Finite Group Databasebeta

← #35216188

Subgroups — #35216188

Order 20864.

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

normal subgroups only

This group has 234 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#23782237C211yesmaximal, characteristic
2#23781646C211yesmaximal, characteristic
3#23781637C211yesmaximal, characteristic
4#23781456C211yesmaximal, characteristic, complemented
5#23781455C211yesmaximal, characteristic, complemented
6#23781454C211yesmaximal, characteristic, complemented
7#352×C163C211yesmaximal, characteristic, complemented
8C2×#14579698C2211yescharacteristic
9C22×Dic326C411yescharacteristic, complemented
10M16×D163—21no—
11#19104734—21no—
12#19104721—21no—
13#19104719C411yescharacteristic, complemented
14#19104718C411yescharacteristic, complemented
15#19104716C2211yescharacteristic
16#19104713—21no—
17#19104712—21no—
18#19104691—21no—
19#19104691C411yescharacteristic, complemented
20#19104653—21no—
21#19104652C2211yescharacteristic
22#19104650C2211yescharacteristic
23#19104649—21no—
24#123×C163C2211yescharacteristic
25#100×C163C2211yescharacteristic
26#99×C163C2211yescharacteristic
27#14579698—41no—
28#14579698—41no—
29C2×Dic326—21no—
30C2×Dic326—21no—
31C2×Dic326—41no—
32C2×Dic326—41no—
33C2×Dic326D411yescharacteristic
34C2×C22⋊D163C2×C411yescharacteristic
35C2×C22⋊D163D411yescharacteristic
36C22×Dic163C2×C411yescharacteristic
37C22×Dic163C2×C411yescharacteristic
38C22×Dic163D411yescharacteristic
39C2×C4×D163D411yescharacteristic
40#14579683—21no—
41#14579682—41no—
42C8×D163—41no—
43C4⋊Dic163—21no—
44Dic163⋊C4—21no—
45Dic163⋊C4—21no—
46Dic163⋊C4—21no—
47Dic163⋊C4—41no—
48C22⋊Dic163—21no—
49C22⋊Dic163—21no—
50C22⋊Dic163C2×C411yescharacteristic
51(C2×D163)⋊C4—21no—
52(C2×D163)⋊C4—21no—
53C4×Dic163—21no—
54C4×Dic163—41no—
55C2×D4×C163C2311yescharacteristic
56C22×C4×C163C2×C411yescharacteristic
57M16×C163—21no—
58C4⋊C4×C163—21no—
59C22⋊C4×C163—21no—
60C22⋊C4×C163—21no—
61C22⋊C4×C163C2×C411yescharacteristic
62C22×D163C22⋊C411yescharacteristic
63Dic326—41no—
64Dic326—42no—
65Dic326—41no—
66Dic326—41no—
67C22⋊D163—41no—
68C22⋊D163—41no—
69C2×Dic163—21no—
70C2×Dic163—21no—
71C2×Dic163—21no—
72C2×Dic163—21no—
73C2×Dic163—21no—
74C2×Dic163—21no—
75C2×Dic163—41no—
76C2×Dic163—41no—
77C2×Dic163C22⋊C411yescharacteristic
78C2×Dic163C22⋊C411yescharacteristic
79C2×Dic163C22⋊C411yescharacteristic
80C4×D163—21no—
81C4×D163—21no—
82C163⋊C8—41no—
83C23×C163C22×C411yescharacteristic
84C23×C163C2×D411yescharacteristic
85D4×C163—41no—
86C2×C4×C163—21no—
87C2×C4×C163—21no—
88C2×C4×C163—41no—
89C2×C4×C163—41no—
90C2×C4×C163C2×D411yescharacteristic
91C8×C163—41no—
92C22×C163—21no—
93C22×C163—21no—
94C22×C163—41no—
95C22×C163—41no—
96C22×C163C2×C22⋊C411yescharacteristic
97C2×D163—21no—
98C2×D163—41no—
99C4×C163—21no—
100C4×C163—41no—
101C4×C163—41no—
102C4×C163—81no—
103Dic163—21no—
104Dic163—21no—
105Dic163—41no—
106Dic163—41no—
107Dic163—81no—
