Finite Group Databasebeta

← #35225998

Subgroups — #35225998

Order 20864.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#23782288C212yesmaximal, complemented
2#23781958C212yesmaximal, complemented
3#23781944C212yesmaximal, complemented
4#23781913C212yesmaximal, complemented
5#23781901C211yesmaximal, characteristic
6#23781889C212yesmaximal, complemented
7#23781883C212yesmaximal, complemented
8#23781691C211yesmaximal, characteristic, complemented
9#467×C163C211yesmaximal, characteristic, complemented
10C2×#14579686C2212yes—
11C2×D1304C2211yescharacteristic
12C2×#14579683C2211yescharacteristic
13#19104734C2212yes—
14#19104730—22no—
15#19104724C2212yes—
16#19104721—22no—
17#19104717—22no—
18#19104715—22no—
19#19104709C2212yes—
20Q8×Dic163C2212yes—
21#19104697C2212yes—
22#19104694C2211yescharacteristic
23#19104669C2211yescharacteristic
24#19104668C2212yes—
25Dic163⋊C8C2212yes—
26Dic163⋊C8C2212yes—
27#19104659C2212yes—
28#19104657C2212yes—
29#19104655C2212yes—
30C2×Q16×C163C2211yescharacteristic
31C22×C8×C163C2211yescharacteristic
32C22⋊Q8×C163C2212yes—
33#107×C163C2211yescharacteristic
34#103×C163C2212yes—
35C2×Dic326—22no—
36C2×D652C2312yes—
37C2×C22⋊D163—22no—
38C2×C4×D163—22no—
39#14579686—42no—
40D1304—41no—
41#14579683—41no—
42C2×C163⋊C8C2312yes—
43C4⋊Dic163—22no—
44C4⋊Dic163D411yescharacteristic
45C4⋊Dic163D411yescharacteristic
46C4⋊Dic163C2312yes—
47Dic163⋊C4—22no—
48C22⋊Dic163D412yes—
49(C2×D163)⋊C4—22no—
50(C2×D163)⋊C4—22no—
51(C2×D163)⋊C4—22no—
52(C2×D163)⋊C4—22no—
53C4×Dic163—22no—
54C4×Dic163C2312yes—
55C2×Q8×C163C2312yes—
56C22×C4×C163C2311yescharacteristic
57Q16×C163—41no—
58C2×C8×C163—41no—
59C2×C8×C163C2311yescharacteristic
60C2×C8×C163C2311yescharacteristic
61C4⋊C4×C163—22no—
62C4⋊C4×C163C2312yes—
63C22⋊C4×C163—22no—
64C22×D163—22no—
65Dic326—82no—
66D652—22no—
67D652—22no—
68C22⋊D163—82no—
69C2×Dic163—22no—
70C2×Dic163C2×D412yes—
71C2×Dic163C2×D412yes—
72C4×D163—82no—
73C163⋊C8—22no—
74C23×C163C2×D411yescharacteristic
75Q8×C163—22no—
76Q8×C163—22no—
77C2×C4×C163—21no—
78C2×C4×C163—21no—
79C2×C4×C163—22no—
80C2×C4×C163—22no—
81C2×C4×C163C2×D411yescharacteristic
82C2×C4×C163C2411yescharacteristic
83C8×C163—21no—
84C8×C163—21no—
85C22×C163—21no—
86C22×C163—21no—
87C22×C163—21no—
88C22×C163C22×D411yescharacteristic
89C2×D163—42no—
90C2×D163—42no—
91C2×D163—42no—
92C4×C163—21no—
93C4×C163—42no—
94C4×C163—42no—
95C4×C163H(5,𝔽2)11yescharacteristic
96C4×C1632−1+411yescharacteristic
97Dic163—22no—
98Dic163—22no—
99Dic163—42no—
100C2×C163—41no—
101C2×C163#53711yescharacteristic
102C2×C163#57611yescharacteristic
103C2×C163#57811yescharacteristic
104D163—82no—
105C163#326211yescharacteristic, complemented, Sylow
106#3262—1631nomaximal, Sylow
107#542—1632no—
108#477—1632no—
109#474—1632no—
110#468—1632no—
111#467D16311yescharacteristic, complemented
112#466—1631no—
113#464—1632no—
114#463—1632no—
115#424—1631no—
116C2×Q16C2×D16311yescharacteristic
117C2×SD16—1632no—
118C2×D8—1631no—
119C2×M16—1631no—
120C22×C8C2×D16311yescharacteristic
121C4⋊Q8—1632no—
122#126—3262no—
123#124—3262no—
124#124—1632no—
125#123—3262no—
126C22⋊Q8—3262no—
127C22⋊Q8C2×D16312yes—
128#121—1632no—
129C4×Q8—1632no—
130C4×D4—1632no—
131#117—1631no—
132#107—1631no—
133#107—1632no—
134#107C2×D16311yescharacteristic
135C4⋊C8—1632no—
136C4⋊C8—1632no—
137#103—1632no—
138#103C2×D16312yes—
139#102—1632no—
140#102—1632no—
141C2×Q8—3262no—
142C2×Q8C22×D16312yes—
143C2×D4—3262no—
144C2×D4—1632no—
145C22×C4—3262no—
146C22×C4C22×D16311yescharacteristic
147Q16—41no—
148SD16—6522no—
149D8—6521no—
150M16—6521no—
151C2×C8—41no—
152C2×C8—1632no—
153C2×C8C22×D16311yescharacteristic
154C2×C8C22×D16311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—3262no—
157C4⋊C4—3262no—
158C4⋊C4—1632no—
159C4⋊C4—1631no—
160C4⋊C4—1631no—
161C4⋊C4C22×D16312yes—
162C22⋊C4—22no—
163C22⋊C4—3262no—
164C22⋊C4—3262no—
165C22⋊C4—3262no—
166C22⋊C4—3262no—
167C22⋊C4—1632no—
168C42—3262no—
169C42—1632no—
170C23—3262no—
171C23D4×D16311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—13042no—
175D4—13042no—
176D4—3262no—
177D4—3262no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—13042no—
183C2×C4—3262no—
184C2×C4—1632no—
185C2×C4—1632no—
186C2×C4D4×D16311yescharacteristic
187C2×C4C23×D16311yescharacteristic
188C8—21no—
189C8—21no—
190C8—3262no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—6522no—
195C22—6522no—
196C22—6522no—
197C22C2×D4×D16311yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—3262no—
202C4—3262no—
203C4—6522no—
204C4#1910478011yescharacteristic
205C4#1910478211yescharacteristic
206C2—41no—
207C2—13042no—
208C2#2378224211yescharacteristic
209C2#2378246711yescharacteristic
210C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#467×C163C2
socleC22×C163C22×D4
prime radical#23781901C2
hypercenter#467D163
hypocenterC163#3262
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23782288C2semidirect
2. #23781958C2semidirect
3. #23781944C2semidirect
4. #23781913C2semidirect
6. #23781889C2semidirect
7. #23781883C2semidirect
8. #23781691C2semidirect
9. #467×C163C2semidirect
105. C163#3262semidirect
111. #467D163semidirect

Maximal subgroups

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

subgroupindex
#237822882
#237819582
#237819442
#237819132
#237819012
#237818892
#237818832
#237816912
#467×C1632
#3262163