Finite Group Databasebeta

← #35213524

Subgroups — #35213524

Order 20864.

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

normal subgroups only

This group has 280 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#23781979C212yesmaximal, complemented
3#23781590C211yesmaximal, characteristic, complemented
4#23781400C212yesmaximal, complemented
5#323×C163C211yesmaximal, characteristic, complemented
6C22×C22⋊D163—21no—
7#19104719—22no—
8#19104709—21no—
9#19104709C2212yes—
10#19104705—22no—
11C2×C4⋊Dic163C2211yescharacteristic, complemented
12C2×C22⋊Dic163—21no—
13#19104669—22no—
14#19104657—22no—
15C163⋊#95—21no—
16C163⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yescomplemented
19C23×D163—21no—
20C2×D652—21no—
21C2×D652D412yes—
22C2×C22⋊D163—22no—
23C2×C22⋊D163—22no—
24C2×C22⋊D163—42no—
25C2×C22⋊D163—42no—
26C22×Dic163—21no—
27C22×Dic163—21no—
28C22×Dic163—21no—
29C4⋊Dic163—21no—
30C4⋊Dic163—22no—
31C4⋊Dic163D412yes—
32C22⋊Dic163—42no—
33(C2×D163)⋊C4—22no—
34(C2×D163)⋊C4—22no—
35C24×C163D411yescharacteristic
36C22×C4×C163D411yescharacteristic
37C22×C4×C163C2311yescharacteristic
38C2×C8×C163—22no—
39C22⋊C4×C163—22no—
40C22×D163—22no—
41C22×D163—42no—
42C22×D163—81no—
43C22×D163—81no—
44D652—42no—
45D652—81no—
46C22⋊D163—44no—
47C22⋊D163—42no—
48C22⋊D163—42no—
49C2×Dic163—22no—
50C2×Dic163—42no—
51C2×Dic163—41no—
52C2×Dic163—41no—
53C2×Dic163—41no—
54C2×Dic163—41no—
55C2×Dic163—41no—
56C2×Dic163—41no—
57C23×C163—21no—
58C23×C163—21no—
59C23×C163—41no—
60C23×C163—41no—
61C23×C163D812yes—
62C23×C163C2×D411yescharacteristic
63C2×C4×C163—21no—
64C2×C4×C163—41no—
65C2×C4×C163—41no—
66C2×C4×C163C2×D412yes—
67C8×C163—42no—
68C22×C163—21no—
69C22×C163—21no—
70C22×C163—22no—
71C22×C163—42no—
72C22×C163—41no—
73C22×C163—41no—
74C22×C163—81no—
75C22×C163#12011yescharacteristic
76C22×C163C2×D811yescharacteristic
77C22×C163AGL(1,ℤ/8)11yescharacteristic
78C2×D163—42no—
79C2×D163—42no—
80C2×D163—42no—
81C2×D163—84no—
82C4×C163—22no—
83C4×C163—41no—
84Dic163—42no—
85Dic163—81no—
86Dic163—81no—
87C2×C163—21no—
88C2×C163—21no—
89C2×C163—42no—
90C2×C163#44711yescharacteristic
91C2×C163#45511yescharacteristic
92C2×C163#45811yescharacteristic
93D163—82no—
94C163#160711yescharacteristic, complemented, Sylow
95#1607—1631nomaximal, Sylow
96#534—1631no—
97#480—1632no—
98#397—1631no—
99#327—1632no—
100#323D16311yescharacteristic, complemented
101C22×D4—3261no—
102#123—3262no—
103#121—3261no—
104#121—1632no—
105#120—3262no—
106C2×C4⋊C4—1631no—
107C2×C22⋊C4—3261no—
108C2×C22⋊C4C2×D16311yescharacteristic
109#107—3262no—
110#102—3262no—
111C22⋊C8C2×D16312yescomplemented
112#95—3261no—
113#95—1631no—
114C24—3261no—
115C24D65211yescharacteristic
116C2×D4—3261no—
117C2×D4—3262no—
118C2×D4—3262no—
119C2×D4—6522no—
120C2×D4—6522no—
121C2×D4—1632no—
122C22×C4—3261no—
123C22×C4—3261no—
124C22×C4—3261no—
125C22×C4D65211yescharacteristic
126C22×C4C22×D16311yescharacteristic
127C2×C8—22no—
128C4⋊C4—3261no—
129C4⋊C4—3262no—
130C4⋊C4—1632no—
131C22⋊C4—22no—
132C22⋊C4—3262no—
133C22⋊C4—3262no—
134C22⋊C4—6522no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—13041no—
140C23—13041no—
141C23—3262no—
142C23—6522no—
143C23D130412yes—
144C23C2×D65211yescharacteristic
145D4—13041no—
146D4—6522no—
147D4—6524no—
148D4—6522no—
149D4—6522no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—3262no—
154C2×C4—6521no—
155C2×C4—6521no—
156C2×C4—6521no—
157C2×C4—6521no—
158C2×C4—6521no—
159C2×C4—6521no—
160C2×C4—6522no—
161C2×C4D4×D16312yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—13044no—
171C22—6522no—
172C22—6522no—
173C22—6522no—
174C22#1910470511yescharacteristic
175C22C2×D130411yescharacteristic
176C22#1910476311yescharacteristic
177C4—22no—
178C4—41no—
179C4—13041no—
180C4—13041no—
181C4—6522no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—13042no—
186C2#2378180111yescharacteristic
187C2#2378184511yescharacteristic
188C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#323×C163C2
socleC22×C163#120
prime radical#23782227C2
hypercenter#323D163
hypocenterC163#1607
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. #23781979C2semidirect
3. #23781590C2semidirect
4. #23781400C2semidirect
5. #323×C163C2semidirect
11. C2×C4⋊Dic163C22semidirect
16. C163⋊#95C22semidirect
18. C22⋊C8×C163C22semidirect
94. C163#1607semidirect
100. #323D163semidirect
111. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822272
#237819792
#237815902
#237814002
#323×C1632
#1607163