Finite Group Databasebeta

← #35213491

Subgroups — #35213491

Order 20864.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#23782290C211yesmaximal, characteristic
2#23781821C211yesmaximal, characteristic, complemented
3#23781576C211yesmaximal, characteristic
4#23781403C211yesmaximal, characteristic, complemented
5#23781402C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic
7#451×C163C211yesmaximal, characteristic, complemented
8C2×#14579689—21no—
9#19104715—21no—
10#19104714C2211yescharacteristic
11#19104713—21no—
12#19104713—21no—
13#19104712—21no—
14#19104711—21no—
15C2×(C2×D163)⋊C4—21no—
16C2×(C2×D163)⋊C4C2211yescharacteristic
17C22⋊C4×D163—21no—
18#19104659—21no—
19C22⋊(C163⋊C8)C2211yescharacteristic
20C163⋊#95—21no—
21C163⋊#95C2211yescharacteristic
22C22×Q8×C163C2211yescharacteristic
23C2×Q16×C163—21no—
24C22⋊Q8×C163C2211yescharacteristic
25#103×C163—21no—
26C22⋊C8×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×Dic326—21no—
29C2×Dic326D411yescharacteristic
30C22×Dic163—21no—
31C22×Dic163D411yescharacteristic
32C2×C4×D163—21no—
33C2×C4×D163—21no—
34#14579689—42no—
35C2×C163⋊C8—21no—
36C4⋊Dic163—21no—
37C4⋊Dic163D411yescharacteristic
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41C22⋊Dic163—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—42no—
47C2×Q8×C163—21no—
48C2×Q8×C163—21no—
49C2×Q8×C163—42no—
50C2×Q8×C163D411yescharacteristic
51C22×C4×C163—21no—
52C22×C4×C163C2311yescharacteristic
53Q16×C163—42no—
54C2×C8×C163—21no—
55C4⋊C4×C163—21no—
56C4⋊C4×C163D411yescharacteristic
57C22⋊C4×C163—21no—
58C22×D163—21no—
59C22×D163—21no—
60C22×D163—41no—
61C22×D163—41no—
62C22×D163Q1612yescomplemented
63Dic326—41no—
64Dic326—81no—
65C2×Dic163—21no—
66C2×Dic163—21no—
67C2×Dic163—21no—
68C2×Dic163—41no—
69C2×Dic163—41no—
70C2×Dic163—41no—
71C2×Dic163—41no—
72C4×D163—81no—
73C4×D163—81no—
74C163⋊C8—41no—
75C23×C163C2×D411yescharacteristic
76Q8×C163—22no—
77Q8×C163—22no—
78Q8×C163—41no—
79Q8×C163—41no—
80Q8×C163—41no—
81C2×C4×C163—22no—
82C2×C4×C163—21no—
83C2×C4×C163—21no—
84C2×C4×C163—41no—
85C2×C4×C163—41no—
86C2×C4×C163—41no—
87C2×C4×C163C2×D411yescharacteristic
88C2×C4×C163C2×D411yescharacteristic
89C8×C163—41no—
90C22×C163—21no—
91C22×C163—21no—
92C22×C163#12011yescharacteristic
93C22×C163C2×Q1611yescharacteristic
94C22×C163#13711yescharacteristic
95C2×D163—22no—
96C2×D163—41no—
97C2×D163—41no—
98C2×D163—42no—
99C2×D163—81no—
100C4×C163—21no—
101C4×C163—21no—
102C4×C163—41no—
103C4×C163—42no—
104C4×C163—41no—
105Dic163—41no—
106Dic163—41no—
107Dic163—41no—
108Dic163—81no—
109C2×C163—21no—
110C2×C163—21no—
111C2×C163#45111yescharacteristic
112C2×C163#45511yescharacteristic
113C2×C163B(4,𝔽2)11yescharacteristic
114D163—42no—
115C163#160211yescharacteristic, complemented, Sylow
116#1602—1631nomaximal, Sylow
117#543—1631no—
118#451—1631no—
119#451D16311yescharacteristic, complemented
120#393—1631no—
121#328—1631no—
122#328—1631no—
123#323—1631no—
124C22×Q8C2×D16311yescharacteristic
125C2×Q16—21no—
126C2×Q16—3261no—
127C22⋊Q8—3261no—
128C22⋊Q8—3261no—
129C22⋊Q8—3261no—
130C22⋊Q8—3261no—
131C22⋊Q8—3261no—
132C22⋊Q8—1631no—
133C22⋊Q8C2×D16311yescharacteristic
134C2×C22⋊C4—3261no—
135C2×C22⋊C4—3261no—
136C2×C22⋊C4—1631no—
137#103—21no—
138#103—3261no—
139C22⋊C8—1631no—
140C22⋊C8C2×D16311yescharacteristic
141#95—3261no—
142#95—1631no—
143C24—1631no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—3261no—
148C2×Q8—1631no—
149C2×Q8C22⋊D16311yescharacteristic
150C22×C4—21no—
151C22×C4—3261no—
152C22×C4—3261no—
153C22×C4—3261no—
154C22×C4—1631no—
155C22×C4C22×D16311yescharacteristic
156Q16—42no—
157Q16—6522no—
158C2×C8—21no—
159C2×C8—3261no—
160C4⋊C4—21no—
161C4⋊C4—3261no—
162C4⋊C4—3261no—
163C4⋊C4—3261no—
164C4⋊C4—3261no—
165C4⋊C4—1631no—
166C4⋊C4C22⋊D16311yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3261no—
169C22⋊C4—3261no—
170C22⋊C4—3261no—
171C22⋊C4—3261no—
172C22⋊C4—3261no—
173C22⋊C4—6522no—
174C23—3261no—
175C23—3261no—
176C23—6521no—
177C23—6521no—
178C23—1632no—
179C23D4×D16311yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—13041no—
186Q8—6521no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—13041no—
194C2×C4—13041no—
195C2×C4—3261no—
196C2×C4—3261no—
197C2×C4—3261no—
198C2×C4—6521no—
199C2×C4—6521no—
200C2×C4—6521no—
201C2×C4—6521no—
202C2×C4D4×D16311yescharacteristic
203C2×C4C2×C22⋊D16311yescharacteristic
204C8—41no—
205C8—6521no—
206C22—21no—
207C22—21no—
208C22—13041no—
209C22—3262no—
210C22—6522no—
211C22—6521no—
212C22—6521no—
213C22#1910470311yescharacteristic
214C22Q16×D16311yescharacteristic
215C22#1910476711yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—13041no—
222C4—6521no—
223C4—6521no—
224C4—6521no—
225C2—21no—
226C2—21no—
227C2—6522no—
228C2#2378181911yescharacteristic
229C2#2378184111yescharacteristic
230C2#2378185111yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#451×C163C2
socleC22×C163#120
prime radicalC23×D163D4
hypercenter#451D163
hypocenterC163#1602
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #23781821C2semidirect
4. #23781403C2semidirect
5. #23781402C2semidirect
7. #451×C163C2semidirect
62. C22×D163Q16semidirect
115. C163#1602semidirect
119. #451D163semidirect

Maximal subgroups

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

subgroupindex
#237822902
#237818212
#237815762
#237814032
#237814022
#237813882
#451×C1632
#1602163