Finite Group Databasebeta

← C4×AΣL(1,𝔽16) #12619

Subgroups — C4×AΣL(1,𝔽16) #12619

Order 256.

422 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 83 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×#120C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)C212yesmaximal
3C4×#100C211yesmaximal, characteristic, complemented
4C4×#99C211yesmaximal, characteristic, complemented
5#1390C212yesmaximal, complemented
6C24×C4C411yescharacteristic, complemented
7C2×#120C2211yescharacteristic
8C2×C4×D4—21no—
9C2×C4×D4C2211yescharacteristic
10C2×#100C2212yes—
11C2×#99C2212yes—
12C4×M16—21no—
13#386—21no—
14#386C2211yescharacteristic
15#385—21no—
16#385C411yescharacteristic, complemented
17#379C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19C4×C22⋊C4—21no—
20C4×C22⋊C4C411yescharacteristic, complemented
21AΣL(1,𝔽16)C418yesdirect product factor
22#343—21no—
23#342—21no—
24C25C2×C411yescharacteristic
25C22×D4—21no—
26C22×D4C2311yescharacteristic
27C23×C4—21no—
28C23×C4—21no—
29C23×C4—44no—
30C23×C4—41no—
31C23×C4—41no—
32C23×C4C2×C411yescharacteristic
33C23×C4C2×C411yescharacteristic
34C23×C4D411yescharacteristic
35C2×M16—22no—
36#120—22no—
37#120C2×C414yes—
38C4×D4—41no—
39C4×D4—41no—
40C4×D4—44no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4C2×C411yescharacteristic
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4D411yescharacteristic
52C2×C42—21no—
53C2×C42—41no—
54C2×C42D411yescharacteristic
55#100—22no—
56#100C2×C414yescomplemented
57#99—22no—
58#99C2×C414yescomplemented
59#97—41no—
60C4×C8—41no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—41no—
65C24—21no—
66C24—21no—
67C24—42no—
68C24—41no—
69C24—42no—
70C24—41no—
71C24C4212yescomplemented
72C24C22×C411yescharacteristic
73C24C2×D411yescharacteristic
74C2×D4—24no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—42no—
78C2×D4—42no—
79C2×D4C22×C412yes—
80C2×D4#4112yes—
81C22×C4—24no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—44no—
103C22×C4—42no—
104C22×C4C22⋊C411yescharacteristic
105C22×C4C22⋊C411yescharacteristic
106C22×C4C22⋊C411yescharacteristic
107C22×C4C22⋊C411yescharacteristic
108C22×C4C2×D411yescharacteristic
109M16—24no—
110M16—42no—
111C2×C8—42no—
112C4⋊C4—41no—
113C4⋊C4—42no—
114C22⋊C4—21no—
115C22⋊C4—24no—
116C22⋊C4—24no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—42no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—41no—
124C22⋊C4C4212yescomplemented
125C42—21no—
126C42—21no—
127C42—42no—
128C42—82no—
129C23—22no—
130C23—22no—
131C23—22no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—41no—
137C23—44no—
138C23—42no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—44no—
143C23—42no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—48no—
150C23—42no—
151C23—41no—
152C23—41no—
153C23C2×C4211yescharacteristic
154C23C2×C22⋊C411yescharacteristic
155C23#11711yescharacteristic
156C23C4×D412yes—
157D4—42no—
158D4—42no—
159D4—48no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—48no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4#9911yescharacteristic
191C2×C4#9911yescharacteristic
192C2×C4C4×D412yes—
193C8—44no—
194C22—21no—
195C22—21no—
196C22—24no—
197C22—22no—
198C22—21no—
199C22—21no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—44no—
208C22—44no—
209C22—44no—
210C22—44no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22—42no—
218C22—81no—
219C22C4×C22⋊C411yescharacteristic
220C22C2×#9911yescharacteristic
221C22#41011yescharacteristic
222C4—21no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—81no—
230C4—84no—
231C4AΣL(1,𝔽16)12yesdirect product factor
232C2—21no—
233C2—21no—
234C2—42no—
235C2—44no—
236C2—41no—
237C2—41no—
238C2C4×#9911yescharacteristic
239C2C2×AΣL(1,𝔽16)11yescharacteristic
240C2#212011yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC23C2×C42
Frattini, cosocleC22×D4C23
socleC22C2×#99
prime radicalC2×#120C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#100C2semidirect
4. C4×#99C2semidirect
5. #1390C2semidirect
6. C24×C4C4semidirect
16. #385C4semidirect
17. #379C4semidirect
20. C4×C22⋊C4C4semidirect
21. AΣL(1,𝔽16)C4direct (N × H)
56. #100C2×C4semidirect
58. #99C2×C4semidirect
71. C24C42semidirect
124. C22⋊C4C42semidirect
231. C4AΣL(1,𝔽16)direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#1202
C2×AΣL(1,𝔽16)2
C4×#1002
C4×#992
#13902