Finite Group Databasebeta

← C127×#11533 #44575206

Subgroups — C127×#11533 #44575206

Order 32512.

504 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2328C211yesmaximal, characteristic
2C127×#1465C212yesmaximal, complemented
3C127×#1324C212yesmaximal, complemented
4C127×#1318C212yesmaximal, complemented
5C2×#127×C127C411yescharacteristic
6C2×C4×D4×C127C411yescharacteristic
7C2×C4×D4×C127C2211yescharacteristic
8#407×C127—22no—
9#390×C127C411yescharacteristic
10#390×C127C2211yescharacteristic
11#386×C127—22no—
12#386×C127—22no—
13#382×C127C2211yescharacteristic
14#376×C127C411yescharacteristic
15#324×C127C2214yes—
16#323×C127—22no—
17C22×D4×C127—22no—
18C22×D4×C127C2×C411yescharacteristic
19C22×D4×C127C2×C411yescharacteristic
20C23×C4×C127—21no—
21C23×C4×C127D412yes—
22C2×M16×C127—42no—
23#127×C127—22no—
24#127×C127—22no—
25C4×D4×C127—42no—
26C4×D4×C127—42no—
27C2×C4⋊C4×C127C2×C411yescharacteristic
28C2×C4⋊C4×C127C2311yescharacteristic
29C2×C22⋊C4×C127—21no—
30C2×C22⋊C4×C127—22no—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127D412yes—
33C2×C42×C127C2×C411yescharacteristic
34C2×C42×C127C2×C411yescharacteristic
35C2×C42×C127C2×C411yescharacteristic
36C22⋊C8×C127—24no—
37C22⋊C8×C127—42no—
38#95×C127—21no—
39#95×C127—21no—
40#95×C127—22no—
41C24×C127—21no—
42C24×C127C22⋊C412yes—
43C2×D4×C127—28no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51C2×D4×C127—82no—
52C2×D4×C127—82no—
53C2×D4×C127M1612yescomplemented
54C22×C4×C127—21no—
55C22×C4×C127—22no—
56C22×C4×C127—22no—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—42no—
60C22×C4×C127—82no—
61C22×C4×C127—82no—
62C22×C4×C127C22⋊C411yescharacteristic
63C22×C4×C127C22⋊C411yescharacteristic
64C22×C4×C127C22×C411yescharacteristic
65C22×C4×C127C2×D412yes—
66M16×C127—84no—
67C2×C8×C127—44no—
68C4⋊C4×C127—41no—
69C4⋊C4×C127—41no—
70C22⋊C4×C127—42no—
71C22⋊C4×C127—42no—
72C22⋊C4×C127—82no—
73C22⋊C4×C127—82no—
74C42×C127—22no—
75C42×C127—21no—
76C42×C127—41no—
77C42×C127M1612yescomplemented
78C23×C127—22no—
79C23×C127—22no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—42no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127—42no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127C2×C22⋊C411yescharacteristic
89D4×C127—42no—
90D4×C127—42no—
91D4×C127—42no—
92D4×C127—42no—
93D4×C127—88no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—21no—
98C2×C4×C127—22no—
99C2×C4×C127—41no—
100C2×C4×C127—44no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—42no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127—42no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—82no—
111C2×C4×C127—84no—
112C2×C4×C127#9912yes—
113C2×C4×C127C2×M1611yescharacteristic
114C2×C4×C127C2×M1611yescharacteristic
115C8×C127—84no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127—44no—
119C22×C127—44no—
120C22×C127—42no—
121C22×C127—42no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—82no—
125C22×C127—82no—
126C22×C127—81no—
127C22×C127—81no—
128C22×C127#40711yescharacteristic
129C22×C127C2×#9911yescharacteristic
130C22×C127#41311yescharacteristic
131C4×C127—22no—
132C4×C127—21no—
133C4×C127—21no—
134C4×C127—42no—
135C4×C127—42no—
136C4×C127—41no—
137C4×C127—81no—
138C4×C127—81no—
139#11533C12711yesmaximal, characteristic, direct product factor, Sylow
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—82no—
143C2×C127—82no—
144C2×C127#146511yescharacteristic
145C2×C127#212711yescharacteristic
146C2×C127#213411yescharacteristic
147#2328C2×C12711yescharacteristic
148#1465C2×C12712yescomplemented
149#1324C2×C12712yescomplemented
150#1318C2×C12712yescomplemented
151C127#1153311yescharacteristic, direct product factor, Sylow
152C2×#127C4×C12711yescharacteristic
153C2×C4×D4C4×C12711yescharacteristic
154C2×C4×D4C22×C12711yescharacteristic
155#407—22no—
156#390C4×C12711yescharacteristic
157#390C22×C12711yescharacteristic
158#386—22no—
159#386—22no—
160#382C22×C12711yescharacteristic
161#376C4×C12711yescharacteristic
162#324C22×C12714yes—
163#323—22no—
164C22×D4—22no—
165C22×D4C2×C4×C12711yescharacteristic
166C22×D4C2×C4×C12711yescharacteristic
167C23×C4—21no—
168C23×C4D4×C12712yes—
169C2×M16—42no—
170#127—22no—
171#127—22no—
172C4×D4—42no—
173C4×D4—42no—
174C2×C4⋊C4C2×C4×C12711yescharacteristic
175C2×C4⋊C4C23×C12711yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4D4×C12712yes—
180C2×C42C2×C4×C12711yescharacteristic
181C2×C42C2×C4×C12711yescharacteristic
182C2×C42C2×C4×C12711yescharacteristic
183C22⋊C8—24no—
184C22⋊C8—42no—
185#95—21no—
186#95—21no—
187#95—22no—
188C24—21no—
189C24C22⋊C4×C12712yes—
190C2×D4—28no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—82no—
199C2×D4—82no—
200C2×D4M16×C12712yescomplemented
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4C22⋊C4×C12711yescharacteristic
210C22×C4C22⋊C4×C12711yescharacteristic
211C22×C4C22×C4×C12711yescharacteristic
212C22×C4C2×D4×C12712yes—
213M16—84no—
214C2×C8—44no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—22no—
222C42—21no—
223C42—41no—
224C42M16×C12712yescomplemented
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊C4×C12711yescharacteristic
236D4—42no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—88no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4#99×C12712yes—
260C2×C4C2×M16×C12711yescharacteristic
261C2×C4C2×M16×C12711yescharacteristic
262C8—84no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22—81no—
274C22—81no—
275C22#407×C12711yescharacteristic
276C22C2×#99×C12711yescharacteristic
277C22#413×C12711yescharacteristic
278C4—22no—
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—41no—
284C4—81no—
285C4—81no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—82no—
290C2C127×#146511yescharacteristic
291C2C127×#212711yescharacteristic
292C2C127×#213411yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC2×#127×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1465C2semidirect
3. C127×#1324C2semidirect
4. C127×#1318C2semidirect
53. C2×D4×C127M16semidirect
77. C42×C127M16semidirect
139. #11533C127direct (N × H)
148. #1465C2×C127semidirect
149. #1324C2×C127semidirect
150. #1318C2×C127semidirect
151. C127#11533direct (N × H)
200. C2×D4M16×C127semidirect
224. C42M16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23282
C127×#14652
C127×#13242
C127×#13182
#11533127