Finite Group Databasebeta

← #35213477

Subgroups — #35213477

Order 20864.

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

normal subgroups only

This group has 266 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#23782294C211yesmaximal, characteristic, complemented
2#23781818C211yesmaximal, characteristic
3#23781815C211yesmaximal, characteristic
4#23781583C211yesmaximal, characteristic
5#23781401C211yesmaximal, characteristic, complemented
6#23781390C211yesmaximal, characteristic, complemented
7#328×C163C211yesmaximal, characteristic, complemented
8C22×D652C2211yescharacteristic
9C2×#14579688—21no—
10C2×#14579686—21no—
11#19104734—21no—
12#19104733—21no—
13#19104725—21no—
14#19104725—21no—
15#19104714C2211yescharacteristic
16#19104712—21no—
17#19104691—21no—
18C2×Dic163⋊C4C2211yescharacteristic
19C2×(C2×D163)⋊C4—21no—
20C2×(C2×D163)⋊C4—21no—
21#19104657—21no—
22#19104655—21no—
23C22⋊(C163⋊C8)C2211yescharacteristic
24C22⋊Q8×C163C2211yescharacteristic
25C22⋊C8×C163C2211yescharacteristic
26#95×C163C2211yescharacteristic
27C23×D163—21no—
28C2×Dic326—21no—
29C2×Dic326D411yescharacteristic
30C2×D652—21no—
31C2×D652—21no—
32C2×D652—42no—
33C22×Dic163D411yescharacteristic
34C22×Dic163D411yescharacteristic
35#14579688—42no—
36#14579686—42no—
37C2×C163⋊C8—21no—
38C4⋊Dic163—21no—
39C4⋊Dic163D411yescharacteristic
40Dic163⋊C4—21no—
41Dic163⋊C4—21no—
42Dic163⋊C4—21no—
43Dic163⋊C4—21no—
44Dic163⋊C4—21no—
45Dic163⋊C4—21no—
46Dic163⋊C4—21no—
47C22⋊Dic163—21no—
48(C2×D163)⋊C4—42no—
49(C2×D163)⋊C4—42no—
50C4×Dic163—21no—
51C4×Dic163—21no—
52C2×Q8×C163D411yescharacteristic
53C22×C4×C163—21no—
54C22×C4×C163C2311yescharacteristic
55C2×C8×C163—21no—
56C4⋊C4×C163—21no—
57C4⋊C4×C163D411yescharacteristic
58C22⋊C4×C163—21no—
59C22×D163—22no—
60C22×D163—41no—
61C22×D163—41no—
62C22×D163—41no—
63C22×D163—41no—
64C22×D163—41no—
65C22×D163—41no—
66Dic326—41no—
67Dic326—81no—
68D652—22no—
69D652—22no—
70D652—41no—
71D652—41no—
72C2×Dic163—21no—
73C2×Dic163—21no—
74C2×Dic163—21no—
75C2×Dic163—21no—
76C2×Dic163—21no—
77C2×Dic163—41no—
78C2×Dic163SD1612yescomplemented
79C163⋊C8—41no—
80C23×C163C2×D411yescharacteristic
81Q8×C163—41no—
82C2×C4×C163—21no—
83C2×C4×C163—21no—
84C2×C4×C163—41no—
85C2×C4×C163—41no—
86C2×C4×C163—41no—
87C2×C4×C163—41no—
88C2×C4×C163C2×D411yescharacteristic
89C2×C4×C163C2×D411yescharacteristic
90C8×C163—41no—
91C22×C163—21no—
92C22×C163—21no—
93C22×C163#12011yescharacteristic
94C22×C163C2×SD1611yescharacteristic
95C22×C163#13711yescharacteristic
96C2×D163—42no—
97C2×D163—42no—
98C2×D163—42no—
99C2×D163—81no—
100C2×D163—81no—
101C2×D163—81no—
102C2×D163—81no—
103C4×C163—21no—
104C4×C163—21no—
105C4×C163—41no—
106C4×C163—41no—
107C4×C163—81no—
108Dic163—22no—
109Dic163—41no—
110Dic163—41no—
111Dic163—41no—
112C2×C163—21no—
113C2×C163—21no—
