Finite Group Databasebeta

← C4×#356×C127 #44576303

Subgroups — C4×#356×C127 #44576303

Order 32512.

500 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C4×C4⋊Q8×C127C211yesmaximal, characteristic
2C2×#356×C127C212yesmaximal
3C4×#101×C127C212yesmaximal
4C127×#1394C212yesmaximal
5C2×C4⋊Q8×C127C2211yescharacteristic
6C2×C4×Q8×C127C411yescharacteristic
7C2×C4×Q8×C127C2211yescharacteristic
8C2×#101×C127C2214yes—
9C4×M16×C127—22no—
10#391×C127C411yescharacteristic
11#391×C127C2211yescharacteristic
12#389×C127—22no—
13#384×C127C411yescharacteristic
14C4×C4⋊C4×C127—22no—
15C43×C127C411yescharacteristic
16#356×C127C418yesdirect product factor
17#343×C127—22no—
18C22×Q8×C127C2×C411yescharacteristic
19C22×Q8×C127C2311yescharacteristic
20C2×M16×C127—24no—
21C4⋊Q8×C127—22no—
22C4⋊Q8×C127C2×C414yes—
23C4×Q8×C127—41no—
24C4×Q8×C127—41no—
25C4×Q8×C127—41no—
26C4×Q8×C127—41no—
27C2×C4⋊C4×C127—22no—
28C2×C4⋊C4×C127—21no—
29C2×C4⋊C4×C127—22no—
30C2×C4⋊C4×C127C2×C411yescharacteristic
31C2×C4⋊C4×C127C2×C411yescharacteristic
32C2×C4⋊C4×C127D412yes—
33C2×C42×C127—21no—
34C2×C42×C127—21no—
35C2×C42×C127—21no—
36C2×C42×C127C2×C411yescharacteristic
37C2×C42×C127C2×C411yescharacteristic
38C2×C42×C127C2×C411yescharacteristic
39C2×C42×C127D412yes—
40#101×C127—24no—
41#101×C127C2×C418yes—
42#97×C127—42no—
43C4×C8×C127—42no—
44#95×C127—22no—
45#95×C127—22no—
46C2×Q8×C127—21no—
47C2×Q8×C127—41no—
48C2×Q8×C127—41no—
49C2×Q8×C127—41no—
50C2×Q8×C127—41no—
51C2×Q8×C127C4212yes—
52C2×Q8×C127C22×C412yes—
53C2×Q8×C127#4112yes—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C22⋊C412yes—
59C22×C4×C127C22⋊C411yescharacteristic
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—28no—
67C4⋊C4×C127—41no—
68C4⋊C4×C127—42no—
69C4⋊C4×C127—41no—
70C4⋊C4×C127—44no—
71C4⋊C4×C127—42no—
72C42×C127—24no—
73C42×C127—22no—
74C42×C127—21no—
75C42×C127—22no—
76C42×C127—21no—
77C42×C127—21no—
78C42×C127—41no—
79C42×C127—42no—
80C42×C127—41no—
81C42×C127—41no—
82C42×C127C4212yes—
83C23×C127C2×C22⋊C411yescharacteristic
84Q8×C127—42no—
85Q8×C127—42no—
86Q8×C127—42no—
87Q8×C127—42no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—42no—
106C2×C4×C127—42no—
107C2×C4×C127—41no—
108C2×C4×C127#9911yescharacteristic
109C2×C4×C127#9911yescharacteristic
110C2×C4×C127C2×C4211yescharacteristic
111C2×C4×C127#11711yescharacteristic
112C2×C4×C127C4×D414yes—
113C8×C127—48no—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127C4×C22⋊C411yescharacteristic
117C22×C127C2×#9911yescharacteristic
118C22×C127#41011yescharacteristic
119C4×C127—24no—
120C4×C127—21no—
121C4×C127—24no—
122C4×C127—21no—
123C4×C127—21no—
124C4×C127—42no—
125C4×C127—41no—
126C4×C127—42no—
127C4×C127—42no—
128C4×C127—81no—
129C4×C127#35612yesdirect product factor
130C4×#356C12711yesmaximal, characteristic, direct product factor, Sylow
131C2×C127—21no—
132C2×C127—21no—
133C2×C127C4×#9911yescharacteristic
134C2×C127C2×#35611yescharacteristic
135C2×C127#213211yescharacteristic
