Finite Group Databasebeta

← #35215676

Subgroups — #35215676

Order 20864.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#19104734C211yesmaximal, characteristic
2C2×#19104660C211yesmaximal, characteristic
3C2×#19104658C211yesmaximal, characteristic
4#23781570C211yesmaximal, characteristic
5#23781427C211yesmaximal, characteristic
6#23781419C211yesmaximal, characteristic
7#340×C163C211yesmaximal, characteristic
8C22×Dic326—21no—
9C22×Dic326C2211yescharacteristic
10C22×C163⋊C8C2211yescharacteristic
11#19104734—22no—
12#19104734—22no—
13C2×C4⋊Dic163C2211yescharacteristic
14C2×Dic163⋊C4—21no—
15C2×C4×Dic163C2211yescharacteristic
16#19104660—22no—
17#19104658—22no—
18C163⋊#95—21no—
19C163⋊#95—21no—
20C22×C8×C163C2211yescharacteristic
21C2×C4⋊C4×C163C2211yescharacteristic
22C2×C4⋊C4×C163C2211yescharacteristic
23C2×Dic326—24no—
24C2×Dic326—21no—
25C2×Dic326—21no—
26C2×Dic326—21no—
27C2×Dic326—21no—
28C2×Dic326—21no—
29C2×Dic326—42no—
30C2×Dic326—42no—
31C2×Dic326Q812yescomplemented
32C22×Dic163—21no—
33C22×Dic163—21no—
34C22×Dic163D411yescharacteristic
35C22×Dic163D411yescharacteristic
36C2×C163⋊C8—21no—
37C2×C163⋊C8—21no—
38C2×C163⋊C8—21no—
39C4⋊Dic163—21no—
40C4⋊Dic163D412yes—
41Dic163⋊C4—24no—
42C4×Dic163—21no—
43C4×Dic163—21no—
44C22×C4×C163—21no—
45C22×C4×C163—21no—
46C22×C4×C163C2311yescharacteristic
47C2×C8×C163—21no—
48C2×C8×C163—21no—
49C2×C8×C163—21no—
50C4⋊C4×C163—21no—
51C4⋊C4×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163D412yes—
54Dic326—22no—
55Dic326—22no—
56Dic326—48no—
57Dic326—41no—
58Dic326—41no—
59C2×Dic163—21no—
60C2×Dic163—21no—
61C2×Dic163—21no—
62C2×Dic163—21no—
63C2×Dic163—22no—
64C2×Dic163—22no—
65C2×Dic163—41no—
66C2×Dic163—41no—
67C2×Dic163—41no—
68C2×Dic163—41no—
69C2×Dic163SD1612yes—
70C2×Dic163Q1612yes—
71C163⋊C8—41no—
72C23×C163C2×D411yescharacteristic
73C2×C4×C163—22no—
74C2×C4×C163—41no—
75C2×C4×C163—41no—
76C2×C4×C163—41no—
77C2×C4×C163—41no—
78C2×C4×C163—41no—
79C2×C4×C163C2×D411yescharacteristic
80C2×C4×C163C2×D411yescharacteristic
81C2×C4×C163C2×Q811yescharacteristic
82C2×C4×C163#4111yescharacteristic
83C2×C4×C163#4111yescharacteristic
84C2×C4×C163#4111yescharacteristic
85C8×C163—41no—
86C22×C163#12011yescharacteristic
87C22×C163C22⋊Q811yescharacteristic
88C22×C163#12411yescharacteristic
89C22×C163C2×SD1611yescharacteristic
90C22×C163C2×Q1611yescharacteristic
91C22×C163AGL(1,ℤ/8)11yescharacteristic
92C22×C163#13711yescharacteristic
93C4×C163—42no—
94C4×C163—81no—
95C4×C163C22⋊Q811yescharacteristic
96C4×C163C22⋊Q811yescharacteristic
97C4×C163#12411yescharacteristic
98C4×C163#12411yescharacteristic
99Dic163—22no—
100Dic163—22no—
101Dic163—42no—
102Dic163—42no—
103C2×C163#39311yescharacteristic
104C2×C163#45011yescharacteristic
105C2×C163#45111yescharacteristic
106C2×C163#47511yescharacteristic
107C2×C163#47711yescharacteristic