114C2×C163#44811yescharacteristic
115C2×C163#45411yescharacteristic
116C2×C163B(4,𝔽2)11yescharacteristic
117D163—82no—
118C163#159911yescharacteristic, complemented, Sylow
119#1599—1631nomaximal, Sylow
120#544—1631no—
121#450—1631no—
122#450—1631no—
123#394—1631no—
124#328—1631no—
125#328D16311yescharacteristic, complemented
126#324—1631no—
127C22×D4—1631no—
128C2×SD16—3261no—
129C2×SD16—3261no—
130C4⋊Q8—3261no—
131C4⋊Q8—3261no—
132#125—3261no—
133#125—3261no—
134C22⋊Q8—3261no—
135C22⋊Q8—1631no—
136C22⋊Q8C2×D16311yescharacteristic
137#117—3261no—
138C2×C4⋊C4—1631no—
139C2×C22⋊C4—3261no—
140C2×C22⋊C4—3261no—
141#102—3261no—
142#102—3261no—
143C22⋊C8—1631no—
144C22⋊C8C2×D16311yescharacteristic
145#95C2×D16311yescharacteristic
146C24—3261no—
147C2×Q8—3261no—
148C2×Q8—1631no—
149C2×Q8D65211yescharacteristic
150C2×D4—3261no—
151C2×D4—3261no—
152C2×D4—6522no—
153C22×C4—21no—
154C22×C4—1631no—
155C22×C4—1631no—
156C22×C4C22×D16311yescharacteristic
157SD16—6522no—
158SD16—6522no—
159C2×C8—21no—
160C2×C8—3261no—
161C4⋊C4—21no—
162C4⋊C4—3261no—
163C4⋊C4—3261no—
164C4⋊C4—3261no—
165C4⋊C4—3261no—
166C4⋊C4—3261no—
167C4⋊C4—3261no—
168C4⋊C4—3261no—
169C4⋊C4—3261no—
170C4⋊C4—1631no—
171C4⋊C4D65211yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—3261no—
174C22⋊C4—6522no—
175C22⋊C4—6522no—
176C42—3261no—
177C42—3261no—
178C23—3262no—
179C23—6521no—
180C23—6521no—
181C23—6521no—
182C23—6521no—
183C23—6521no—
184C23—6521no—
185C23D4×D16311yescharacteristic
186Q8—41no—
187Q8—13041no—
188Q8—6521no—
189D4—3262no—
190D4—3262no—
191D4—6521no—
192D4—6521no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3261no—
200C2×C4—3261no—
201C2×C4—3261no—
202C2×C4—3261no—
203C2×C4—3261no—
204C2×C4—6521no—
205C2×C4—1632no—
206C2×C4D4×D16311yescharacteristic
207C2×C4C2×D65211yescharacteristic
208C8—41no—
209C8—6521no—
210C22—21no—
211C22—21no—
212C22—13041no—
213C22—13041no—
214C22—13041no—
215C22—13041no—
216C22—6522no—
217C22—6522no—
218C22—6522no—
219C22#1910470511yescharacteristic
220C22SD16×D16311yescharacteristic
221C22#1910476711yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—3262no—
228C4—6521no—
229C4—6521no—
230C4—6521no—
231C2—21no—
232C2—21no—
233C2—13042no—
234C2#2378180511yescharacteristic
235C2#2378183911yescharacteristic
236C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#328×C163C2
socleC22×C163#120
prime radicalC22×D652C22
hypercenter#328D163
hypocenterC163#1599
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #23782294C2semidirect
5. #23781401C2semidirect
6. #23781390C2semidirect
7. #328×C163C2semidirect
78. C2×Dic163SD16semidirect
118. C163#1599semidirect
125. #328D163semidirect

Maximal subgroups

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

subgroupindex
#237822942
#237818182
#237818152
#237815832
#237814012
#237813902
#328×C1632
#1599163