Finite Group Databasebeta

← C127×#20710 #44584383

Subgroups — C127×#20710 #44584383

Order 32512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3428C211yesmaximal, characteristic
2C2×#410×C127C212yesmaximal
3C127×#1828C218yesmaximal, complemented
4C127×#1740C214yesmaximal, complemented
5C22×#41×C127C2211yescharacteristic
6#520×C127—24no—
7#519×C127C412yescomplemented
8#518×C127C412yescomplemented
9#518×C127C2214yes—
10C4×#41×C127—24no—
11C2×#117×C127C414yescomplemented
12C2×#117×C127C2214yes—
13C2×#117×C127C2212yes—
14#410×C127—24no—
15#410×C127—24no—
16C2×#99×C127C2218yes—
17#342×C127C22116yes—
18C2×#41×C127—24no—
19C2×#41×C127—24no—
20C2×#41×C127D414yes—
21C2×#41×C127D414yes—
22C22×Q8×C127C2×C411yescharacteristic
23C22×D4×C127C2×C411yescharacteristic
24C22×D4×C127C2312yes—
25C23×C4×C127C2×C412yes—
26C23×C4×C127C2311yescharacteristic
27C4×Q8×C127—28no—
28C4×D4×C127—28no—
29C4×D4×C127—216no—
30#117×C127—24no—
31#117×C127—24no—
32#117×C127—22no—
33#117×C127—22no—
34#117×C127—24no—
35#117×C127—24no—
36#117×C127—24no—
37#117×C127—24no—
38#117×C127C2×C414yes—
39#117×C127C2×C418yes—
40#117×C127C2×C414yes—
41C2×C4⋊C4×C127—24no—
42C2×C4⋊C4×C127—22no—
43C2×C4⋊C4×C127—24no—
44C2×C22⋊C4×C127C2×C418yes—
45C2×C22⋊C4×C127C2318yes—
46C2×C22⋊C4×C127C2314yes—
47C2×C42×C127—24no—
48C2×C42×C127—22no—
49C2×C42×C127—24no—
50#99×C127—216no—
51C24×C127C22×C412yes—
52C24×C127C2411yescharacteristic
53#41×C127—28no—
54#41×C127—28no—
55#41×C127—28no—
56#41×C127—28no—
57C2×Q8×C127—24no—
58C2×Q8×C127C22⋊C414yes—
59C2×D4×C127—28no—
60C2×D4×C127—24no—
61C2×D4×C127C22⋊C414yes—
62C2×D4×C127C2×D418yes—
63C22×C4×C127—24no—
64C22×C4×C127—24no—
65C22×C4×C127—28no—
66C22×C4×C127—24no—
67C22×C4×C127—28no—
68C22×C4×C127C22⋊C414yes—
69C22×C4×C127C22⋊C414yes—
70C22×C4×C127C22×C412yes—
71C22×C4×C127C22×C411yescharacteristic
72C22×C4×C127C22×C411yescharacteristic
73C22×C4×C127C2×D412yes—
74C22×C4×C127C2×D412yes—
75C22×C4×C127#4114yes—
76C22×C4×C127#4114yes—
77C4⋊C4×C127—24no—
78C4⋊C4×C127—28no—
79C4⋊C4×C127—24no—
80C4⋊C4×C127—44no—
81C4⋊C4×C127—44no—
82C22⋊C4×C127—28no—
83C22⋊C4×C127—24no—
84C22⋊C4×C127—28no—
85C22⋊C4×C127—28no—
86C22⋊C4×C127C22×C418yes—
87C42×C127—24no—
88C42×C127—28no—
89C42×C127—24no—
90C42×C127—44no—
91C42×C127—44no—
92C23×C127—24no—
93C23×C127—81no—
94C23×C127—82no—
95C23×C127C2×C22⋊C414yes—
96C23×C127C23×C411yescharacteristic
97C23×C127C22×D412yes—
98C23×C127C2×#4114yes—
99Q8×C127—28no—
100D4×C127—216no—
101D4×C127—28no—
102C2×C4×C127—24no—
103C2×C4×C127—24no—
104C2×C4×C127—28no—
105C2×C4×C127—24no—
106C2×C4×C127—28no—
107C2×C4×C127—48no—
108C2×C4×C127—84no—
109C2×C4×C127—88no—
110C2×C4×C127—88no—
111C2×C4×C127C2×C22⋊C414yes—
112C2×C4×C127C2×C22⋊C412yes—
113C2×C4×C127C2×C22⋊C411yescharacteristic
114C2×C4×C127C2×C22⋊C411yescharacteristic
115C2×C4×C127#11712yes—
116C2×C4×C127#11714yes—
117C2×C4×C127#11712yes—
118C2×C4×C127#12314yes—
119C2×C4×C127#12314yes—
120C2×C4×C127#12314yes—
121C2×C4×C127#12314yes—
122C22×C127—24no—
123C22×C127—42no—
124C22×C127—44no—
125C22×C127—41no—
126C22×C127—44no—
127C22×C127—84no—
128C22×C127C22×C22⋊C411yescharacteristic
129C22×C127C2×#11712yes—
130C22×C127C2×#12314yes—
131C4×C127—24no—
132C4×C127—48no—
133C4×C127—88no—
134C4×C127#38112yes—
135C4×C127#38111yescharacteristic
136C4×C127#38114yes—
137C4×C127#38111yescharacteristic
138#20710C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—21no—
140C2×C127—24no—
141C2×C127—22no—
142C2×C127—44no—
143C2×C127C2×#38111yescharacteristic
144#3428C2×C12711yescharacteristic
