Finite Group Databasebeta

← #35216174

Subgroups — #35216174

Order 20864.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#23782171C211yesmaximal, characteristic
2#23781646C211yesmaximal, characteristic, complemented
3#23781637C211yesmaximal, characteristic, complemented
4#23781456C211yesmaximal, characteristic, complemented
5#23781454C211yesmaximal, characteristic, complemented
6#23781452C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C163C211yesmaximal, characteristic, complemented
8C2×#14579698C2211yescharacteristic
9C23×Dic163C411yescharacteristic, complemented
10M16×D163—21no—
11#19104719C411yescharacteristic, complemented
12#19104716C2211yescharacteristic
13#19104712—21no—
14#19104707—21no—
15#19104704C411yescharacteristic, complemented
16D4×Dic163—21no—
17(C22⋊D163)⋊C4—21no—
18#19104691—21no—
19#19104691C411yescharacteristic, complemented
20#19104653—21no—
21#19104652C2211yescharacteristic
22#19104650C2211yescharacteristic
23#19104649—21no—
24#120×C163C2211yescharacteristic
25#100×C163C2211yescharacteristic
26#99×C163C2211yescharacteristic
27#14579698—41no—
28#14579698—41no—
29C2×Dic326D411yescharacteristic
30C2×C22⋊D163—21no—
31C2×C22⋊D163C2×C411yescharacteristic
32C2×C22⋊D163D411yescharacteristic
33C22×Dic163—21no—
34C22×Dic163—21no—
35C22×Dic163—41no—
36C22×Dic163—41no—
37C22×Dic163C2×C411yescharacteristic
38C22×Dic163C2×C411yescharacteristic
39C22×Dic163D411yescharacteristic
40C2×C4×D163D411yescharacteristic
41#14579683—21no—
42#14579682—41no—
43C8×D163—41no—
44C4⋊Dic163—21no—
45Dic163⋊C4—21no—
46Dic163⋊C4—21no—
47Dic163⋊C4—41no—
48C22⋊Dic163—21no—
49C22⋊Dic163—21no—
50C22⋊Dic163—21no—
51C22⋊Dic163C2×C411yescharacteristic
52(C2×D163)⋊C4—21no—
53C4×Dic163—21no—
54C4×Dic163—41no—
55C24×C163C2×C411yescharacteristic
56C2×D4×C163—21no—
57C2×D4×C163C2311yescharacteristic
58M16×C163—21no—
59C22⋊C4×C163—21no—
60C22⋊C4×C163—21no—
61C22⋊C4×C163C2×C411yescharacteristic
62C22×D163C22⋊C411yescharacteristic
63Dic326—41no—
64C22⋊D163—41no—
65C22⋊D163—41no—
66C22⋊D163—42no—
67C2×Dic163—21no—
68C2×Dic163—21no—
69C2×Dic163—21no—
70C2×Dic163—21no—
71C2×Dic163—21no—
72C2×Dic163—21no—
73C2×Dic163—42no—
74C2×Dic163—41no—
75C2×Dic163—41no—
76C2×Dic163—41no—
77C2×Dic163—41no—
78C2×Dic163C22⋊C411yescharacteristic
79C2×Dic163C22⋊C411yescharacteristic
80C2×Dic163C22⋊C411yescharacteristic
81C4×D163—21no—
82C4×D163—21no—
83C163⋊C8—41no—
84C23×C163—21no—
85C23×C163—42no—
86C23×C163—41no—
87C23×C163C22×C411yescharacteristic
88C23×C163C2×D411yescharacteristic
89D4×C163—41no—
90D4×C163—42no—
91C2×C4×C163—21no—
92C2×C4×C163—41no—
93C2×C4×C163C2×D411yescharacteristic
94C8×C163—41no—
95C22×C163—22no—
96C22×C163—21no—
97C22×C163—21no—
98C22×C163—41no—
99C22×C163—41no—
100C22×C163—41no—
101C22×C163—42no—
102C22×C163—42no—
103C22×C163—41no—
104C22×C163C2×C22⋊C411yescharacteristic
105C2×D163—21no—
106C2×D163—41no—
107C4×C163—21no—
108C4×C163—41no—
109C4×C163—81no—
110Dic163—21no—
111Dic163—21no—
112Dic163—41no—
113Dic163—41no—
114Dic163—81no—
115Dic163#9911yescharacteristic
116Dic163#9911yescharacteristic
