Finite Group Databasebeta

← #35213490

Subgroups — #35213490

Order 20864.

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

normal subgroups only

This group has 308 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#23782227C211yesmaximal, characteristic
2#23781806C212yesmaximal, complemented
3#23781577C211yesmaximal, characteristic, complemented
4#23781400C212yesmaximal, complemented
5#323×C163C211yesmaximal, characteristic, complemented
6C22×Dic326C2211yescharacteristic, complemented
7C22×C22⋊D163—21no—
8C2×#14579688—22no—
9#19104719—22no—
10#19104709—21no—
11#19104709C2212yes—
12#19104705—22no—
13C2×C22⋊Dic163—21no—
14#19104660—22no—
15C163⋊#95—21no—
16C163⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yescomplemented
19C23×D163—21no—
20C2×Dic326—22no—
21C2×Dic326—42no—
22C2×D652—21no—
23C2×D652D412yes—
24C2×C22⋊D163—22no—
25C2×C22⋊D163—22no—
26C2×C22⋊D163—42no—
27C2×C22⋊D163—42no—
28C22×Dic163—21no—
29C22×Dic163—21no—
30C22×Dic163—21no—
31#14579688—44no—
32C4⋊Dic163—21no—
33C4⋊Dic163D412yes—
34C22⋊Dic163—42no—
35(C2×D163)⋊C4—22no—
36(C2×D163)⋊C4—22no—
37C24×C163D411yescharacteristic
38C22×C4×C163D411yescharacteristic
39C22×C4×C163C2311yescharacteristic
40C2×C8×C163—22no—
41C22⋊C4×C163—22no—
42C22×D163—22no—
43C22×D163—42no—
44C22×D163—81no—
45C22×D163—81no—
46Dic326—24no—
47Dic326—42no—
48D652—42no—
49D652—81no—
50C22⋊D163—44no—
51C22⋊D163—42no—
52C22⋊D163—42no—
53C2×Dic163—22no—
54C2×Dic163—22no—
55C2×Dic163—42no—
56C2×Dic163—41no—
57C2×Dic163—41no—
58C2×Dic163—41no—
59C2×Dic163—41no—
60C2×Dic163—41no—
61C23×C163—21no—
62C23×C163—21no—
63C23×C163—41no—
64C23×C163—41no—
65C23×C163SD1612yescomplemented
66C23×C163C2×D411yescharacteristic
67C2×C4×C163—21no—
68C2×C4×C163—41no—
69C2×C4×C163—41no—
70C2×C4×C163C2×D412yes—
71C8×C163—42no—
72C22×C163—21no—
73C22×C163—21no—
74C22×C163—22no—
75C22×C163—42no—
76C22×C163—41no—
77C22×C163—41no—
78C22×C163—81no—
79C22×C163#12011yescharacteristic
80C22×C163C2×SD1611yescharacteristic
81C22×C163AGL(1,ℤ/8)11yescharacteristic
82C2×D163—42no—
83C2×D163—42no—
84C2×D163—42no—
85C2×D163—84no—
86C4×C163—22no—
87C4×C163—41no—
88Dic163—42no—
89Dic163—42no—
90Dic163—81no—
91C2×C163—21no—
92C2×C163—21no—
93C2×C163—42no—
94C2×C163#45011yescharacteristic
95C2×C163#45511yescharacteristic
96C2×C163B(4,𝔽2)11yescharacteristic
97D163—82no—
98C163#160111yescharacteristic, complemented, Sylow
99#1601—1631nomaximal, Sylow
100#534—1631no—
101#448—1632no—
102#393—1631no—
103#327—1632no—
104#323D16311yescharacteristic, complemented
105C22×Q8—1631no—
106C22×D4—3261no—
107C2×SD16—3262no—
108#123—3262no—
109#121—3261no—
110#121—1632no—
111#120—3262no—
112C2×C22⋊C4—3261no—
113C2×C22⋊C4C2×D16311yescharacteristic
114#103—3262no—
115C22⋊C8C2×D16312yescomplemented
116#95—3261no—
117#95—1631no—
118C24—3261no—
119C24D65211yescharacteristic
120C2×Q8—3262no—
121C2×Q8—6522no—
122C2×D4—3261no—
123C2×D4—3262no—
124C2×D4—3262no—
125C2×D4—6522no—
126C2×D4—6522no—
127C2×D4—1632no—
128C22×C4—3261no—
129C22×C4—3261no—
130C22×C4—3261no—
131C22×C4D65211yescharacteristic
132C22×C4C22×D16311yescharacteristic
133SD16—6524no—
134C2×C8—22no—
135C4⋊C4—3261no—
136C4⋊C4—1632no—
137C22⋊C4—22no—
138C22⋊C4—3262no—
139C22⋊C4—3262no—
140C22⋊C4—6522no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—13041no—
146C23—13041no—
147C23—3262no—
148C23—6522no—
149C23#1457968812yescomplemented
150C23C2×D65211yescharacteristic
151Q8—3264no—
152Q8—6522no—
153D4—13041no—
154D4—6522no—
155D4—6524no—
156D4—6522no—
157D4—6522no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3262no—
162C2×C4—3262no—
163C2×C4—6521no—
164C2×C4—6521no—
165C2×C4—6521no—
166C2×C4—6522no—
167C2×C4—6521no—
168C2×C4—6521no—
169C2×C4D4×D16312yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—13044no—
179C22—6522no—
180C22—6522no—
181C22—6522no—
182C22#1910470511yescharacteristic
183C22C2×#1457968811yescharacteristic
184C22#1910476311yescharacteristic
185C4—22no—
186C4—41no—
187C4—13041no—
188C4—6522no—
189C4—6522no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—13042no—
194C2#2378181811yescharacteristic
195C2#2378184511yescharacteristic
196C2#2378185311yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#323×C163C2
socleC22×C163#120
prime radical#23782227C2
hypercenter#323D163
hypocenterC163#1601
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #23781806C2semidirect
3. #23781577C2semidirect
4. #23781400C2semidirect
5. #323×C163C2semidirect
6. C22×Dic326C22semidirect
16. C163⋊#95C22semidirect
18. C22⋊C8×C163C22semidirect
65. C23×C163SD16semidirect
98. C163#1601semidirect
104. #323D163semidirect
115. C22⋊C8C2×D163semidirect
149. C23#14579688semidirect

Maximal subgroups

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

subgroupindex
#237822272
#237818062
#237815772
#237814002
#323×C1632
#1601163