Finite Group Databasebeta

← #35224065

Subgroups — #35224065

Order 20864.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19104768C211yesmaximal, characteristic
2C22×#14579689C211yesmaximal, characteristic, complemented
3#23782239C211yesmaximal, characteristic
4#23782142C211yesmaximal, characteristic, complemented
5#23781890C214yesmaximal, complemented
6#23781874C214yesmaximal, complemented
7#23781693C211yesmaximal, characteristic, complemented
8#23781659C211yesmaximal, characteristic
9#419×C163C211yesmaximal, characteristic, complemented
10C2×#14579699C2211yescharacteristic
11C22×Dic326C2211yescharacteristic
12#19104768—24no—
13C2×Dic652C2212yes—
14C2×#14579689—24no—
15C2×#14579689C2214yes—
16C2×#14579688C2212yes—
17C22×C163⋊C8C2211yescharacteristic
18#19104733C2212yes—
19#19104724C2212yes—
20#19104718—22no—
21#19104711—22no—
22C4×Dic326—22no—
23#19104700C2214yes—
24#19104694C2211yescharacteristic
25#19104691—22no—
26Dic163⋊C8C2214yes—
27#19104658C2212yes—
28#19104655C2212yes—
29C22×Q8×C163C2211yescharacteristic
30C2×M16×C163C2211yescharacteristic
31#117×C163C2211yescharacteristic
32#103×C163C2214yes—
33#14579699—22no—
34#14579699—22no—
35C2×Dic326—21no—
36C2×Dic326—22no—
37C2×Dic326D414yes—
38C2×Dic326C2311yescharacteristic
39C2×Dic326C2312yes—
40C2×D652C2311yescharacteristic
41C2×C22⋊D163—21no—
42C22×Dic163—21no—
43C2×C4×D163—21no—
44Dic652—24no—
45#14579689—28no—
46#14579688—24no—
47C2×C163⋊C8—22no—
48C2×C163⋊C8C2312yes—
49C4⋊Dic163D412yes—
50Dic163⋊C4—24no—
51Dic163⋊C4—24no—
52C22⋊Dic163D412yes—
53(C2×D163)⋊C4—24no—
54C4×Dic163—24no—
55C4×Dic163C2312yes—
56C2×Q8×C163—21no—
57C2×Q8×C163—22no—
58C2×Q8×C163—22no—
59C2×Q8×C163C2312yes—
60C22×C4×C163—21no—
61C22×C4×C163C2311yescharacteristic
62M16×C163—22no—
63C2×C8×C163C2312yes—
64C4⋊C4×C163C2312yes—
65C22⋊C4×C163—21no—
66C42×C163—21no—
67C22×D163—21no—
68Dic326—24no—
69Dic326—22no—
70Dic326—21no—
71Dic326—21no—
72Dic326C2×D414yes—
73D652—21no—
74D652—21no—
75C22⋊D163—42no—
76C2×Dic163—24no—
77C2×Dic163—22no—
78C2×Dic163—21no—
79C2×Dic163C2×D414yes—
80C4×D163—42no—
81C163⋊C8—22no—
82C23×C163C2×D411yescharacteristic
83Q8×C163—22no—
84Q8×C163—22no—
85Q8×C163—42no—
86C2×C4×C163—22no—
87C2×C4×C163—22no—
88C2×C4×C163—42no—
89C2×C4×C163C2×D411yescharacteristic
90C2×C4×C163C2×D412yes—
91C2×C4×C163#4112yes—
92C2×C4×C163C2411yescharacteristic
93C8×C163—22no—
94C22×C163—22no—
95C22×C163#12112yes—
96C22×C163C22×D411yescharacteristic
97C2×D163—41no—
98C2×D163—42no—
99C4×C163—42no—
100C4×C163—42no—
101C4×C163#12112yes—
102C4×C163C22×D411yescharacteristic
103C4×C163C2×#4111yescharacteristic
104Dic163—24no—
105Dic163—24no—
106Dic163—41no—
107C2×C163—22no—
108C2×C163C2×#12111yescharacteristic
109C2×C163#57812yes—
110D163—81no—
111C163#304411yescharacteristic, complemented, Sylow
112#3044—1631nomaximal, Sylow
113C2×#137—1631no—
114C22×Q16—1631no—
115#536—1631no—
116#519—1631no—
117#464—1634no—
118#462—1634no—
119#424—1631no—
120#419D16311yescharacteristic, complemented
121#416—1631no—
122C2×#41—1631no—
123C22×Q8—1631no—
124C22×Q8C2×D16311yescharacteristic
125#137—3264no—
126C2×Q16—3264no—
127C2×Q16—1634no—
128C2×Q16—1632no—
129C2×SD16—1632no—
130C2×M16C2×D16311yescharacteristic
131C22×C8—1631no—
132C4⋊Q8—1632no—
133#124—1632no—
134#123—3262no—
135C22⋊Q8—3262no—
136C4×Q8—3262no—
137C4×Q8—1634no—
138#117—3262no—
139#117—1631no—
140#117C2×D16311yescharacteristic
141C4⋊C8—1634no—
142#103—1632no—
143#103C2×D16314yes—
144#102—1632no—
145#41—3262no—
146#41—3262no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—3261no—
151C2×Q8—3262no—
152C2×Q8—1634no—
153C2×Q8—1631no—
154C2×Q8—1632no—
155C2×Q8C22×D16312yes—
156C2×D4—3261no—
157C2×D4—1631no—
158C22×C4—21no—
159C22×C4—3261no—
160C22×C4—3261no—
161C22×C4C22×D16311yescharacteristic
162Q16—3264no—
163Q16—3268no—
164SD16—3264no—
165M16—22no—
166C2×C8—3262no—
167C2×C8—1632no—
168C2×C8C22×D16312yes—
169C4⋊C4—3264no—
170C4⋊C4—3264no—
171C4⋊C4—1632no—
172C4⋊C4C22×D16312yes—
173C22⋊C4—21no—
174C22⋊C4—3264no—
175C22⋊C4—1632no—
176C42—21no—
177C42—3264no—
178C42—1632no—
179C23—3261no—
180C23D4×D16311yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—3262no—
185Q8—3264no—
186Q8—3261no—
187Q8—3261no—
188Q8—1634no—
189D4—3261no—
190D4—3261no—
191D4—6522no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—3262no—
196C2×C4—3264no—
197C2×C4—3261no—
198C2×C4—6522no—
199C2×C4—1634no—
200C2×C4D4×D16312yes—
201C2×C4D4×D16311yescharacteristic
202C2×C4#1457969912yes—
203C2×C4C23×D16311yescharacteristic
204C8—22no—
205C8—3262no—
206C22—22no—
207C22—6521no—
208C22—6522no—
209C22#1910470812yes—
210C22C2×D4×D16311yescharacteristic
211C4—42no—
212C4—42no—
213C4—3264no—
214C4—3264no—
215C4—6521no—
216C4#1910470812yes—
217C4C2×D4×D16311yescharacteristic
218C4C2×#1457969911yescharacteristic
219C2—22no—
220C2—13041no—
221C2C2×#1910470811yescharacteristic
222C2#2378248112yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#419×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579699C22
hypercenter#419D163
hypocenterC163#3044
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#14579689C2semidirect
4. #23782142C2semidirect
5. #23781890C2semidirect
6. #23781874C2semidirect
7. #23781693C2semidirect
9. #419×C163C2semidirect
111. C163#3044semidirect
120. #419D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
C22×#145796892
#237822392
#237821422
#237818902
#237818742
#237816932
#237816592
#419×C1632
#3044163