Finite Group Databasebeta

← #35213498

Subgroups — #35213498

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#23782264C211yesmaximal, characteristic
2#23781804C211yesmaximal, characteristic
3#23781584C211yesmaximal, characteristic, complemented
4#23781402C211yesmaximal, characteristic, complemented
5#23781400C211yesmaximal, characteristic
6#23781390C211yesmaximal, characteristic, complemented
7#451×C163C211yesmaximal, characteristic, complemented
8C2×#14579686—21no—
9#19104730—21no—
10#19104726—21no—
11#19104720—21no—
12#19104719—21no—
13#19104711—21no—
14#19104709C2211yescharacteristic
15(C22⋊D163)⋊C4—21no—
16C2×Dic163⋊C4—21no—
17C2×Dic163⋊C4C2211yescharacteristic
18#19104659—21no—
19C22⋊(C163⋊C8)C2211yescharacteristic
20C163⋊#95—21no—
21C163⋊#95C2211yescharacteristic
22C22×Q8×C163C2211yescharacteristic
23C2×Q16×C163—21no—
24C22⋊Q8×C163C2211yescharacteristic
25#103×C163—21no—
26C22⋊C8×C163C2211yescharacteristic
27C2×D652D411yescharacteristic
28C2×C22⋊D163—21no—
29C22×Dic163—21no—
30C22×Dic163D411yescharacteristic
31C22×Dic163D411yescharacteristic
32C2×C4×D163—21no—
33#14579686—42no—
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163—21no—
37C4⋊Dic163D411yescharacteristic
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41Dic163⋊C4—42no—
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—
50C2×Q8×C163D411yescharacteristic
51C22×C4×C163—21no—
52C22×C4×C163C2311yescharacteristic
53Q16×C163—42no—
54C2×C8×C163—21no—
55C4⋊C4×C163—21no—
56C4⋊C4×C163D411yescharacteristic
57C22⋊C4×C163—21no—
58C22×D163—21no—
59D652—41no—
60C22⋊D163—41no—
61C22⋊D163—41no—
62C2×Dic163—22no—
63C2×Dic163—21no—
64C2×Dic163—21no—
65C2×Dic163—21no—
66C2×Dic163—21no—
67C2×Dic163—21no—
68C2×Dic163—41no—
69C2×Dic163—41no—
70C2×Dic163—41no—
71C4×D163—81no—
72C163⋊C8—41no—
73C23×C163C2×D411yescharacteristic
74Q8×C163—22no—
75Q8×C163—22no—
76Q8×C163—41no—
77Q8×C163—41no—
78Q8×C163—41no—
79C2×C4×C163—22no—
80C2×C4×C163—21no—
81C2×C4×C163—21no—
82C2×C4×C163—41no—
83C2×C4×C163—41no—
84C2×C4×C163—41no—
85C2×C4×C163C2×D411yescharacteristic
86C2×C4×C163C2×D411yescharacteristic
87C8×C163—41no—
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—41no—
99C4×C163—42no—
100C4×C163—41no—
101Dic163—42no—
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#451D16311yescharacteristic, complemented
115#448—1631no—
116#395—1631no—
117#328—1631no—
118#327—1631no—
119#324—1631no—
120C22×Q8C2×D16311yescharacteristic
121C2×Q16—21no—
122C2×SD16—3261no—
123#126—3261no—
124#125—3261no—
125#123—3261no—
126#123—3261no—
127C22⋊Q8—3261no—
128C22⋊Q8C2×D16311yescharacteristic
129#121—1631no—
130C4×D4—3261no—
131C2×C4⋊C4—3261no—
132C2×C4⋊C4—1631no—
133#103—21no—
134#103—3261no—
135C22⋊C8—1631no—
136C22⋊C8C2×D16311yescharacteristic
137#95—3261no—
138#95—1631no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D16311yescharacteristic
143C2×D4—3261no—
144C2×D4—1631no—
145C22×C4—21no—
146C22×C4—3261no—
147C22×C4—3261no—
148C22×C4—1631no—
149C22×C4—1631no—
150C22×C4C22×D16311yescharacteristic
151Q16—42no—
152SD16—6522no—
153C2×C8—21no—
154C2×C8—3261no—
155C4⋊C4—21no—
156C4⋊C4—3261no—
157C4⋊C4—3261no—
158C4⋊C4—3261no—
159C4⋊C4—3261no—
160C4⋊C4—3261no—
161C4⋊C4—6522no—
162C4⋊C4—1631no—
163C4⋊C4C22⋊D16311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3261no—
166C22⋊C4—3261no—
167C22⋊C4—3261no—
168C22⋊C4—3261no—
169C42—3261no—
170C23—3261no—
171C23D4×D16311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—6521no—
178D4—6521no—
179D4—6521no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—13041no—
187C2×C4—3262no—
188C2×C4—3261no—
189C2×C4—3261no—
190C2×C4—3261no—
191C2×C4—3261no—
192C2×C4—3261no—
193C2×C4—6521no—
194C2×C4—6521no—
195C2×C4—6521no—
196C2×C4D4×D16311yescharacteristic
197C2×C4C2×C22⋊D16311yescharacteristic
198C8—41no—
199C8—6521no—
200C22—21no—
201C22—21no—
202C22—6521no—
203C22—6521no—
204C22—6521no—
205C22#1910470311yescharacteristic
206C22#1910475811yescharacteristic
207C22#1910476711yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—6522no—
214C4—6521no—
215C4—6521no—
216C4—6521no—
217C2—21no—
218C2—21no—
219C2—13041no—
220C2#2378182611yescharacteristic
221C2#2378184611yescharacteristic
222C2#2378185111yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#451×C163C2
socleC22×C163#120
prime radical#19104709C22
hypercenter#451D163
hypocenterC163#1603
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #23781584C2semidirect
4. #23781402C2semidirect
6. #23781390C2semidirect
7. #451×C163C2semidirect
111. C163#1603semidirect
114. #451D163semidirect

Maximal subgroups

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

subgroupindex
#237822642
#237818042
#237815842
#237814022
#237814002
#237813902
#451×C1632
#1603163