Finite Group Databasebeta

← #35213581

Subgroups — #35213581

Order 20864.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782252C211yesmaximal, characteristic
2#23782000C211yesmaximal, characteristic, complemented
3#23781600C211yesmaximal, characteristic, complemented
4#23781403C211yesmaximal, characteristic, complemented
5#23781399C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic, complemented
7#481×C163C211yesmaximal, characteristic, complemented
8#19104730—21no—
9#19104721—21no—
10#19104719—21no—
11#19104714C2211yescharacteristic
12#19104707—21no—
13#19104703—21no—
14D4×Dic163—21no—
15C2×Dic163⋊C4—21no—
16C2×(C2×D163)⋊C4—21no—
17C2×(C2×D163)⋊C4C2211yescharacteristic
18#19104668—21no—
19#19104658—21no—
20C22⋊(C163⋊C8)C2211yescharacteristic
21C163⋊#95C2211yescharacteristic
22#121×C163C2211yescharacteristic
23C2×C4⋊C4×C163C2211yescharacteristic
24#106×C163—21no—
25#102×C163—21no—
26C22⋊C8×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×Dic326D411yescharacteristic
29C2×C22⋊D163—21no—
30C2×C22⋊D163—21no—
31C22×Dic163—21no—
32C22×Dic163—21no—
33C22×Dic163D411yescharacteristic
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163D411yescharacteristic
37Dic163⋊C4—21no—
38Dic163⋊C4—42no—
39C22⋊Dic163—21no—
40C22⋊Dic163—21no—
41(C2×D163)⋊C4—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—42no—
46C4×Dic163—21no—
47C2×D4×C163—21no—
48C2×D4×C163D411yescharacteristic
49C22×C4×C163—21no—
50C22×C4×C163C2311yescharacteristic
51C2×C8×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163—21no—
54C4⋊C4×C163D411yescharacteristic
55C22⋊C4×C163—21no—
56C22×D163—21no—
57C22×D163—21no—
58C22×D163—21no—
59C22×D163—41no—
60C22×D163—41no—
61Dic326—41no—
62C22⋊D163—81no—
63C22⋊D163—81no—
64C2×Dic163—22no—
65C2×Dic163—21no—
66C2×Dic163—21no—
67C2×Dic163—21no—
68C2×Dic163—41no—
69C2×Dic163—41no—
70C2×Dic163—41no—
71C2×Dic163—81no—
72C163⋊C8—41no—
73C23×C163—21no—
74C23×C163C2×D411yescharacteristic
75D4×C163—41no—
76D4×C163—41no—
77D4×C163—41no—
78C2×C4×C163—21no—
79C2×C4×C163—41no—
80C2×C4×C163—41no—
81C2×C4×C163—41no—
82C2×C4×C163—41no—
83C2×C4×C163C2×D411yescharacteristic
84C2×C4×C163C2×D411yescharacteristic
85C8×C163—41no—
86C22×C163—21no—
87C22×C163—21no—
88C22×C163—41no—
89C22×C163—41no—
90C22×C163—41no—
91C22×C163#12011yescharacteristic
92C22×C163#13511yescharacteristic
93C22×C163#13711yescharacteristic
94C2×D163—42no—
95C2×D163—41no—
96C2×D163—41no—
97C2×D163—41no—
98C2×D163—81no—
99C4×C163—21no—
100C4×C163—21no—
101C4×C163—41no—
102C4×C163—81no—
103Dic163—42no—
104Dic163—41no—
105Dic163—41no—
106Dic163—41no—
107C2×C163—21no—
108C2×C163—21no—
109C2×C163—81no—
110C2×C163#45211yescharacteristic
111C2×C163#45311yescharacteristic
112C2×C163#45811yescharacteristic
113D163—81no—
114C163#161811yescharacteristic, complemented, Sylow
115#1618—1631nomaximal, Sylow
116#538—1631no—
117#484—1631no—
118#481D16311yescharacteristic, complemented
119#399—1631no—
120#328—1631no—
121#327—1631no—
122#323—1631no—
123#126—3261no—
124#124—3261no—
125#123—3261no—
126C22⋊Q8—1631no—
127#121—3261no—
128#121C2×D16311yescharacteristic
129#120—3261no—
130C4×D4—3261no—
131C2×C4⋊C4—3261no—
132C2×C4⋊C4C2×D16311yescharacteristic
133C2×C22⋊C4—3261no—
134C2×C22⋊C4—1631no—
135#107—3261no—
136#106—21no—
137#103—3261no—
138#102—21no—
139C22⋊C8—1631no—
140C22⋊C8C2×D16311yescharacteristic
141#95—1631no—
142C24—1631no—
143C2×Q8—1631no—
144C2×D4—21no—
145C2×D4—3261no—
146C2×D4—3261no—
147C2×D4C22⋊D16311yescharacteristic
148C22×C4—21no—
149C22×C4—3261no—
150C22×C4—3261no—
151C22×C4—1631no—
152C22×C4C22×D16311yescharacteristic
153C2×C8—21no—
154C2×C8—3261no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3261no—
158C4⋊C4—3261no—
159C4⋊C4—6522no—
160C4⋊C4—1631no—
161C4⋊C4C22⋊D16311yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—3261no—
164C22⋊C4—3261no—
165C22⋊C4—3261no—
166C22⋊C4—3261no—
167C22⋊C4—3261no—
168C22⋊C4—3261no—
169C22⋊C4—6522no—
170C42—3261no—
171C23—21no—
172C23—3261no—
173C23—3261no—
174C23—3261no—
175C23—6521no—
176C23—6521no—
177C23D4×D16311yescharacteristic
178Q8—6521no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—13041no—
183D4—13041no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—13041no—
190C2×C4—3262no—
191C2×C4—3261no—
192C2×C4—3261no—
193C2×C4—3261no—
194C2×C4—6521no—
195C2×C4—6521no—
196C2×C4—6521no—
197C2×C4D4×D16311yescharacteristic
198C2×C4C2×C22⋊D16311yescharacteristic
199C8—41no—
200C8—6521no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—13041no—
207C22—6522no—
208C22—6521no—
209C22—6521no—
210C22—6521no—
211C22#1910470311yescharacteristic
212C22#1910475611yescharacteristic
213C22#1910476411yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—81no—
218C4—6522no—
219C4—6521no—
220C4—6521no—
221C4—6521no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—13041no—
226C2#2378182411yescharacteristic
227C2#2378183111yescharacteristic
228C2#2378185411yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#481×C163C2
socleC22×C163#120
prime radical#23782252C2
hypercenter#481D163
hypocenterC163#1618
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #23782000C2semidirect
3. #23781600C2semidirect
4. #23781403C2semidirect
5. #23781399C2semidirect
6. #23781388C2semidirect
7. #481×C163C2semidirect
114. C163#1618semidirect
118. #481D163semidirect

Maximal subgroups

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

subgroupindex
#237822522
#237820002
#237816002
#237814032
#237813992
#237813882
#481×C1632
#1618163