Finite Group Databasebeta

← #35216204

Subgroups — #35216204

Order 20864.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#23782233C211yesmaximal, characteristic
2#99×D163C212yesmaximal, complemented
3#23781460C211yesmaximal, characteristic, complemented
4#23781457C212yesmaximal, complemented
5#354×C163C211yesmaximal, characteristic, complemented
6C2×#14579700C411yescharacteristic, complemented
7C2×D4×D163C2211yescharacteristic
8#19104732C411yescharacteristic, complemented
9#19104731C2211yescharacteristic
10#19104724C411yescharacteristic, complemented
11#19104708—22no—
12#19104705—22no—
13#19104692C411yescharacteristic, complemented
14C22⋊C4×D163—22no—
15#19104650C2212yes—
16#19104649—22no—
17#124×C163C2211yescharacteristic
18#99×C163C2212yes—
19C23×D163D412yes—
20#14579700—42no—
21C2×D652—21no—
22C2×D652—22no—
23C2×D652C2×C411yescharacteristic
24C2×D652C2×C411yescharacteristic, complemented
25C2×C22⋊D163—22no—
26C2×C22⋊D163D412yes—
27D4×D163—42no—
28C2×C4×D163—21no—
29C2×C4×D163—42no—
30C2×C4×D163C2×C411yescharacteristic
31Dic163⋊C4—21no—
32C22⋊Dic163—22no—
33(C2×D163)⋊C4—21no—
34(C2×D163)⋊C4—22no—
35(C2×D163)⋊C4—42no—
36C4×Dic163C2×C411yescharacteristic, complemented
37C2×Q8×C163C2×C411yescharacteristic, complemented
38C2×D4×C163C2311yescharacteristic
39C22⋊C4×C163—22no—
40C22⋊C4×C163—22no—
41C42×C163C2×C411yescharacteristic, complemented
42C22×D163—22no—
43C22×D163—22no—
44C22×D163—21no—
45C22×D163—82no—
46C22×D163C22⋊C412yes—
47C22×D163C22⋊C411yescharacteristic
48D652—41no—
49D652—41no—
50D652—42no—
51D652—82no—
52C22⋊D163—42no—
53C22⋊D163—82no—
54C2×Dic163—21no—
55C2×Dic163—42no—
56C2×Dic163C22⋊C411yescharacteristic
57C4×D163—22no—
58C4×D163—42no—
59C4×D163—84no—
60C23×C163C2×D412yes—
61Q8×C163—41no—
62D4×C163—41no—
63C2×C4×C163—21no—
64C2×C4×C163—21no—
65C2×C4×C163—42no—
66C2×C4×C163C22×C411yescharacteristic
67C22×C163—22no—
68C22×C163—42no—
69C22×C163C2×C22⋊C411yescharacteristic
70C2×D163—22no—
71C2×D163—41no—
72C2×D163—44no—
73C2×D163—42no—
74C2×D163—42no—
75C2×D163—41no—
76C2×D163—82no—
77C2×D163#9912yes—
78C4×C163—21no—
79C4×C163—41no—
80C4×C163—41no—
81C4×C163—82no—
82Dic163—21no—
83Dic163—41no—
84Dic163—82no—
85C2×C163—21no—
86C2×C163—42no—
87C2×C163C2×#9911yescharacteristic
88D163—22no—
89D163—42no—
90D163—81no—
91C163#212711yescharacteristic, complemented, Sylow
92#2127—1631nomaximal, Sylow
93#535—1631no—
94C2×#99—1632no—
95#354—1631no—
96#354D16311yescharacteristic, complemented
97#353—1632no—
98C2×#41—1631no—
99C22×D4—1631no—
100#127—1631no—
101#127—1631no—
102#124—1631no—
103#124C2×D16311yescharacteristic
104#121—3262no—
105#120—3262no—
106#117—1631no—
107C2×C22⋊C4—3262no—
108#99—3262no—
109#99—1632no—
110#99C2×D16312yes—
111C24—1632no—
112#41—6522no—
113C2×Q8C4×D16311yescharacteristic, complemented
114C2×D4—3262no—
115C2×D4—3261no—
116C2×D4—3262no—
117C2×D4—6522no—
118C2×D4—1632no—
119C2×D4—1631no—
120C2×D4—1631no—
121C2×D4C22×D16311yescharacteristic
122C22×C4—3261no—
123C22×C4—6522no—
124C22×C4—1631no—
125C4⋊C4—3261no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—3261no—
129C22⋊C4—3262no—
130C22⋊C4—3262no—
131C22⋊C4—6522no—
132C42—1631no—
133C42C4×D16311yescharacteristic, complemented
134C23—13042no—
135C23—3262no—
136C23—3262no—
137C23—3261no—
138C23—1632no—
139C23—1631no—
140C23D4×D16312yes—
141Q8—41no—
142D4—41no—
143D4—13042no—
144D4—13042no—
145D4—6521no—
146D4—6521no—
147D4—6522no—
148D4—6522no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—13044no—
153C2×C4—3262no—
154C2×C4—3261no—
155C2×C4—6522no—
156C2×C4—6522no—
157C2×C4—1631no—
158C2×C4C2×C4×D16311yescharacteristic
159C22—22no—
160C22—42no—
161C22—13042no—
162C22—3262no—
163C22—6522no—
164C22—6521no—
165C22—6522no—
166C22—6524no—
167C22—6521no—
168C22—1632no—
169C22C22⋊C4×D16311yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—13042no—
175C4—3261no—
176C4—6521no—
177C2—21no—
178C2—42no—
179C2—13041no—
180C2—3262no—
181C2—6522no—
182C2#99×D16311yescharacteristic
subgroupG/H
centerC2#99×D163
commutatorC2×C4×C163C22×C4
FrattiniC2×D4C22×D163
Fitting#354×C163C2
socleC2×C163C2×#99
prime radical#23782233C2
cosocleC2×D4×C163C23
hypercenter#354D163
hypocenterC163#2127
LCSG ▷ C2×C4×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D163C2semidirect
3. #23781460C2semidirect
4. #23781457C2semidirect
5. #354×C163C2semidirect
6. C2×#14579700C4semidirect
8. #19104732C4semidirect
10. #19104724C4semidirect
13. #19104692C4semidirect
24. C2×D652C2×C4semidirect
36. C4×Dic163C2×C4semidirect
37. C2×Q8×C163C2×C4semidirect
41. C42×C163C2×C4semidirect
91. C163#2127semidirect
96. #354D163semidirect
113. C2×Q8C4×D163semidirect
133. C42C4×D163semidirect

Maximal subgroups

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

subgroupindex
#237822332
#99×D1632
#237814602
#237814572
#354×C1632
#2127163