Finite Group Databasebeta

← #35213559

Subgroups — #35213559

Order 20864.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#23782290C211yesmaximal, characteristic
2#23782000C211yesmaximal, characteristic, complemented
3#23781600C211yesmaximal, characteristic
4#23781403C211yesmaximal, characteristic, complemented
5#23781402C211yesmaximal, characteristic, complemented
6#23781388C211yesmaximal, characteristic
7#484×C163C211yesmaximal, characteristic, complemented
8#19104715—21no—
9#19104714C2211yescharacteristic
10#19104713—21no—
11#19104713—21no—
12#19104712—21no—
13#19104711—21no—
14C2×Dic163⋊C4—21no—
15C2×(C2×D163)⋊C4—21no—
16C2×(C2×D163)⋊C4C2211yescharacteristic
17C22⋊C4×D163—21no—
18#19104668—21no—
19#19104658—21no—
20C22⋊(C163⋊C8)C2211yescharacteristic
21C163⋊#95C2211yescharacteristic
22C22⋊Q8×C163C2211yescharacteristic
23C2×C4⋊C4×C163C2211yescharacteristic
24#107×C163—21no—
25#103×C163—21no—
26C22⋊C8×C163C2211yescharacteristic
27C23×D163D411yescharacteristic
28C2×Dic326—21no—
29C2×Dic326D411yescharacteristic
30C22×Dic163—21no—
31C22×Dic163D411yescharacteristic
32C2×C4×D163—21no—
33C2×C4×D163—21no—
34C2×C163⋊C8—21no—
35C4⋊Dic163—21no—
36C4⋊Dic163D411yescharacteristic
37Dic163⋊C4—21no—
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—42no—
41C22⋊Dic163—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—42no—
47C2×Q8×C163D411yescharacteristic
48C22×C4×C163—21no—
49C22×C4×C163C2311yescharacteristic
50C2×C8×C163—21no—
51C4⋊C4×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163—21no—
54C4⋊C4×C163D411yescharacteristic
55C22⋊C4×C163—21no—
56C22×D163—21no—
57C22×D163—21no—
58C22×D163—41no—
59C22×D163—41no—
60C22×D163Q1612yes—
61Dic326—41no—
62Dic326—81no—
63C2×Dic163—22no—
64C2×Dic163—21no—
65C2×Dic163—21no—
66C2×Dic163—21no—
67C2×Dic163—41no—
68C2×Dic163—41no—
69C2×Dic163—41no—
70C4×D163—81no—
71C4×D163—81no—
72C163⋊C8—41no—
73C23×C163C2×D411yescharacteristic
74Q8×C163—41no—
75C2×C4×C163—21no—
76C2×C4×C163—21no—
77C2×C4×C163—41no—
78C2×C4×C163—41no—
79C2×C4×C163—41no—
80C2×C4×C163—41no—
81C2×C4×C163C2×D411yescharacteristic
82C2×C4×C163C2×D411yescharacteristic
83C8×C163—41no—
84C22×C163—21no—
85C22×C163—21no—
86C22×C163#12011yescharacteristic
87C22×C163C2×Q1611yescharacteristic
88C22×C163#13711yescharacteristic
89C2×D163—22no—
90C2×D163—41no—
91C2×D163—41no—
92C2×D163—42no—
93C2×D163—81no—
94C4×C163—21no—
95C4×C163—21no—
96C4×C163—41no—
97C4×C163—41no—
98C4×C163—81no—
99Dic163—42no—
100Dic163—41no—
101Dic163—41no—
102Dic163—41no—
103C2×C163—21no—
104C2×C163—21no—
105C2×C163#45111yescharacteristic
106C2×C163#45311yescharacteristic
107C2×C163#45811yescharacteristic
108D163—42no—
109C163#161411yescharacteristic, complemented, Sylow
110#1614—1631nomaximal, Sylow
111#543—1631no—
112#484—1631no—
113#484D16311yescharacteristic, complemented
114#399—1631no—
115#328—1631no—
116#328—1631no—
117#323—1631no—
118C22⋊Q8—3261no—
119C22⋊Q8—3261no—
120C22⋊Q8—3261no—
121C22⋊Q8—3261no—
122C22⋊Q8—3261no—
123C22⋊Q8—1631no—
124C22⋊Q8C2×D16311yescharacteristic
125C2×C4⋊C4—3261no—
126C2×C4⋊C4C2×D16311yescharacteristic
127C2×C22⋊C4—3261no—
128C2×C22⋊C4—3261no—
129C2×C22⋊C4—1631no—
130#107—21no—
131#107—3261no—
132#103—21no—
133#103—3261no—
134C22⋊C8—1631no—
135C22⋊C8C2×D16311yescharacteristic
136#95—1631no—
137C24—1631no—
138C2×Q8—3261no—
139C2×Q8—1631no—
140C2×Q8C22⋊D16311yescharacteristic
141C22×C4—21no—
142C22×C4—3261no—
143C22×C4—3261no—
144C22×C4—3261no—
145C22×C4—1631no—
146C22×C4C22×D16311yescharacteristic
147C2×C8—21no—
148C2×C8—3261no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—3261no—
153C4⋊C4—3261no—
154C4⋊C4—3261no—
155C4⋊C4—3261no—
156C4⋊C4—6522no—
157C4⋊C4—1631no—
158C4⋊C4C22⋊D16311yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—3261no—
161C22⋊C4—3261no—
162C22⋊C4—3261no—
163C22⋊C4—3261no—
164C22⋊C4—3261no—
165C22⋊C4—6522no—
166C23—3261no—
167C23—3261no—
168C23—6521no—
169C23—6521no—
170C23—1632no—
171C23D4×D16311yescharacteristic
172Q8—41no—
173Q8—13041no—
174Q8—6521no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—13041no—
182C2×C4—13041no—
183C2×C4—3262no—
184C2×C4—3261no—
185C2×C4—3261no—
186C2×C4—3261no—
187C2×C4—6521no—
188C2×C4—6521no—
189C2×C4—6521no—
190C2×C4D4×D16311yescharacteristic
191C2×C4C2×C22⋊D16311yescharacteristic
192C8—41no—
193C8—6521no—
194C22—21no—
195C22—21no—
196C22—13041no—
197C22—3262no—
198C22—6522no—
199C22—6521no—
200C22—6521no—
201C22#1910470311yescharacteristic
202C22Q16×D16311yescharacteristic
203C22#1910476711yescharacteristic
204C4—21no—
205C4—21no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—6522no—
210C4—6521no—
211C4—6521no—
212C4—6521no—
213C2—21no—
214C2—21no—
215C2—6522no—
216C2#2378181911yescharacteristic
217C2#2378183111yescharacteristic
218C2#2378185411yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#484×C163C2
socleC22×C163#120
prime radicalC23×D163D4
hypercenter#484D163
hypocenterC163#1614
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #23782000C2semidirect
4. #23781403C2semidirect
5. #23781402C2semidirect
7. #484×C163C2semidirect
109. C163#1614semidirect
113. #484D163semidirect

Maximal subgroups

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

subgroupindex
#237822902
#237820002
#237816002
#237814032
#237814022
#237813882
#484×C1632
#1614163