108Dic163#9911yescharacteristic
109Dic163#9911yescharacteristic
110C2×C163—21no—
111C2×C163—41no—
112C2×C163—41no—
113C2×C163C2×#9911yescharacteristic
114D163—41no—
115C163#212311yescharacteristic, complemented, Sylow
116#2123—1631nomaximal, Sylow
117#536—1631no—
118#413—1631no—
119#410—1631no—
120#352—1631no—
121#352—1631no—
122#352—1631no—
123#352D16311yescharacteristic, complemented
124C2×#41—1631no—
125C22×Q8—1631no—
126C2×M16—3261no—
127C4⋊Q8—3261no—
128#124—3261no—
129#123—1631no—
130#123—1631no—
131#123—1631no—
132#123C2×D16311yescharacteristic
133C22⋊Q8—3261no—
134C22⋊Q8—3261no—
135#117—3261no—
136#117—1631no—
137#101—3261no—
138#100—1631no—
139#100C2×D16311yescharacteristic
140#99—3261no—
141#99—1631no—
142#99C2×D16311yescharacteristic
143#41—6521no—
144#41—6521no—
145C2×Q8—3261no—
146C2×Q8—3261no—
147C2×Q8—6521no—
148C2×Q8—6521no—
149C2×Q8—1631no—
150C2×D4—1631no—
151C2×D4—1631no—
152C2×D4C22×D16311yescharacteristic
153C22×C4—1631no—
154C22×C4—1631no—
155C22×C4—1631no—
156C22×C4—1631no—
157C22×C4C4×D16311yescharacteristic
158M16—21no—
159M16—3261no—
160M16—6521no—
161C2×C8—6521no—
162C4⋊C4—21no—
163C4⋊C4—3261no—
164C4⋊C4—3261no—
165C4⋊C4—3261no—
166C4⋊C4—3261no—
167C4⋊C4—6521no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3261no—
171C22⋊C4—3261no—
172C22⋊C4—3261no—
173C22⋊C4—3261no—
174C22⋊C4—1631no—
175C22⋊C4C4×D16311yescharacteristic
176C42—3261no—
177C42—6521no—
178C23—1631no—
179C23C2×C4×D16311yescharacteristic
180C23D4×D16311yescharacteristic
181Q8—6521no—
182Q8—6522no—
183Q8—6521no—
184Q8—6521no—
185D4—41no—
186D4—6521no—
187D4—6521no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3261no—
193C2×C4—3261no—
194C2×C4—3261no—
195C2×C4—3261no—
196C2×C4—3261no—
197C2×C4—3261no—
198C2×C4—3261no—
199C2×C4—3261no—
200C2×C4—6521no—
201C2×C4—6521no—
202C2×C4—1631no—
203C2×C4—1631no—
204C2×C4—1631no—
205C2×C4D4×D16311yescharacteristic
206C8—41no—
207C8—6521no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3261no—
213C22—6521no—
214C22C22⋊C4×D16311yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—13041no—
220C4—3261no—
221C4—3261no—
222C4—6521no—
223C4—6521no—
224C4—1631no—
225C4—1631no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—6521no—
230C2#99×D16311yescharacteristic
subgroupG/H
centerC2#99×D163
commutatorC23×C163C22×C4
FrattiniC2×D4C22×D163
Fitting#352×C163C2
socleC2×C163C2×#99
prime radicalC2×#14579698C22
cosocleC2×D4×C163C23
hypercenter#352D163
hypocenterC163#2123
LCSG ▷ C23×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #23781456C2semidirect
5. #23781455C2semidirect
6. #23781454C2semidirect
7. #352×C163C2semidirect
9. C22×Dic326C4semidirect
13. #19104719C4semidirect
14. #19104718C4semidirect
19. #19104691C4semidirect
115. C163#2123semidirect
123. #352D163semidirect

Maximal subgroups

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

subgroupindex
#237822372
#237816462
#237816372
#237814562
#237814552
#237814542
#352×C1632
#2123163