Finite Group Databasebeta

← #35213497

Subgroups — #35213497

Order 20864.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782262C211yesmaximal, characteristic
2#23781821C211yesmaximal, characteristic, complemented
3#23781804C211yesmaximal, characteristic
4#23781401C211yesmaximal, characteristic, complemented
5#23781398C211yesmaximal, characteristic
6#23781391C211yesmaximal, characteristic, complemented
7#395×C163C211yesmaximal, characteristic, complemented
8C2×#14579689—21no—
9C2×#14579686—21no—
10#19104730—21no—
11#19104725—21no—
12#19104720—21no—
13#19104718—21no—
14#19104714C2211yescharacteristic
15#19104713—21no—
16#19104709C2211yescharacteristic
17C4×C22⋊D163—21no—
18#19104659—21no—
19#19104659—21no—
20C22⋊(C163⋊C8)C2211yescharacteristic
21C22⋊(C163⋊C8)C2211yescharacteristic
22C22×Q8×C163C2211yescharacteristic
23C2×C4⋊C4×C163—21no—
24C2×C4⋊C4×C163C2211yescharacteristic
25#95×C163—21no—
26#95×C163C2211yescharacteristic
27C2×Dic326D411yescharacteristic
28C2×D652D411yescharacteristic
29C2×C22⋊D163—21no—
30C2×C4×D163—21no—
31#14579689—42no—
32#14579686—42no—
33C2×C163⋊C8—21no—
34C2×C163⋊C8—21no—
35C4⋊Dic163D411yescharacteristic
36C4⋊Dic163D411yescharacteristic
37Dic163⋊C4—21no—
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41C22⋊Dic163—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46C4×Dic163—21no—
47C2×Q8×C163—21no—
48C2×Q8×C163—21no—
49C2×Q8×C163—42no—
50C22×C4×C163—21no—
51C22×C4×C163—21no—
52C22×C4×C163D411yescharacteristic
53C22×C4×C163D411yescharacteristic
54C22×C4×C163C2311yescharacteristic
55C4⋊C4×C163—21no—
56C4⋊C4×C163—21no—
57C4⋊C4×C163—42no—
58C22×D163—21no—
59Dic326—41no—
60D652—41no—
61C22⋊D163—41no—
62C22⋊D163—41no—
63C2×Dic163—21no—
64C2×Dic163—21no—
65C2×Dic163—21no—
66C4×D163—81no—
67C163⋊C8—41no—
68C163⋊C8—41no—
69C23×C163C2×D411yescharacteristic
70Q8×C163—22no—
71Q8×C163—22no—
72Q8×C163—41no—
73Q8×C163—41no—
74C2×C4×C163—22no—
75C2×C4×C163—22no—
76C2×C4×C163—21no—
77C2×C4×C163—21no—
78C2×C4×C163—21no—
79C2×C4×C163—21no—
80C2×C4×C163—41no—
81C2×C4×C163—41no—
82C2×C4×C163—41no—
83C2×C4×C163—41no—
84C2×C4×C163—41no—
85C2×C4×C163—41no—
86C2×C4×C163C2×D411yescharacteristic
87C2×C4×C163C2×D411yescharacteristic
88C22×C163—21no—
89C22×C163—21no—
90C22×C163#12011yescharacteristic
91C22×C163#13511yescharacteristic
92C22×C163#13711yescharacteristic
93C2×D163—41no—
94C2×D163—41no—
95C2×D163—41no—
96C4×C163—21no—
97C4×C163—21no—
98C4×C163—42no—
99C4×C163—41no—
100C4×C163—42no—
101C4×C163—41no—
102Dic163—41no—
103Dic163—41no—
104Dic163—41no—
105C2×C163—21no—
106C2×C163—21no—
107C2×C163#45211yescharacteristic
108C2×C163#45611yescharacteristic
109C2×C163B(4,𝔽2)11yescharacteristic
110D163—81no—
111C163#160311yescharacteristic, complemented, Sylow
112#1603—1631nomaximal, Sylow
113#539—1631no—
114#451—1631no—
115#448—1631no—
116#395D16311yescharacteristic, complemented
117#328—1631no—
118#327—1631no—
119#324—1631no—
120C22×Q8C2×D16311yescharacteristic
121C2×Q16—3261no—
122C2×SD16—3261no—
123#126—3261no—
124#125—3261no—
125#123—3261no—
126#123—3261no—
127C22⋊Q8—3261no—
128C22⋊Q8—1631no—
129#121—1631no—
130C4×D4—3261no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2×D16311yescharacteristic
133#103—3261no—
134#103—3261no—
135C22⋊C8—1631no—
136C22⋊C8—1631no—
137#95—21no—
138#95C2×D16311yescharacteristic
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8—1631no—
143C2×D4—3261no—
144C2×D4—1631no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—3261no—
148C22×C4C22⋊D16311yescharacteristic
149C22×C4C22⋊D16311yescharacteristic
150C22×C4C22×D16311yescharacteristic
151Q16—6522no—
152SD16—6522no—
153C2×C8—3261no—
154C2×C8—3261no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—3261no—
159C4⋊C4—3261no—
160C4⋊C4—3261no—
161C4⋊C4—3261no—
162C4⋊C4—1631no—
163C4⋊C4—1631no—
164C22⋊C4—3261no—
165C22⋊C4—3261no—
166C22⋊C4—3261no—
167C22⋊C4—3261no—
168C22⋊C4—3261no—
169C42—3261no—
170C23—3261no—
171C23C2×C22⋊D16311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—6521no—
177D4—6521no—
178D4—6521no—
179D4—6521no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—13041no—
193C2×C4—3261no—
194C2×C4—3261no—
195C2×C4—3261no—
196C2×C4D4×D16311yescharacteristic
197C2×C4D4×D16311yescharacteristic
198C8—6521no—
199C8—6521no—
200C22—21no—
201C22—21no—
202C22—6521no—
203C22—6521no—
204C22—6521no—
205C22#1910470311yescharacteristic
206C22#1910475511yescharacteristic
207C22#1910476611yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—42no—
213C4—41no—
214C4—6521no—
215C4—6521no—
216C4—6521no—
217C2—21no—
218C2—21no—
219C2—13041no—
220C2#2378182511yescharacteristic
221C2#2378184711yescharacteristic
222C2#2378185111yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#395×C163C2
socleC22×C163#120
prime radical#19104709C22
hypercenter#395D163
hypocenterC163#1603
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #23781821C2semidirect
4. #23781401C2semidirect
6. #23781391C2semidirect
7. #395×C163C2semidirect
111. C163#1603semidirect
116. #395D163semidirect

Maximal subgroups

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

subgroupindex
#237822622
#237818212
#237818042
#237814012
#237813982
#237813912
#395×C1632
#1603163