212no—
197C2×#41C2×C4×C127112yes—
198C2×#41D4×C127112yes—
199C22×Q8—26no—
200C22×Q8D4×C12713yes—
201C22×D4—26no—
202C22×D4—212no—
203C22×D4D4×C12713yes—
204C22×D4C23×C12716yes—
205C23×C4—24no—
206C23×C4D4×C12716yes—
207C23×C4C23×C12712yes—
208C23×C4C23×C12711yescharacteristic
209#117—212no—
210C2×C4⋊C4—23no—
211C2×C4⋊C4—23no—
212C2×C4⋊C4—23no—
213C2×C22⋊C4—212no—
214C2×C22⋊C4C23×C12716yes—
215C2×C42—23no—
216#99—224no—
217#95—212no—
218C24—24no—
219C24C2×D4×C12716yes—
220C24C24×C12711yescharacteristic
221#41—232no—
222#41—248no—
223#41—296no—
224#41—224no—
225#41—248no—
226#41—224no—
227#41—224no—
228#41—224no—
229C2×Q8—24no—
230C2×Q8—248no—
231C2×Q8—26no—
232C2×Q8—26no—
233C2×Q8C22⋊C4×C127112yescomplemented
234C2×D4—212no—
235C2×D4—248no—
236C2×D4—248no—
237C2×D4—248no—
238C2×D4—212no—
239C2×D4—212no—
240C2×D4—26no—
241C2×D4—26no—
242C2×D4C22⋊C4×C127112yescomplemented
243C2×D4C22×C4×C127112yes—
244C2×D4C2×D4×C127112yes—
245C22×C4—212no—
246C22×C4—232no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—212no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—82no—
253C22×C4C22⋊C4×C127124yescomplemented
254C22×C4C22×C4×C12712yes—
255C22×C4C2×D4×C12716yes—
256C22×C4C2×D4×C12713yes—
257C22×C4C2×D4×C12713yes—
258C22×C4C2×D4×C12716yes—
259C4⋊C4—41no—
260C4⋊C4—43no—
261C4⋊C4—46no—
262C22⋊C4—212no—
263C22⋊C4—812no—
264C42—46no—
265C23—216no—
266C23—26no—
267C23—26no—
268C23—41no—
269C23—41no—
270C23—46no—
271C23—46no—
272C23—84no—
273C23—84no—
274C23C2×C22⋊C4×C127112yes—
275C23C23×C4×C12711yescharacteristic
276C23C22×D4×C12713yes—
277C23C22×D4×C12713yes—
278Q8—216no—
279Q8—224no—
280D4—248no—
281D4—224no—
282D4—224no—
283D4—224no—
284C2×C4—248no—
285C2×C4—26no—
286C2×C4—26no—
287C2×C4—224no—
288C2×C4—224no—
289C2×C4—46no—
290C2×C4—412no—
291C2×C4—46no—
292C2×C4—46no—
293C2×C4—42no—
294C2×C4C2×C22⋊C4×C127112yes—
295C2×C4C2×C22⋊C4×C12716yes—
296C2×C4C2×C22⋊C4×C12716yes—
297C2×C4#120×C12716yes—
298C2×C4#120×C12711yescharacteristic
299C2×C4#120×C12713yes—
300C2×C4#120×C12716yes—
301C22—23no—
302C22—23no—
303C22—212no—
304C22—41no—
305C22—43no—
306C22—412no—
307C22—46no—
308C22—46no—
309C22—86no—
310C22—816no—
311C22C22×C22⋊C4×C12713yes—
312C22C2×#120×C12713yes—
313C22C2×#120×C12711yescharacteristic
314C4—212no—
315C4—86no—
316C4—82no—
317C4#379×C12712yes—
318C4#379×C12716yes—
319C2—23no—
320C2—23no—
321C2—412no—
322C2C2×#379×C12711yescharacteristic
323C2C127×#302711yescharacteristic
324C2C127×#302811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC2×#585×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×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×#2311C2semidirect
4. C127×#1790C2semidirect
6. #585×C127C4semidirect
12. C2×#117×C127C22semidirect
13. C22×C4⋊C4×C127C22semidirect
15. C2×#99×C127C22semidirect
18. #386×C127C22semidirect
70. C2×Q8×C127C22⋊C4semidirect
79. C2×D4×C127C22⋊C4semidirect
90. C22×C4×C127C22⋊C4semidirect
156. #20610C127direct (N × H)
164. C2×#410C2×C127semidirect
165. #2311C2×C127semidirect
166. #1790C2×C127semidirect
167. C127#20610direct (N × H)
169. #585C4×C127semidirect
175. C2×#117C22×C127semidirect
176. C22×C4⋊C4C22×C127semidirect
178. C2×#99C22×C127semidirect
181. #386C22×C127semidirect
233. C2×Q8C22⋊C4×C127semidirect
242. C2×D4C22⋊C4×C127semidirect
253. 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×#23112
C127×#17902
#20610127