Finite Group Databasebeta

← #35213257

Subgroups — #35213257

Order 20864.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#23782216C211yesmaximal, characteristic
2#23781721C211yesmaximal, characteristic, complemented
3#23781687C211yesmaximal, characteristic, complemented
4#23781411C211yesmaximal, characteristic, complemented
5#23781408C211yesmaximal, characteristic, complemented
6#23781408C211yesmaximal, characteristic, complemented
7#332×C163C211yesmaximal, characteristic, complemented
8C2×#14579700—21no—
9C2×#14579682—21no—
10#19104732C2211yescharacteristic
11#19104724—21no—
12#19104710—21no—
13#19104710C412yes—
14#19104697—21no—
15#19104697C412yes—
16C42×D163C2211yescharacteristic
17C4⋊(C163⋊C8)C2211yescharacteristic
18C4⋊(C163⋊C8)C2211yescharacteristic
19Dic163⋊C8—21no—
20(C2×D163)⋊C8—21no—
21#19104645—21no—
22C4⋊Q8×C163C2211yescharacteristic
23C4⋊C8×C163C2211yescharacteristic
24C4⋊C8×C163C2211yescharacteristic
25#14579700—42no—
26#14579700—42no—
27C2×D652—22no—
28C2×D652—21no—
29C2×D652—21no—
30C2×D652C2×C412yes—
31C2×C4×D163—22no—
32C2×C4×D163C2×C411yescharacteristic
33C2×C4×D163D411yescharacteristic
34C2×C4×D163D411yescharacteristic
35#14579682—42no—
36C2×C163⋊C8—21no—
37C2×C163⋊C8—21no—
38(C2×D163)⋊C4—22no—
39(C2×D163)⋊C4—22no—
40C4×Dic163C2×C411yescharacteristic
41C4×Dic163D411yescharacteristic
42C4×Dic163D411yescharacteristic
43C2×Q8×C163—21no—
44C2×C8×C163—21no—
45C2×C8×C163—21no—
46C4⋊C4×C163—21no—
47C4⋊C4×C163C2×C412yes—
48C42×C163C2311yescharacteristic
49C22×D163—22no—
50C22×D163C22⋊C411yescharacteristic
51D652—42no—
52D652—44no—
53D652—41no—
54D652—41no—
55D652—41no—
56D652—41no—
57C2×Dic163C22⋊C411yescharacteristic
58C2×Dic163C22⋊C411yescharacteristic
59C2×Dic163C22⋊C411yescharacteristic
60C4×D163—21no—
61C4×D163—21no—
62C4×D163—21no—
63C4×D163—21no—
64C4×D163—41no—
65C4×D163—44no—
66C163⋊C8—41no—
67C163⋊C8—41no—
68Q8×C163—41no—
69Q8×C163—41no—
70C2×C4×C163—22no—
71C2×C4×C163C22×C411yescharacteristic
72C2×C4×C163C2×D411yescharacteristic
73C2×C4×C163C2×D411yescharacteristic
74C8×C163—41no—
75C8×C163—41no—
76C22×C163C2×C22⋊C411yescharacteristic
77C2×D163—21no—
78C2×D163—21no—
79C2×D163—21no—
80C2×D163—42no—
81C2×D163—42no—
82C2×D163—42no—
83C4×C163—21no—
84C4×C163—42no—
85C4×C163AGL(1,ℤ/8)11yescharacteristic
86C4×C163AGL(1,ℤ/8)11yescharacteristic
87C4×C163AGL(1,ℤ/8)11yescharacteristic
88C4×C163#13711yescharacteristic
89Dic163—21no—
90Dic163—21no—
91Dic163—21no—
92C2×C163#41311yescharacteristic
93C2×C163#41711yescharacteristic
94C2×C163#41811yescharacteristic
95D163—41no—
96D163—82no—
97C163#155711yescharacteristic, complemented, Sylow
98#1557—1631nomaximal, Sylow
99#532—1631no—
100#432—1631no—
101#423—1631no—
102#332—1631no—
103#332D16311yescharacteristic, complemented
104#331—1631no—
105#331—1631no—
106C2×#41—3261no—
107C2×M16—3261no—
108C4⋊Q8C2×D16311yescharacteristic
109#127—1631no—
110#124—3261no—
111#121—3261no—
112#121—1632no—
113C4×D4—3261no—
114C4×D4—1632no—
115C2×C42—1631no—
116C4⋊C8—3261no—
117C4⋊C8—1631no—
118C4⋊C8—1631no—
119C4⋊C8C2×D16311yescharacteristic
120C4⋊C8C2×D16311yescharacteristic
121C22⋊C8—3261no—
122#97—3261no—
123#41—6522no—
124#41—6522no—
125C2×Q8—21no—
126C2×D4—3262no—
127C2×D4—3261no—
128C2×D4—3261no—
129C2×D4—1632no—
130C22×C4—3262no—
131C22×C4—1631no—
132C22×C4—1631no—
133C22×C4—1631no—
134M16—6522no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—3261no—
138C2×C8—3261no—
139C4⋊C4—21no—
140C4⋊C4C4×D16312yes—
141C22⋊C4—3262no—
142C22⋊C4—3262no—
143C42—1631no—
144C42—1631no—
145C42—1631no—
146C42C22×D16311yescharacteristic
147C23—3262no—
148C23—1631no—
149Q8—41no—
150Q8—41no—
151D4—6522no—
152D4—6524no—
153D4—6521no—
154D4—6521no—
155D4—6521no—
156D4—6521no—
157C2×C4—22no—
158C2×C4—3261no—
159C2×C4—3261no—
160C2×C4—3261no—
161C2×C4—3261no—
162C2×C4—6521no—
163C2×C4—6524no—
164C2×C4—1631no—
165C2×C4—1631no—
166C2×C4—1631no—
167C2×C4C2×C4×D16311yescharacteristic
168C2×C4D4×D16311yescharacteristic
169C2×C4D4×D16311yescharacteristic
170C8—41no—
171C8—41no—
172C8—6521no—
173C8—6521no—
174C22—3261no—
175C22—3261no—
176C22—3261no—
177C22—6522no—
178C22—6522no—
179C22—6522no—
180C22C22⋊C4×D16311yescharacteristic
181C4—21no—
182C4—42no—
183C4—3261no—
184C4—3261no—
185C4—3261no—
186C4#1910475911yescharacteristic
187C4#1910476211yescharacteristic
188C4#1910476211yescharacteristic
189C4#1910476711yescharacteristic
190C2—13042no—
191C2—6521no—
192C2#2378164811yescharacteristic
193C2#2378166511yescharacteristic
194C2#2378166811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D163
commutatorC2×C4×C163C22×C4
FrattiniC42C22×D163
Fitting#332×C163C2
socleC22×C163C2×C22⋊C4
prime radical#23782216C2
cosocleC42×C163C23
hypercenter#332D163
hypocenterC163#1557
LCSG ▷ C2×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #23781721C2semidirect
3. #23781687C2semidirect
4. #23781411C2semidirect
5. #23781408C2semidirect
6. #23781408C2semidirect
7. #332×C163C2semidirect
97. C163#1557semidirect
103. #332D163semidirect

Maximal subgroups

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

subgroupindex
#237822162
#237817212
#237816872
#237814112
#237814082
#332×C1632
#1557163