Finite Group Databasebeta

← C127×#11448 #44575121

Subgroups — C127×#11448 #44575121

Order 32512.

454 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2317C211yesmaximal, characteristic
2C2×#324×C127C212yesmaximal
3C127×#1325C212yesmaximal, complemented
4C127×#1324C212yesmaximal, complemented
5C2×#117×C127C411yescharacteristic
6C2×#117×C127C411yescharacteristic
7C22×C4⋊C4×C127C2211yescharacteristic
8C2×C22⋊C8×C127—22no—
9#389×C127—22no—
10#385×C127—22no—
11#384×C127C411yescharacteristic
12#384×C127C2211yescharacteristic
13#382×C127C411yescharacteristic
14#382×C127C2211yescharacteristic
15#324×C127—22no—
16#324×C127C2214yes—
17C23×C4×C127C2×C411yescharacteristic
18C23×C4×C127D412yes—
19C22×C8×C127—42no—
20#117×C127—22no—
21#117×C127—42no—
22#117×C127C814yescomplemented
23C2×C4⋊C4×C127—21no—
24C2×C4⋊C4×C127—22no—
25C2×C4⋊C4×C127—22no—
26C2×C4⋊C4×C127—42no—
27C2×C4⋊C4×C127C2×C411yescharacteristic
28C2×C4⋊C4×C127C2×C411yescharacteristic
29C2×C4⋊C4×C127C2×C411yescharacteristic
30C2×C4⋊C4×C127D412yes—
31C2×C4⋊C4×C127C2311yescharacteristic
32C2×C22⋊C4×C127—21no—
33C2×C22⋊C4×C127—21no—
34C2×C42×C127—21no—
35C2×C42×C127C2×C411yescharacteristic
36C2×C42×C127C2×C411yescharacteristic
37C22⋊C8×C127—24no—
38C22⋊C8×C127—42no—
39#95×C127—22no—
40#95×C127—22no—
41C24×C127C22⋊C411yescharacteristic
42C22×C4×C127—22no—
43C22×C4×C127—22no—
44C22×C4×C127—22no—
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127—22no—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—82no—
51C22×C4×C127C22⋊C411yescharacteristic
52C22×C4×C127C22⋊C412yes—
53C22×C4×C127C2×C812yes—
54C22×C4×C127M1612yes—
55C22×C4×C127C22×C411yescharacteristic
56C22×C4×C127C2×D412yes—
57C2×C8×C127—44no—
58C2×C8×C127—84no—
59C4⋊C4×C127—24no—
60C4⋊C4×C127—41no—
61C4⋊C4×C127—42no—
62C4⋊C4×C127—41no—
63C4⋊C4×C127—41no—
64C4⋊C4×C127—41no—
65C4⋊C4×C127—82no—
66C4⋊C4×C127—82no—
67C22⋊C4×C127—24no—
68C22⋊C4×C127—42no—
69C42×C127—21no—
70C42×C127—21no—
71C42×C127—42no—
72C42×C127C2×C812yescomplemented
73C42×C127C2×C812yescomplemented
74C23×C127—41no—
75C23×C127—41no—
76C23×C127C22⋊C812yes—
77C23×C127#9912yes—
78C23×C127#10112yes—
79C23×C127C2×C22⋊C411yescharacteristic
80C2×C4×C127—24no—
81C2×C4×C127—24no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—44no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—42no—
96C2×C4×C127—42no—
97C2×C4×C127—42no—
98C2×C4×C127—42no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—82no—
102C2×C4×C127C22⋊C812yes—
103C2×C4×C127C22×C811yescharacteristic
104C2×C4×C127C2×M1611yescharacteristic
105C8×C127—84no—
106C22×C127—21no—
107C22×C127—22no—
108C22×C127—21no—
109C22×C127—22no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127#32414yes—
113C22×C127C2×C22⋊C811yescharacteristic
114C22×C127C2×#9911yescharacteristic
115C22×C127C2×#10111yescharacteristic
116C4×C127—22no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—42no—
120C4×C127—42no—
121C4×C127—42no—
122C4×C127—42no—
123C4×C127—81no—
124C4×C127—81no—
125#11448C12711yesmaximal, characteristic, direct product factor, Sylow
126C2×C127—21no—
127C2×C127—24no—
128C2×C127—21no—
129C2×C127C2×#32411yescharacteristic
130C2×C127#212711yescharacteristic
131C2×C127#213311yescharacteristic
132#2317C2×C12711yescharacteristic
133C2×#324C2×C12712yes—
134#1325C2×C12712yescomplemented
135#1324C2×C12712yescomplemented
136C127#1144811yescharacteristic, direct product factor, Sylow
137C2×#117C4×C12711yescharacteristic
138C2×#117C4×C12711yescharacteristic
139C22×C4⋊C4C22×C12711yescharacteristic
140C2×C22⋊C8—22no—
141#389—22no—
142#385—22no—
143#384C4×C12711yescharacteristic
144#384C22×C12711yescharacteristic
145#382C4×C12711yescharacteristic
146#382C22×C12711yescharacteristic
147#324—22no—
148#324C22×C12714yes—
149C23×C4C2×C4×C12711yescharacteristic
150C23×C4D4×C12712yes—
151C22×C8—42no—
152#117—22no—
153#117—42no—
154#117C8×C12714yescomplemented
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×C4×C12711yescharacteristic
160C2×C4⋊C4C2×C4×C12711yescharacteristic
161C2×C4⋊C4C2×C4×C12711yescharacteristic
162C2×C4⋊C4D4×C12712yes—
163C2×C4⋊C4C23×C12711yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42C2×C4×C12711yescharacteristic
168C2×C42C2×C4×C12711yescharacteristic
169C22⋊C8—24no—
170C22⋊C8—42no—
171#95—22no—
172#95—22no—
173C24C22⋊C4×C12711yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—82no—
183C22×C4C22⋊C4×C12711yescharacteristic
184C22×C4C22⋊C4×C12712yes—
185C22×C4C2×C8×C12712yes—
186C22×C4M16×C12712yes—
187C22×C4C22×C4×C12711yescharacteristic
188C22×C4C2×D4×C12712yes—
189C2×C8—44no—
190C2×C8—84no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42C2×C8×C12712yescomplemented
205C42C2×C8×C12712yescomplemented
206C23—41no—
207C23—41no—
208C23C22⋊C8×C12712yes—
209C23#99×C12712yes—
210C23#101×C12712yes—
211C23C2×C22⋊C4×C12711yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4C22⋊C8×C12712yes—
235C2×C4C22×C8×C12711yescharacteristic
236C2×C4C2×M16×C12711yescharacteristic
237C8—84no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22#324×C12714yes—
245C22C2×C22⋊C8×C12711yescharacteristic
246C22C2×#99×C12711yescharacteristic
247C22C2×#101×C12711yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—24no—
259C2—21no—
260C2C2×#324×C12711yescharacteristic
261C2C127×#212711yescharacteristic
262C2C127×#213311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC2×C4C22×C8×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1325C2semidirect
4. C127×#1324C2semidirect
22. #117×C127C8semidirect
72. C42×C127C2×C8semidirect
73. C42×C127C2×C8semidirect
125. #11448C127direct (N × H)
134. #1325C2×C127semidirect
135. #1324C2×C127semidirect
136. C127#11448direct (N × H)
154. #117C8×C127semidirect
204. C42C2×C8×C127semidirect
205. C42C2×C8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23172
C2×#324×C1272
C127×#13252
C127×#13242
#11448127