Finite Group Databasebeta

← #35224190

Subgroups — #35224190

Order 20864.

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

normal subgroups only

This group has 460 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
1C22×#14579689C211yesmaximal, characteristic, complemented
2C2×#19104743C211yesmaximal, characteristic
3#23782241C211yesmaximal, characteristic
4#23782239C211yesmaximal, characteristic
5#23781922C214yesmaximal, complemented
6#23781905C214yesmaximal, complemented
7#23781701C211yesmaximal, characteristic, complemented
8#23781675C211yesmaximal, characteristic, complemented
9#419×C163C211yesmaximal, characteristic, complemented
10C2×#14579699C2211yescharacteristic
11C22×Dic326C2211yescharacteristic
12C2×#14579689—24no—
13C2×#14579689C2212yes—
14C2×#14579689C2212yes—
15#19104743—24no—
16C2×#14579682C2212yes—
17C22×C163⋊C8C2211yescharacteristic
18C2×C8×D163C2212yes—
19#19104735—21no—
20#19104733—21no—
21#19104724—21no—
22#19104723—21no—
23#19104719—22no—
24#19104718—22no—
25#19104715C2214yes—
26#19104711C2214yes—
27#19104694C2211yescharacteristic
28#19104668C2212yes—
29#19104666C2212yes—
30#19104660C2214yes—
31C22×Q8×C163C2211yescharacteristic
32C2×M16×C163C2211yescharacteristic
33#117×C163C2211yescharacteristic
34#103×C163C2214yes—
35#14579699—22no—
36#14579699—22no—
37C2×Dic326—21no—
38C2×Dic326—22no—
39C2×Dic326—22no—
40C2×Dic326D411yescharacteristic
41C2×Dic326C2312yes—
42C2×D652D411yescharacteristic
43C2×C22⋊D163D412yes—
44C22×Dic163—21no—
45C2×C4×D163C2312yes—
46#14579689—24no—
47#14579689—24no—
48#14579682—24no—
49C2×C163⋊C8D414yes—
50C2×C163⋊C8C2311yescharacteristic
51C2×C163⋊C8C2311yescharacteristic
52C8×D163—24no—
53C4⋊Dic163—24no—
54C4⋊Dic163C2312yes—
55Dic163⋊C4—24no—
56C22⋊Dic163—21no—
57(C2×D163)⋊C4—24no—
58(C2×D163)⋊C4—24no—
59C4×Dic163—21no—
60C2×Q8×C163—21no—
61C2×Q8×C163—22no—
62C2×Q8×C163—22no—
63C2×Q8×C163C2312yes—
64C22×C4×C163—21no—
65C22×C4×C163C2311yescharacteristic
66M16×C163—22no—
67C2×C8×C163C2312yes—
68C4⋊C4×C163C2312yes—
69C22⋊C4×C163—21no—
70C42×C163—21no—
71C22×D163C2×D412yes—
72Dic326—22no—
73Dic326—22no—
74Dic326—22no—
75Dic326—42no—
76D652—22no—
77C22⋊D163—24no—
78C2×Dic163—22no—
79C2×Dic163—22no—
80C2×Dic163—42no—
81C2×Dic163C2×D412yes—
82C4×D163—22no—
83C4×D163—22no—
84C163⋊C8C2×D412yes—
85C163⋊C8C2×D412yes—
86C23×C163C2×D411yescharacteristic
87Q8×C163—22no—
88Q8×C163—22no—
89Q8×C163—42no—
90C2×C4×C163—22no—
91C2×C4×C163—22no—
92C2×C4×C163—42no—
93C2×C4×C163C2×D411yescharacteristic
94C2×C4×C163C2×D412yes—
95C2×C4×C163#4112yes—
96C2×C4×C163C2411yescharacteristic
97C8×C163—22no—
98C22×C163—22no—
99C22×C163#12112yes—
100C22×C163C22×D411yescharacteristic
101C2×D163—22no—
102C2×D163—24no—
103C4×C163—42no—
104C4×C163—42no—
105C4×C163#12112yes—
106C4×C163C22×D411yescharacteristic
107C4×C163C2×#4111yescharacteristic
