Finite Group Databasebeta

← #35213473

Subgroups — #35213473

Order 20864.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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#23782246C211yesmaximal, characteristic, complemented
2#23781801C212yesmaximal, complemented
3#23781582C211yesmaximal, characteristic, complemented
4#23781400C212yesmaximal, complemented
5#324×C163C211yesmaximal, characteristic, complemented
6C2×#14579699—21no—
7C22×D652C2211yescharacteristic
8C2×D1304—22no—
9#19104715—22no—
10#19104710—22no—
11#19104709—21no—
12#19104709C2212yes—
13C2×(C2×D163)⋊C4—21no—
14C2×(C2×D163)⋊C4—21no—
15#19104657—22no—
16C163⋊#95C2211yescharacteristic
17C2×C4⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yes—
19C23×D163—21no—
20#14579699—42no—
21#14579699—42no—
22C2×Dic326—21no—
23C2×D652—21no—
24C2×D652—21no—
25C2×D652—22no—
26C2×D652—42no—
27C2×D652D412yes—
28C2×C22⋊D163—22no—
29C22×Dic163—21no—
30C2×C4×D163—22no—
31D1304—44no—
32C4⋊Dic163—21no—
33C4⋊Dic163D412yes—
34(C2×D163)⋊C4—22no—
35(C2×D163)⋊C4—22no—
36(C2×D163)⋊C4—42no—
37(C2×D163)⋊C4—42no—
38C22×C4×C163D411yescharacteristic
39C22×C4×C163D411yescharacteristic
40C22×C4×C163C2311yescharacteristic
41C2×C8×C163—22no—
42C4⋊C4×C163—22no—
43C22×D163—22no—
44C22×D163—22no—
45C22×D163—41no—
46C22×D163—41no—
47C22×D163—41no—
48C22×D163—41no—
49C22×D163—41no—
50C22×D163—41no—
51Dic326—42no—
52D652—24no—
53D652—42no—
54D652—42no—
55D652—42no—
56D652—81no—
57C22⋊D163—42no—
58C22⋊D163—42no—
59C2×Dic163—22no—
60C2×Dic163—41no—
61C2×Dic163—41no—
62C2×Dic163—41no—
63C4×D163—44no—
64C23×C163C2×D411yescharacteristic
65C2×C4×C163—21no—
66C2×C4×C163—21no—
67C2×C4×C163—21no—
68C2×C4×C163—41no—
69C2×C4×C163—41no—
70C2×C4×C163D812yescomplemented
71C2×C4×C163C2×D412yes—
72C8×C163—42no—
73C22×C163—21no—
74C22×C163—21no—
75C22×C163#12011yescharacteristic
76C22×C163C2×D811yescharacteristic
77C22×C163AGL(1,ℤ/8)11yescharacteristic
78C2×D163—42no—
79C2×D163—42no—
80C2×D163—42no—
81C2×D163—42no—
82C2×D163—42no—
83C2×D163—42no—
84C2×D163—81no—
85C2×D163—81no—
86C2×D163—81no—
87C2×D163—81no—
88C4×C163—22no—
89C4×C163—22no—
90C4×C163—41no—
91Dic163—42no—
92Dic163—81no—
93C2×C163—21no—
94C2×C163—21no—
95C2×C163#44711yescharacteristic
96C2×C163#45411yescharacteristic
97C2×C163B(4,𝔽2)11yescharacteristic
98D163—82no—
99D163—82no—
100C163#159811yescharacteristic, complemented, Sylow
101#1598—1631nomaximal, Sylow
102#537—1631no—
103#447—1632no—
104#394—1631no—
105#327—1632no—
106#324D16311yescharacteristic, complemented
107C2×#41—3261no—
108C22×D4—1631no—
109C2×D8—3262no—
110C22⋊Q8—3262no—
111#121—3262no—
112#121—3261no—
113#121—1632no—
114C2×C4⋊C4C2×D16311yescharacteristic
115C2×C22⋊C4—3261no—
116C2×C22⋊C4—3261no—
117#102—3262no—
118C22⋊C8C2×D16312yes—
119#95—1631no—
120C24—3261no—
121#41—6522no—
122#41—6522no—
123C2×Q8—3261no—
124C2×D4—3261no—
125C2×D4—3261no—
126C2×D4—3262no—
127C2×D4—3262no—
128C2×D4—6522no—
129C2×D4—1632no—
130C22×C4—3262no—
131C22×C4—3261no—
132C22×C4D65211yescharacteristic
133C22×C4D65211yescharacteristic
134C22×C4C22×D16311yescharacteristic
135D8—6524no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—3261no—
139C4⋊C4—1632no—
140C22⋊C4—3262no—
141C22⋊C4—3262no—
142C22⋊C4—6522no—
143C22⋊C4—6522no—
144C23—3262no—
145C23—3262no—
146C23—6521no—
147C23—6521no—
148C23—6521no—
149C23—6521no—
150C23—6521no—
151C23—6521no—
152C23C2×D65211yescharacteristic
153Q8—6522no—
154D4—13041no—
155D4—3264no—
156D4—6522no—
157D4—6522no—
158D4—6522no—
159D4—6522no—
160D4—6522no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—3262no—
167C2×C4—6524no—
168C2×C4—6521no—
169C2×C4—6521no—
170C2×C4—6521no—
171C2×C4D130412yescomplemented
172C2×C4D4×D16312yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—13041no—
177C22—13041no—
178C22—13041no—
179C22—13041no—
180C22—6522no—
181C22—6522no—
182C22—6522no—
183C22—6522no—
184C22—6522no—
185C22—6522no—
186C22#1910470511yescharacteristic
187C22C2×D130411yescharacteristic
188C22#1910476311yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—13041no—
193C4—6522no—
194C2—21no—
195C2—21no—
196C2—13042no—
197C2—13042no—
198C2#2378180111yescharacteristic
199C2#2378184011yescharacteristic
200C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#324×C163C2
socleC22×C163#120
hypercenter#324D163
hypocenterC163#1598
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #23782246C2semidirect
2. #23781801C2semidirect
3. #23781582C2semidirect
4. #23781400C2semidirect
5. #324×C163C2semidirect
70. C2×C4×C163D8semidirect
100. C163#1598semidirect
106. #324D163semidirect
171. C2×C4D1304semidirect

Maximal subgroups

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

subgroupindex
#237822462
#237818012
#237815822
#237814002
#324×C1632
#1598163