136C4×C4⋊Q8C2×C12711yescharacteristic
137C2×#356C2×C12712yes—
138C4×#101C2×C12712yes—
139#1394C2×C12712yes—
140C127C4×#35611yescharacteristic, direct product factor, Sylow
141C2×C4⋊Q8C22×C12711yescharacteristic
142C2×C4×Q8C4×C12711yescharacteristic
143C2×C4×Q8C22×C12711yescharacteristic
144C2×#101C22×C12714yes—
145C4×M16—22no—
146#391C4×C12711yescharacteristic
147#391C22×C12711yescharacteristic
148#389—22no—
149#384C4×C12711yescharacteristic
150C4×C4⋊C4—22no—
151C43C4×C12711yescharacteristic
152#356C4×C12718yesdirect product factor
153#343—22no—
154C22×Q8C2×C4×C12711yescharacteristic
155C22×Q8C23×C12711yescharacteristic
156C2×M16—24no—
157C4⋊Q8—22no—
158C4⋊Q8C2×C4×C12714yes—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C2×C4×C12711yescharacteristic
167C2×C4⋊C4C2×C4×C12711yescharacteristic
168C2×C4⋊C4D4×C12712yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42C2×C4×C12711yescharacteristic
173C2×C42C2×C4×C12711yescharacteristic
174C2×C42C2×C4×C12711yescharacteristic
175C2×C42D4×C12712yes—
176#101—24no—
177#101C2×C4×C12718yes—
178#97—42no—
179C4×C8—42no—
180#95—22no—
181#95—22no—
182C2×Q8—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8C42×C12712yes—
188C2×Q8C22×C4×C12712yes—
189C2×Q8#41×C12712yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4C22⋊C4×C12711yescharacteristic
195C22×C4C22⋊C4×C12712yes—
196C22×C4C22⋊C4×C12711yescharacteristic
197C22×C4C22×C4×C12711yescharacteristic
198C22×C4C2×D4×C12712yes—
199M16—28no—
200M16—44no—
201C2×C8—44no—
202C4⋊C4—28no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C42—24no—
209C42—22no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—41no—
215C42—42no—
216C42—41no—
217C42—41no—
218C42C42×C12712yes—
219C23C2×C22⋊C4×C12711yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—42no—
223Q8—42no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4#99×C12711yescharacteristic
245C2×C4#99×C12711yescharacteristic
246C2×C4C2×C42×C12711yescharacteristic
247C2×C4#117×C12711yescharacteristic
248C2×C4C4×D4×C12714yes—
249C8—48no—
250C22—21no—
251C22—21no—
252C22C4×C22⋊C4×C12711yescharacteristic
253C22C2×#99×C12711yescharacteristic
254C22#410×C12711yescharacteristic
255C4—24no—
256C4—21no—
257C4—24no—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—41no—
262C4—42no—
263C4—42no—
264C4—81no—
265C4#356×C12712yesdirect product factor
266C2—21no—
267C2—21no—
268C2C4×#99×C12711yescharacteristic
269C2C2×#356×C12711yescharacteristic
270C2C127×#213211yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC2×C4C2×C42×C127
Frattini, cosocleC22×Q8C23×C127
socleC22×C127C2×#99
prime radicalC23×C127C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
16. #356×C127C4direct (N × H)
129. C4×C127#356direct (N × H)
130. C4×#356C127direct (N × H)
140. C127C4×#356direct (N × H)
152. #356C4×C127direct (N × H)
265. C4#356×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C4⋊Q8×C1272
C2×#356×C1272
C4×#101×C1272
C127×#13942
C4×#356127