108Dic163—22no—
109Dic163—42no—
110Dic163—42no—
111C2×C163—22no—
112C2×C163C2×#12111yescharacteristic
113C2×C163#57812yes—
114D163—42no—
115C163#305911yescharacteristic, complemented, Sylow
116#3059—1631nomaximal, Sylow
117C22×Q16—1631no—
118C2×#131—1631no—
119#536—1631no—
120#536—1631no—
121#470—1634no—
122#467—1634no—
123#427—1631no—
124#419—1631no—
125#419D16311yescharacteristic, complemented
126C2×#41—1631no—
127C22×Q8—1631no—
128C22×Q8C2×D16311yescharacteristic
129C2×Q16—3264no—
130C2×Q16—1632no—
131C2×Q16—1632no—
132#131—3264no—
133C2×M16—1632no—
134C2×M16C2×D16311yescharacteristic
135C22×C8—1632no—
136C22×C8—1631no—
137C4⋊Q8—3261no—
138C4⋊Q8—3261no—
139#124—3261no—
140#124—3261no—
141#123—3262no—
142#123—3262no—
143C22⋊Q8—1634no—
144C22⋊Q8—1634no—
145#117—1631no—
146#117C2×D16311yescharacteristic
147#107—1632no—
148#106—1632no—
149#103—1634no—
150#103C2×D16314yes—
151#41—3262no—
152#41—3262no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—3261no—
157C2×Q8—3262no—
158C2×Q8—3262no—
159C2×Q8—1632no—
160C2×Q8—1631no—
161C2×Q8C22×D16312yes—
162C2×D4—1632no—
163C2×D4—1631no—
164C22×C4—21no—
165C22×C4—3261no—
166C22×C4—1632no—
167C22×C4C22×D16311yescharacteristic
168Q16—3264no—
169Q16—3264no—
170M16—22no—
171M16—3264no—
172C2×C8—3264no—
173C2×C8—1634no—
174C2×C8—1631no—
175C2×C8—1631no—
176C2×C8C22×D16312yes—
177C4⋊C4—3264no—
178C4⋊C4—3264no—
179C4⋊C4—1632no—
180C4⋊C4C22×D16312yes—
181C22⋊C4—21no—
182C22⋊C4—3264no—
183C22⋊C4—3264no—
184C22⋊C4—3261no—
185C42—21no—
186C42—3261no—
187C23—1632no—
188C23D4×D16311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3262no—
193Q8—3262no—
194Q8—3262no—
195Q8—6522no—
196D4—3264no—
197D4—3262no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—3262no—
202C2×C4—3262no—
203C2×C4—3262no—
204C2×C4—3262no—
205C2×C4—6522no—
206C2×C4—1632no—
207C2×C4D4×D16312yes—
208C2×C4D4×D16311yescharacteristic
209C2×C4#1457969912yes—
210C2×C4C23×D16311yescharacteristic
211C8—22no—
212C8—1632no—
213C8—1632no—
214C22—22no—
215C22—3262no—
216C22—3264no—
217C22#1910470812yes—
218C22C2×D4×D16311yescharacteristic
219C4—42no—
220C4—42no—
221C4—3262no—
222C4—6522no—
223C4—6522no—
224C4#1910470812yes—
225C4C2×D4×D16311yescharacteristic
226C4C2×#1457969911yescharacteristic
227C2—22no—
228C2—6522no—
229C2C2×#1910470811yescharacteristic
230C2#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#3059
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
1. C22×#14579689C2semidirect
5. #23781922C2semidirect
6. #23781905C2semidirect
7. #23781701C2semidirect
8. #23781675C2semidirect
9. #419×C163C2semidirect
115. C163#3059semidirect
125. #419D163semidirect

Maximal subgroups

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

subgroupindex
C22×#145796892
C2×#191047432
#237822412
#237822392
#237819222
#237819052
#237817012
#237816752
#419×C1632
#3059163