Finite Group Databasebeta

← #35215580

Subgroups — #35215580

Order 20864.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19104763C212yesmaximal, complemented
2#23782246C211yesmaximal, characteristic, complemented
3C2×#19104661C212yesmaximal, complemented
4#23781566C211yesmaximal, characteristic, complemented
5#343×C163C211yesmaximal, characteristic, complemented
6C2×#14579699C2212yes—
7C22×D652C2211yescharacteristic
8#19104763—24no—
9#19104763—24no—
10C2×#14579688—22no—
11C2×D1304—22no—
12#19104715—22no—
13#19104710—22no—
14#19104709—22no—
15C2×(C2×D163)⋊C4—22no—
16C2×C4×Dic163C2211yescharacteristic
17#19104661—24no—
18C2×M16×C163C2212yes—
19C2×C4⋊C4×C163C2211yescharacteristic
20C23×D163—21no—
21#14579699—22no—
22#14579699—22no—
23#14579699—22no—
24#14579699—22no—
25C2×Dic326D412yes—
26C2×D652—22no—
27C2×D652—22no—
28C2×D652—22no—
29C2×D652—21no—
30C2×D652D412yes—
31C2×D652D412yescomplemented
32C2×C22⋊D163—22no—
33C22×Dic163—21no—
34C2×C4×D163—22no—
35#14579688—28no—
36D1304—28no—
37C4⋊Dic163—22no—
38(C2×D163)⋊C4—24no—
39(C2×D163)⋊C4—44no—
40C4×Dic163—22no—
41C22×C4×C163D412yes—
42C22×C4×C163C2311yescharacteristic
43M16×C163—22no—
44M16×C163D414yescomplemented
45C2×C8×C163—22no—
46C4⋊C4×C163—22no—
47C22×D163—22no—
48C22×D163—22no—
49C22×D163—42no—
50C22×D163—42no—
51C22×D163—41no—
52C22×D163—41no—
53Dic326—22no—
54Dic326—22no—
55D652—22no—
56D652—24no—
57D652—22no—
58D652—42no—
59D652—82no—
60C22⋊D163—42no—
61C22⋊D163—42no—
62C2×Dic163—22no—
63C2×Dic163—22no—
64C2×Dic163—41no—
65C2×Dic163—41no—
66C4×D163—42no—
67C4×D163—42no—
68C23×C163C2×D411yescharacteristic
69C2×C4×C163—22no—
70C2×C4×C163—42no—
71C2×C4×C163C2×D411yescharacteristic
72C2×C4×C163C2×D412yes—
73C2×C4×C163C2×D412yes—
74C2×C4×C163#4111yescharacteristic
75C8×C163—24no—
76C22×C163—21no—
77C22×C163—21no—
78C22×C163#12011yescharacteristic
79C22×C163#12111yescharacteristic
80C22×C163#12711yescharacteristic
81C2×D163—44no—
82C2×D163—42no—
83C2×D163—42no—
84C2×D163—42no—
85C2×D163—42no—
86C2×D163—82no—
87C2×D163—81no—
88C2×D163—81no—
89C4×C163—42no—
90C4×C163#12012yes—
91C4×C163#12112yes—
92Dic163—42no—
93Dic163—42no—
94C2×C163—21no—
95C2×C163—21no—
96C2×C163#39211yescharacteristic
97C2×C163#45311yescharacteristic
98C2×C163#45411yescharacteristic
99D163—82no—
100D163—82no—
101C163#200611yescharacteristic, complemented, Sylow
102#2006—1631nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1632no—
104#537—1631no—
105C2×#104—1632no—
106#390—1631no—
107#343D16311yescharacteristic, complemented
108C2×#41—1632no—
109C22×D4—1631no—
110AGL(1,ℤ/8)—3264no—
111AGL(1,ℤ/8)—3264no—
112C2×SD16—3262no—
113C2×D8—3262no—
114C2×M16C2×D16312yes—
115C22⋊Q8—3262no—
116#121—3262no—
117#121—3262no—
118C2×C4⋊C4C2×D16311yescharacteristic
119C2×C22⋊C4—3262no—
120C2×C42—1631no—
121#104—3264no—
122C24—3261no—
123#41—3262no—
124#41—3262no—
125#41—3262no—
126#41—3262no—
127C2×Q8—1632no—
128C2×D4—3262no—
129C2×D4—3262no—
130C2×D4—3262no—
131C2×D4—3262no—
132C2×D4—3261no—
133C2×D4—1632no—
134C2×D4—1632no—
135C22×C4—3261no—
136C22×C4—3262no—
137C22×C4D65212yes—
138C22×C4C22×D16311yescharacteristic
139SD16—3268no—
140D8—3268no—
141M16—22no—
142M16D65214yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3262no—
146C22⋊C4—3264no—
147C22⋊C4—6524no—
148C42—3262no—
149C23—3262no—
150C23—3262no—
151C23—6522no—
152C23—6522no—
153C23—6521no—
154C23—6521no—
155C23C2×D65211yescharacteristic
156Q8—3262no—
157Q8—3262no—
158D4—13042no—
159D4—3262no—
160D4—3264no—
161D4—3262no—
162D4—6522no—
163D4—6522no—
164D4—6522no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—3262no—
168C2×C4—3262no—
169C2×C4—6521no—
170C2×C4—6521no—
171C2×C4—6522no—
172C2×C4—6522no—
173C2×C4D4×D16311yescharacteristic
174C2×C4D4×D16312yes—
175C2×C4C2×D65212yes—
176C2×C4#1457970011yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—13042no—
181C22—13041no—
182C22—13041no—
183C22—6524no—
184C22—6522no—
185C22—6522no—
186C22—6522no—
187C22—6522no—
188C22#1910470511yescharacteristic
189C22#1910471011yescharacteristic
190C22#1910473211yescharacteristic
191C4—42no—
192C4—6522no—
193C4—6522no—
194C4#1910470512yes—
195C4#1910471012yes—
196C2—21no—
197C2—21no—
198C2—13042no—
199C2—13042no—
200C2#2378157511yescharacteristic
201C2#2378183511yescharacteristic
202C2#2378184011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#343×C163C2
socleC22×C163#120
hypercenter#343D163
hypocenterC163#2006
LCSG ▷ C22×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. #23782246C2semidirect
3. C2×#19104661C2semidirect
4. #23781566C2semidirect
5. #343×C163C2semidirect
31. C2×D652D4semidirect
44. M16×C163D4semidirect
101. C163#2006semidirect
107. #343D163semidirect
142. M16D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
#237822462
C2×#191046612
#237815662
#343×C1632
#2006163