Finite Group Databasebeta

← #35225983

Subgroups — #35225983

Order 20864.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#23782277C211yesmaximal, characteristic, complemented
2#23782270C211yesmaximal, characteristic
3#23781973C212yesmaximal, complemented
4#23781966C212yesmaximal, complemented
5#23781911C212yesmaximal, complemented
6#23781894C212yesmaximal, complemented
7#23781871C212yesmaximal, complemented
8#23781866C212yesmaximal, complemented
9#424×C163C211yesmaximal, characteristic, complemented
10C2×#14579688C2212yes—
11C2×#14579688C2212yes—
12#19104735C2211yescharacteristic
13#19104732C2211yescharacteristic
14#19104728—22no—
15#19104727C2211yescharacteristic
16#19104727C2211yescharacteristic
17#19104725—22no—
18#19104720—22no—
19#19104714C2212yes—
20#19104709C2212yes—
21#19104708—22no—
22C4×Dic326C2212yes—
23C4×D652C2212yes—
24#19104669C2212yes—
25#19104667C2212yes—
26#19104660C2212yes—
27#19104660C2212yes—
28#19104657C2212yes—
29#19104657C2212yes—
30C2×M16×C163C2211yescharacteristic
31C22×C8×C163C2211yescharacteristic
32#117×C163C2211yescharacteristic
33C4⋊C8×C163C2212yes—
34C4⋊C8×C163C2212yes—
35C2×Dic326C2312yes—
36C2×D652—22no—
37C2×D652C2312yes—
38C2×C22⋊D163—22no—
39C2×C4×D163—22no—
40#14579688—42no—
41#14579688—42no—
42C4⋊Dic163—22no—
43C4⋊Dic163C2312yes—
44C4⋊Dic163C2312yes—
45Dic163⋊C4—22no—
46Dic163⋊C4—22no—
47Dic163⋊C4—22no—
48Dic163⋊C4—22no—
49C22⋊Dic163—22no—
50(C2×D163)⋊C4—22no—
51(C2×D163)⋊C4—22no—
52C4×Dic163—22no—
53C22×C4×C163C2311yescharacteristic
54M16×C163—41no—
55C2×C8×C163—41no—
56C2×C8×C163C2312yes—
57C2×C8×C163C2312yes—
58C4⋊C4×C163D412yes—
59C22⋊C4×C163D412yes—
60C42×C163C2311yescharacteristic
61C42×C163C2311yescharacteristic
62C22×D163—22no—
63Dic326—22no—
64Dic326—22no—
65D652—22no—
66D652—22no—
67D652—82no—
68C22⋊D163—82no—
69C2×Dic163—22no—
70C2×Dic163—22no—
71C2×Dic163—22no—
72C4×D163—82no—
73C23×C163C2×D411yescharacteristic
74C2×C4×C163—21no—
75C2×C4×C163—21no—
76C2×C4×C163C2×D412yes—
77C2×C4×C163C2×D412yes—
78C2×C4×C163C2×D411yescharacteristic
79C2×C4×C163C2411yescharacteristic
80C8×C163—22no—
81C8×C163—22no—
82C22×C163—21no—
83C22×C163—22no—
84C22×C163C22×D411yescharacteristic
85C2×D163—42no—
86C2×D163—42no—
87C2×D163—42no—
88C4×C163—21no—
89C4×C163—22no—
90C4×C163—22no—
91C4×C163H(5,𝔽2)11yescharacteristic
92C4×C1632−1+411yescharacteristic
93Dic163—42no—
94Dic163—42no—
95Dic163—42no—
96C2×C163—41no—
97C2×C163#53711yescharacteristic
98C2×C163#57712yes—
99D163—82no—
100C163#326011yescharacteristic, complemented, Sylow
101#3260—1631nomaximal, Sylow
102#541—1631no—
103#540—1631no—
104#479—1632no—
105#478—1632no—
106#468—1632no—
107#465—1632no—
108#461—1632no—
109#460—1632no—
110#424D16311yescharacteristic, complemented
111C2×SD16—1632no—
112C2×SD16—1632no—
113C2×M16C2×D16311yescharacteristic
114C22×C8C2×D16311yescharacteristic
115C4⋊Q8—1631no—
116#127—1631no—
117#126—3262no—
118#125—3262no—
119#125—1631no—
120#125—1631no—
121#123—3262no—
122C22⋊Q8—1632no—
123#121—3262no—
124#121—1632no—
125C4×Q8—1632no—
126C4×D4—1632no—
127#117C2×D16311yescharacteristic
128#107—1632no—
129#106—1632no—
130C4⋊C8C2×D16312yes—
131C4⋊C8C2×D16312yes—
132#103—1632no—
133#103—1632no—
134#102—1632no—
135#102—1632no—
136C2×Q8—1632no—
137C2×D4—3262no—
138C2×D4—3262no—
139C2×D4—1632no—
140C22×C4—3262no—
141C22×C4C22×D16311yescharacteristic
142SD16—6522no—
143SD16—6522no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D16312yes—
147C2×C8C22×D16312yes—
148C4⋊C4—3262no—
149C4⋊C4—3262no—
150C4⋊C4—3262no—
151C4⋊C4—3262no—
152C4⋊C4—3262no—
153C4⋊C4—1632no—
154C4⋊C4—1632no—
155C4⋊C4D65212yes—
156C22⋊C4—3262no—
157C22⋊C4—3262no—
158C22⋊C4—3262no—
159C22⋊C4D65212yes—
160C42—3262no—
161C42C22×D16311yescharacteristic
162C42C22×D16311yescharacteristic
163C23—3262no—
164C23C2×D65211yescharacteristic
165Q8—3262no—
166Q8—3262no—
167D4—13042no—
168D4—13042no—
169D4—3262no—
170D4—3262no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—13042no—
174C2×C4—3262no—
175C2×C4—3262no—
176C2×C4—3262no—
177C2×C4C2×D65212yes—
178C2×C4C2×D65212yes—
179C2×C4C2×D65211yescharacteristic
180C2×C4C23×D16311yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—6522no—
186C22—6522no—
187C22—6522no—
188C22C22×D65211yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—6522no—
193C4—6522no—
194C4—6522no—
195C4#1910478011yescharacteristic
196C4#1910478311yescharacteristic
197C2—41no—
198C2—13042no—
199C2#2378224611yescharacteristic
200C2#2378247712yes—
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#424×C163C2
socleC22×C163C22×D4
prime radical#23782270C2
hypercenter#424D163
hypocenterC163#3260
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #23782277C2semidirect
3. #23781973C2semidirect
4. #23781966C2semidirect
5. #23781911C2semidirect
6. #23781894C2semidirect
7. #23781871C2semidirect
8. #23781866C2semidirect
9. #424×C163C2semidirect
100. C163#3260semidirect
110. #424D163semidirect

Maximal subgroups

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

subgroupindex
#237822772
#237822702
#237819732
#237819662
#237819112
#237818942
#237818712
#237818662
#424×C1632
#3260163