117C2×C163—21no—
118C2×C163—41no—
119C2×C163—42no—
120C2×C163—41no—
121C2×C163C2×#9911yescharacteristic
122D163—41no—
123C163#212011yescharacteristic, complemented, Sylow
124#2120—1631nomaximal, Sylow
125#525—1631no—
126#413—1631no—
127#410—1631no—
128#352—1631no—
129#352—1631no—
130AΣL(1,𝔽16)—1631no—
131AΣL(1,𝔽16)D16311yescharacteristic, complemented
132C2×#41—1631no—
133C23×C4—1631no—
134C2×M16—3261no—
135#123—1631no—
136#123—1631no—
137C22⋊Q8—3261no—
138#121—3261no—
139#120—1631no—
140#120C2×D16311yescharacteristic
141C4×D4—3261no—
142C4×D4—3261no—
143#117—3261no—
144#117—1631no—
145#101—3261no—
146#100—1631no—
147#100C2×D16311yescharacteristic
148#99—3261no—
149#99—1631no—
150#99C2×D16311yescharacteristic
151C24C4×D16311yescharacteristic
152#41—6521no—
153#41—6521no—
154C2×Q8—1631no—
155C2×D4—21no—
156C2×D4—3261no—
157C2×D4—1631no—
158C2×D4—1631no—
159C2×D4C22×D16311yescharacteristic
160C22×C4—3261no—
161C22×C4—3261no—
162C22×C4—6521no—
163C22×C4—6521no—
164C22×C4—1631no—
165C22×C4—1631no—
166C22×C4—1631no—
167C22×C4—1631no—
168M16—21no—
169M16—3261no—
170M16—6521no—
171C2×C8—6521no—
172C4⋊C4—3261no—
173C4⋊C4—3261no—
174C4⋊C4—3261no—
175C4⋊C4—6521no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—3261no—
179C22⋊C4—3261no—
180C22⋊C4—3261no—
181C22⋊C4—3261no—
182C22⋊C4—1631no—
183C22⋊C4C4×D16311yescharacteristic
184C42—3261no—
185C42—6521no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—1631no—
190C23C2×C4×D16311yescharacteristic
191C23D4×D16311yescharacteristic
192Q8—6521no—
193D4—41no—
194D4—42no—
195D4—6521no—
196D4—6521no—
197D4—6522no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—3261no—
201C2×C4—3261no—
202C2×C4—3261no—
203C2×C4—3261no—
204C2×C4—3261no—
205C2×C4—3261no—
206C2×C4—3261no—
207C2×C4—3261no—
208C2×C4—6521no—
209C2×C4—6522no—
210C2×C4—6521no—
211C2×C4—6521no—
212C2×C4—6521no—
213C2×C4—1631no—
214C2×C4—1631no—
215C2×C4—1631no—
216C2×C4D4×D16311yescharacteristic
217C8—41no—
218C8—6521no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—3261no—
229C22—6521no—
230C22C22⋊C4×D16311yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—13041no—
235C4—3261no—
236C4—3261no—
237C4—6521no—
238C4—6521no—
239C4—1631no—
240C4—1631no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—6521no—
246C2#99×D16311yescharacteristic
subgroupG/H
centerC2#99×D163
commutatorC23×C163C22×C4
FrattiniC2×D4C22×D163
FittingAΣL(1,𝔽16)×C163C2
socleC2×C163C2×#99
prime radical#23782171C2
cosocleC2×D4×C163C23
hypercenterAΣL(1,𝔽16)D163
hypocenterC163#2120
LCSG ▷ C23×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #23781646C2semidirect
3. #23781637C2semidirect
4. #23781456C2semidirect
5. #23781454C2semidirect
6. #23781452C2semidirect
7. AΣL(1,𝔽16)×C163C2semidirect
9. C23×Dic163C4semidirect
11. #19104719C4semidirect
15. #19104704C4semidirect
19. #19104691C4semidirect
123. C163#2120semidirect
131. AΣL(1,𝔽16)D163semidirect

Maximal subgroups

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

subgroupindex
#237821712
#237816462
#237816372
#237814562
#237814542
#237814522
AΣL(1,𝔽16)×C1632
#2120163