108C2×C163#48711yescharacteristic
109C2×C163#48911yescharacteristic
110C163#202411yescharacteristic, complemented, Sylow
111#2024—1631nomaximal, Sylow
112C2×C4⋊Q8—1631no—
113C2×#103—1631no—
114C2×#103—1631no—
115#391—1631no—
116#340—1631no—
117#340D16311yescharacteristic
118#336—1631no—
119C22×Q8—3261no—
120C22×Q8—1631no—
121C22×C8—1631no—
122C22×C8C2×D16311yescharacteristic
123C4⋊Q8—3262no—
124C4⋊Q8—3262no—
125C2×C4⋊C4—3261no—
126C2×C4⋊C4—1631no—
127C2×C4⋊C4C2×D16311yescharacteristic
128C2×C4⋊C4C2×D16311yescharacteristic
129C2×C42—1631no—
130#103—3262no—
131#103—3262no—
132#95—3261no—
133#95—3261no—
134C2×Q8—3264no—
135C2×Q8—3261no—
136C2×Q8—3261no—
137C2×Q8—3261no—
138C2×Q8—3261no—
139C2×Q8—3261no—
140C2×Q8—6522no—
141C2×Q8—6522no—
142C2×Q8—1632no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3261no—
146C22×C4—3261no—
147C22×C4—1631no—
148C22×C4—1631no—
149C22×C4C22×D16311yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—3261no—
154C2×C8—3261no—
155C2×C8—3261no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—3264no—
160C4⋊C4—3261no—
161C4⋊C4—1632no—
162C4⋊C4D65212yes—
163C42—3261no—
164C42—3261no—
165C23D4×D16311yescharacteristic
166Q8—3262no—
167Q8—3262no—
168Q8—6528no—
169Q8—6521no—
170Q8—6521no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—3261no—
178C2×C4—3261no—
179C2×C4—3261no—
180C2×C4—3261no—
181C2×C4—3262no—
182C2×C4—3262no—
183C2×C4—6521no—
184C2×C4—6521no—
185C2×C4—6521no—
186C2×C4—6521no—
187C2×C4—1632no—
188C2×C4—1632no—
189C2×C4D4×D16311yescharacteristic
190C2×C4C2×D65211yescharacteristic
191C2×C4Q8×D16311yescharacteristic
192C2×C4#1457969811yescharacteristic
193C2×C4#1457969911yescharacteristic
194C2×C4#1457969911yescharacteristic
195C8—41no—
196C8—6521no—
197C22#1910470511yescharacteristic
198C22#1910471311yescharacteristic
199C22#1910472111yescharacteristic
200C22SD16×D16311yescharacteristic
201C22Q16×D16311yescharacteristic
202C22#1910475911yescharacteristic
203C22#1910476411yescharacteristic
204C4—42no—
205C4—81no—
206C4—3262no—
207C4—3262no—
208C4—6522no—
209C4—6522no—
210C4#1910471311yescharacteristic
211C4#1910471511yescharacteristic
212C4#1910472111yescharacteristic
213C4#1910472311yescharacteristic
214C2#2378157811yescharacteristic
215C2#2378181711yescharacteristic
216C2#2378182211yescharacteristic
217C2#2378194911yescharacteristic
218C2#2378195911yescharacteristic
219C2#2378201411yescharacteristic
220C2#2378202311yescharacteristic
subgroupG/H
centerC23D4×D163
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#340×C163C2
socle, prime radicalC23×C163C2×D4
hypercenter#340D163
hypocenterC163#2024
LCSG ▷ C22×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic326Q8semidirect
110. C163#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#191047342
C2×#191046602
C2×#191046582
#237815702
#237814272
#237814192
#340×C1632
#2024163