Finite Group Databasebeta

← C127×#12632 #44576305

Subgroups — C127×#12632 #44576305

Order 32512.

512 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2326C211yesmaximal, characteristic
2C2×#355×C127C212yesmaximal
3C4×#101×C127C212yesmaximal, complemented
4C127×#1394C212yesmaximal, complemented
5C2×#124×C127C2211yescharacteristic
6C2×C4×Q8×C127C2211yescharacteristic
7C2×C4×D4×C127C411yescharacteristic
8C2×#101×C127C2214yes—
9C4×M16×C127—22no—
10#391×C127C2211yescharacteristic
11#390×C127C411yescharacteristic
12#388×C127—22no—
13#385×C127—22no—
14#384×C127C411yescharacteristic
15#376×C127C411yescharacteristic
16#355×C127C418yescomplemented
17#343×C127—22no—
18C22×Q8×C127C2311yescharacteristic
19C22×D4×C127C2×C411yescharacteristic
20C23×C4×C127—21no—
21C2×M16×C127—24no—
22#124×C127—22no—
23#124×C127C2×C414yes—
24C4×Q8×C127—42no—
25C4×D4×C127—42no—
26C2×C4⋊C4×C127—22no—
27C2×C4⋊C4×C127C2×C411yescharacteristic
28C2×C4⋊C4×C127C2×C411yescharacteristic
29C2×C4⋊C4×C127D412yes—
30C2×C22⋊C4×C127—22no—
31C2×C22⋊C4×C127—21no—
32C2×C22⋊C4×C127—22no—
33C2×C42×C127C2×C411yescharacteristic
34C2×C42×C127C2×C411yescharacteristic
35C2×C42×C127C2×C411yescharacteristic
36C2×C42×C127D412yes—
37#101×C127—24no—
38#101×C127C2×C418yescomplemented
39#97×C127—42no—
40C4×C8×C127—42no—
41#95×C127—21no—
42#95×C127—21no—
43#95×C127—22no—
44C24×C127—21no—
45C2×Q8×C127—41no—
46C2×Q8×C127—41no—
47C2×Q8×C127C22×C412yes—
48C2×Q8×C127#4112yes—
49C2×D4×C127—21no—
50C2×D4×C127—41no—
51C2×D4×C127—41no—
52C2×D4×C127C4212yes—
53C22×C4×C127—21no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—42no—
57C22×C4×C127—82no—
58C22×C4×C127C22⋊C411yescharacteristic
59C22×C4×C127C22⋊C412yes—
60C22×C4×C127C22⋊C411yescharacteristic
61C22×C4×C127C22×C411yescharacteristic
62C22×C4×C127C2×D412yes—
63M16×C127—28no—
64M16×C127—44no—
65C2×C8×C127—44no—
66C4⋊C4×C127—41no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127—41no—
69C4⋊C4×C127—82no—
70C22⋊C4×C127—28no—
71C22⋊C4×C127—42no—
72C22⋊C4×C127—82no—
73C42×C127—22no—
74C42×C127—22no—
75C42×C127—22no—
76C42×C127—21no—
77C42×C127—41no—
78C42×C127C4212yes—
79C23×C127—22no—
80C23×C127—42no—
81C23×C127—42no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127C2×C22⋊C411yescharacteristic
85Q8×C127—42no—
86Q8×C127—42no—
87D4×C127—42no—
88D4×C127—42no—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—21no—
95C2×C4×C127—22no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127—42no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—42no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—84no—
107C2×C4×C127#9912yes—
108C2×C4×C127C2×C4211yescharacteristic
109C2×C4×C127#11711yescharacteristic
110C2×C4×C127C4×D414yes—
111C8×C127—48no—
112C22×C127—21no—
113C22×C127—21no—
114C22×C127—44no—
115C22×C127—42no—
116C22×C127—82no—
117C22×C127—81no—
118C22×C127—81no—
119C22×C127C4×C22⋊C411yescharacteristic
120C22×C127C2×#9911yescharacteristic
121C22×C127#41011yescharacteristic
122C4×C127—22no—
123C4×C127—24no—
124C4×C127—21no—
125C4×C127—21no—
126C4×C127—42no—
127C4×C127—42no—
128C4×C127—41no—
129C4×C127—81no—
130C4×C127—81no—
131#12632C12711yesmaximal, characteristic, direct product factor, Sylow
132C2×C127—21no—
133C2×C127—21no—
134C2×C127—82no—
135C2×C127C4×#9911yescharacteristic
136C2×C127#213311yescharacteristic
137C2×C127#213411yescharacteristic
138#2326C2×C12711yescharacteristic
139C2×#355C2×C12712yes—
140C4×#101C2×C12712yescomplemented
141#1394C2×C12712yescomplemented
142C127#1263211yescharacteristic, direct product factor, Sylow
143C2×#124C22×C12711yescharacteristic
144C2×C4×Q8C22×C12711yescharacteristic
145C2×C4×D4C4×C12711yescharacteristic
146C2×#101C22×C12714yes—
147C4×M16—22no—
148#391C22×C12711yescharacteristic
149#390C4×C12711yescharacteristic
150#388—22no—
151#385—22no—
152#384C4×C12711yescharacteristic
153#376C4×C12711yescharacteristic
154#355C4×C12718yescomplemented
155#343—22no—
156C22×Q8C23×C12711yescharacteristic
157C22×D4C2×C4×C12711yescharacteristic
158C23×C4—21no—
159C2×M16—24no—
160#124—22no—
161#124C2×C4×C12714yes—
162C4×Q8—42no—
163C4×D4—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×C4×C12711yescharacteristic
166C2×C4⋊C4C2×C4×C12711yescharacteristic
167C2×C4⋊C4D4×C12712yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C42C2×C4×C12711yescharacteristic
172C2×C42C2×C4×C12711yescharacteristic
173C2×C42C2×C4×C12711yescharacteristic
174C2×C42D4×C12712yes—
175#101—24no—
176#101C2×C4×C12718yescomplemented
177#97—42no—
178C4×C8—42no—
179#95—21no—
180#95—21no—
181#95—22no—
182C24—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C22×C4×C12712yes—
186C2×Q8#41×C12712yes—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C42×C12712yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—82no—
196C22×C4C22⋊C4×C12711yescharacteristic
197C22×C4C22⋊C4×C12712yes—
198C22×C4C22⋊C4×C12711yescharacteristic
199C22×C4C22×C4×C12711yescharacteristic
200C22×C4C2×D4×C12712yes—
201M16—28no—
202M16—44no—
203C2×C8—44no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—41no—
216C42C42×C12712yes—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊C4×C12711yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4#99×C12712yes—
246C2×C4C2×C42×C12711yescharacteristic
247C2×C4#117×C12711yescharacteristic
248C2×C4C4×D4×C12714yes—
249C8—48no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊C4×C12711yescharacteristic
258C22C2×#99×C12711yescharacteristic
259C22#410×C12711yescharacteristic
260C4—22no—
261C4—24no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—41no—
266C4—42no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#99×C12711yescharacteristic
273C2C127×#213311yescharacteristic
274C2C127×#213411yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC2×C4C2×C42×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#101×C127C2semidirect
4. C127×#1394C2semidirect
16. #355×C127C4semidirect
38. #101×C127C2×C4semidirect
131. #12632C127direct (N × H)
140. C4×#101C2×C127semidirect
141. #1394C2×C127semidirect
142. C127#12632direct (N × H)
154. #355C4×C127semidirect
176. #101C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23262
C2×#355×C1272
C4×#101×C1272
C127×#13942
#12632127