Finite Group Databasebeta

← C127×#11534 #44575207

Subgroups — C127×#11534 #44575207

Order 32512.

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

normal subgroups only

This group has 424 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×#2326C211yesmaximal, characteristic
2C127×#1467C212yesmaximal, complemented
3C127×#1325C212yesmaximal, complemented
4C127×#1318C212yesmaximal
5C2×#124×C127C411yescharacteristic
6C2×C4×Q8×C127C2211yescharacteristic
7C2×C4×D4×C127C411yescharacteristic
8#432×C127—22no—
9#391×C127C411yescharacteristic
10#390×C127C2211yescharacteristic
11#388×C127—22no—
12#385×C127—22no—
13#384×C127C2211yescharacteristic
14#376×C127C411yescharacteristic
15#324×C127—22no—
16#324×C127C2214yes—
17C22×Q8×C127C2×C411yescharacteristic
18C22×D4×C127C2×C411yescharacteristic
19C23×C4×C127—21no—
20#124×C127—22no—
21#124×C127—22no—
22C4×Q8×C127—42no—
23C4×D4×C127—42no—
24C2×C4⋊C4×C127—22no—
25C2×C4⋊C4×C127C2×C411yescharacteristic
26C2×C4⋊C4×C127D412yes—
27C2×C4⋊C4×C127C2311yescharacteristic
28C2×C22⋊C4×C127—22no—
29C2×C22⋊C4×C127—21no—
30C2×C22⋊C4×C127—22no—
31C2×C42×C127C2×C411yescharacteristic
32C2×C42×C127C2×C411yescharacteristic
33C2×C42×C127C2×C411yescharacteristic
34C2×C42×C127D412yes—
35C4⋊C8×C127—42no—
36C22⋊C8×C127—24no—
37#97×C127—42no—
38#95×C127—21no—
39#95×C127—22no—
40#95×C127—21no—
41C24×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—41no—
45C2×Q8×C127—41no—
46C2×D4×C127—21no—
47C2×D4×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127M1612yes—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127—42no—
54C22×C4×C127—82no—
55C22×C4×C127C22⋊C412yes—
56C22×C4×C127C22⋊C411yescharacteristic
57C22×C4×C127C22⋊C411yescharacteristic
58C22×C4×C127C22×C411yescharacteristic
59C22×C4×C127C2×D412yes—
60C2×C8×C127—44no—
61C4⋊C4×C127—42no—
62C4⋊C4×C127—41no—
63C4⋊C4×C127—41no—
64C4⋊C4×C127—82no—
65C22⋊C4×C127—28no—
66C22⋊C4×C127—42no—
67C22⋊C4×C127—82no—
68C42×C127—22no—
69C42×C127—22no—
70C42×C127—22no—
71C42×C127—21no—
72C42×C127—41no—
73C42×C127M1612yes—
74C23×C127—22no—
75C23×C127—42no—
76C23×C127—42no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127C2×C22⋊C411yescharacteristic
80Q8×C127—42no—
81Q8×C127—42no—
82D4×C127—42no—
83D4×C127—42no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—42no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127—42no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—84no—
103C2×C4×C127#9912yes—
104C2×C4×C127C2×M1611yescharacteristic
105C2×C4×C127C2×M1611yescharacteristic
106C8×C127—84no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127—44no—
110C22×C127—42no—
111C22×C127—82no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127#40711yescharacteristic
115C22×C127C2×#9911yescharacteristic
116C22×C127#41311yescharacteristic
117C4×C127—22no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127—42no—
121C4×C127—42no—
122C4×C127—41no—
123C4×C127—42no—
124C4×C127—81no—
125C4×C127—81no—
126#11534C12711yesmaximal, characteristic, direct product factor, Sylow
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—82no—
130C2×C127#146511yescharacteristic
131C2×C127#212811yescharacteristic
132C2×C127#213311yescharacteristic
133#2326C2×C12711yescharacteristic
134#1467C2×C12712yescomplemented
135#1325C2×C12712yescomplemented
136#1318C2×C12712yes—
137C127#1153411yescharacteristic, direct product factor, Sylow
138C2×#124C4×C12711yescharacteristic
139C2×C4×Q8C22×C12711yescharacteristic
140C2×C4×D4C4×C12711yescharacteristic
141#432—22no—
142#391C4×C12711yescharacteristic
143#390C22×C12711yescharacteristic
144#388—22no—
145#385—22no—
146#384C22×C12711yescharacteristic
147#376C4×C12711yescharacteristic
148#324—22no—
149#324C22×C12714yes—
150C22×Q8C2×C4×C12711yescharacteristic
151C22×D4C2×C4×C12711yescharacteristic
152C23×C4—21no—
153#124—22no—
154#124—22no—
155C4×Q8—42no—
156C4×D4—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×C4×C12711yescharacteristic
159C2×C4⋊C4D4×C12712yes—
160C2×C4⋊C4C23×C12711yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42C2×C4×C12711yescharacteristic
165C2×C42C2×C4×C12711yescharacteristic
166C2×C42C2×C4×C12711yescharacteristic
167C2×C42D4×C12712yes—
168C4⋊C8—42no—
169C22⋊C8—24no—
170#97—42no—
171#95—21no—
172#95—21no—
173#95—22no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4M16×C12712yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊C4×C12712yes—
189C22×C4C22⋊C4×C12711yescharacteristic
190C22×C4C22⋊C4×C12711yescharacteristic
191C22×C4C22×C4×C12711yescharacteristic
192C22×C4C2×D4×C12712yes—
193C2×C8—44no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C22⋊C4—28no—
199C22⋊C4—42no—
200C22⋊C4—82no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42M16×C12712yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊C4×C12711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—84no—
236C2×C4#99×C12712yes—
237C2×C4C2×M16×C12711yescharacteristic
238C2×C4C2×M16×C12711yescharacteristic
239C8—84no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#407×C12711yescharacteristic
248C22C2×#99×C12711yescharacteristic
249C22#413×C12711yescharacteristic
250C4—22no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2C127×#146511yescharacteristic
263C2C127×#212811yescharacteristic
264C2C127×#213311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C22×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
2. C127×#1467C2semidirect
3. C127×#1325C2semidirect
126. #11534C127direct (N × H)
134. #1467C2×C127semidirect
135. #1325C2×C127semidirect
137. C127#11534direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#23262
C127×#14672
C127×#13252
C127×#13182
#11534127