145C2×#410C2×C12712yes—
146#1828C2×C12718yescomplemented
147#1740C2×C12714yescomplemented
148C127#2071011yescharacteristic, direct product factor, Sylow
149C22×#41C22×C12711yescharacteristic
150#520—24no—
151#519C4×C12712yescomplemented
152#518C4×C12712yescomplemented
153#518C22×C12714yes—
154C4×#41—24no—
155C2×#117C4×C12714yescomplemented
156C2×#117C22×C12714yes—
157C2×#117C22×C12712yes—
158#410—24no—
159#410—24no—
160C2×#99C22×C12718yes—
161#342C22×C127116yes—
162C2×#41—24no—
163C2×#41—24no—
164C2×#41D4×C12714yes—
165C2×#41D4×C12714yes—
166C22×Q8C2×C4×C12711yescharacteristic
167C22×D4C2×C4×C12711yescharacteristic
168C22×D4C23×C12712yes—
169C23×C4C2×C4×C12712yes—
170C23×C4C23×C12711yescharacteristic
171C4×Q8—28no—
172C4×D4—28no—
173C4×D4—216no—
174#117—24no—
175#117—24no—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117C2×C4×C12714yes—
183#117C2×C4×C12718yes—
184#117C2×C4×C12714yes—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C22⋊C4C2×C4×C12718yes—
189C2×C22⋊C4C23×C12718yes—
190C2×C22⋊C4C23×C12714yes—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—24no—
194#99—216no—
195C24C22×C4×C12712yes—
196C24C24×C12711yescharacteristic
197#41—28no—
198#41—28no—
199#41—28no—
200#41—28no—
201C2×Q8—24no—
202C2×Q8C22⋊C4×C12714yes—
203C2×D4—28no—
204C2×D4—24no—
205C2×D4C22⋊C4×C12714yes—
206C2×D4C2×D4×C12718yes—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4C22⋊C4×C12714yes—
213C22×C4C22⋊C4×C12714yes—
214C22×C4C22×C4×C12712yes—
215C22×C4C22×C4×C12711yescharacteristic
216C22×C4C22×C4×C12711yescharacteristic
217C22×C4C2×D4×C12712yes—
218C22×C4C2×D4×C12712yes—
219C22×C4#41×C12714yes—
220C22×C4#41×C12714yes—
221C4⋊C4—24no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C22⋊C4—28no—
227C22⋊C4—24no—
228C22⋊C4—28no—
229C22⋊C4—28no—
230C22⋊C4C22×C4×C12718yes—
231C42—24no—
232C42—28no—
233C42—24no—
234C42—44no—
235C42—44no—
236C23—24no—
237C23—81no—
238C23—82no—
239C23C2×C22⋊C4×C12714yes—
240C23C23×C4×C12711yescharacteristic
241C23C22×D4×C12712yes—
242C23C2×#41×C12714yes—
243Q8—28no—
244D4—216no—
245D4—28no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—28no—
249C2×C4—24no—
250C2×C4—28no—
251C2×C4—48no—
252C2×C4—84no—
253C2×C4—88no—
254C2×C4—88no—
255C2×C4C2×C22⋊C4×C12714yes—
256C2×C4C2×C22⋊C4×C12711yescharacteristic
257C2×C4C2×C22⋊C4×C12712yes—
258C2×C4C2×C22⋊C4×C12711yescharacteristic
259C2×C4#117×C12712yes—
260C2×C4#117×C12712yes—
261C2×C4#117×C12714yes—
262C2×C4#123×C12714yes—
263C2×C4#123×C12714yes—
264C2×C4#123×C12714yes—
265C2×C4#123×C12714yes—
266C22—24no—
267C22—42no—
268C22—44no—
269C22—41no—
270C22—44no—
271C22—84no—
272C22C22×C22⋊C4×C12711yescharacteristic
273C22C2×#117×C12712yes—
274C22C2×#123×C12714yes—
275C4—24no—
276C4—48no—
277C4—88no—
278C4#381×C12711yescharacteristic
279C4#381×C12712yes—
280C4#381×C12711yescharacteristic
281C4#381×C12714yes—
282C2—21no—
283C2—24no—
284C2—22no—
285C2—44no—
286C2C2×#381×C12711yescharacteristic
subgroupG/H
centerC4×C127#381
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
socleC2×C127C2×#381
prime radicalC22×#41×C127C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1828C2semidirect
4. C127×#1740C2semidirect
7. #519×C127C4semidirect
8. #518×C127C4semidirect
11. C2×#117×C127C4semidirect
138. #20710C127direct (N × H)
146. #1828C2×C127semidirect
147. #1740C2×C127semidirect
148. C127#20710direct (N × H)
151. #519C4×C127semidirect
152. #518C4×C127semidirect
155. C2×#117C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34282
C2×#410×C1272
C127×#18282
C127×#17402
#20710127