Finite Group Databasebeta

← #35227948

Subgroups — #35227948

Order 20864.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#23782264C216yesmaximal, complemented
2#23782255C216yesmaximal, complemented
3#23782227C213yesmaximal, complemented
4#23782201C216yesmaximal, complemented
5#23782178C216yesmaximal, complemented
6C2×(C22⋊D163)⋊C4C213yesmaximal, complemented
7#543×C163C211yesmaximal, characteristic, complemented
8C2×#14579700C2212yescomplemented
9C22×C22⋊D163C2213yes—
10C23×Dic163C2211yescharacteristic
11C22×C4×D163C2213yescomplemented
12#19104730C22112yescomplemented
13#19104726C2216yescomplemented
14#19104724C2216yescomplemented
15#19104720C22112yescomplemented
16#19104719C22112yescomplemented
17#19104711C2216yescomplemented
18#19104710C2216yescomplemented
19#19104709C2216yes—
20#19104709C2216yes—
21#19104705C22112yescomplemented
22Q8×Dic163C2212yescomplemented
23#19104697C2216yescomplemented
24(C22⋊D163)⋊C4C22112yescomplemented
25(C22⋊D163)⋊C4C22112yescomplemented
26#19104693C2216yescomplemented
27C2×Dic163⋊C4C2213yescomplemented
28C2×(C2×D163)⋊C4C2213yescomplemented
29C2×C4×Dic163C2213yescomplemented
30C22⋊Q8×C163C2216yescomplemented
31C22⋊Q8×C163C2216yescomplemented
32C2×C22⋊C4×C163C2213yes—
33C23×D163C2313yes—
34#14579700—44no—
35C2×D652C2316yes—
36C2×D652C2316yes—
37C2×C22⋊D163—212no—
38C2×C22⋊D163C23112yes—
39C22×Dic163—42no—
40C22×Dic163C2313yes—
41C22×Dic163C2313yes—
42C22×Dic163C2313yes—
43C2×C4×D163—46no—
44C2×C4×D163C2316yes—
45C2×C4×D163C2316yes—
46C4⋊Dic163C2316yes—
47C4⋊Dic163C2316yes—
48Dic163⋊C4C23112yes—
49(C2×D163)⋊C4C23112yes—
50(C2×D163)⋊C4C23112yes—
51(C2×D163)⋊C4C23112yes—
52C4×Dic163C2316yes—
53C4×Dic163C2316yes—
54C24×C163C2311yescharacteristic
55C2×Q8×C163C2312yes—
56C2×Q8×C163C2312yes—
57C22×C4×C163C2313yes—
58C22×C4×C163C2313yes—
59C4⋊C4×C163C2316yes—
60C4⋊C4×C163C2316yes—
61C22⋊C4×C163C23112yes—
62C22×D163—26no—
63C22×D163—43no—
64C22×D163—43no—
65C22×D163C2416yes—
66D652—46no—
67D652—46no—
68C22⋊D163—224no—
69C2×Dic163—26no—
70C2×Dic163—26no—
71C2×Dic163—43no—
72C2×Dic163C2413yes—
73C2×Dic163C2416yes—
74C2×Dic163C2411yescharacteristic
75C4×D163—212no—
76C4×D163—46no—
77C23×C163—41no—
78C23×C163—41no—
79C23×C163#4114yescomplemented
80C23×C163C2413yes—
81Q8×C163—42no—
82Q8×C163—42no—
83C2×C4×C163—43no—
84C2×C4×C163—43no—
85C2×C4×C163C2416yes—
86C2×C4×C163C2416yes—
87C22×C163—23no—
88C22×C163—23no—
89C22×C163—44no—
90C22×C163C2×#4116yes—
91C22×C163C2511yescharacteristic
92C2×D163—26no—
93C2×D163—26no—
94C2×D163—26no—
95C2×D163—412no—
96C4×C163—26no—
97C4×C163—26no—
98Dic163—26no—
99Dic163—23no—
100Dic163H(5,𝔽2)12yes—
101C2×C163—26no—
102C2×C163C22×#4111yescharacteristic
103C2×C163C2×H(5,𝔽2)11yescharacteristic
104C2×C163#58511yescharacteristic
105D163—46no—
106C163#345911yescharacteristic, complemented, Sylow
107#3459—1631nomaximal, Sylow
108#543D16311yescharacteristic, complemented
109#539—1636no—
110#538—1636no—
111#534—1633no—
112#529—1636no—
113#525—1636no—
114C2×C4×D4—1633no—
115C2×#41—1632no—
116C22×D4—1633no—
117C23×C4—1631no—
118C23×C4—1633no—
119#126—16312no—
120#125—1636no—
121#124—1636no—
122#123—16312no—
123#123—16312no—
124C22⋊Q8—1636no—
125C22⋊Q8C2×D16316yescomplemented
126C22⋊Q8C2×D16316yescomplemented
127#121—1636no—
128#121—1636no—
129#121—1636no—
130#120—16312no—
131C4×Q8—1632no—
132C4×D4—1636no—
133C4×D4—16312no—
134C4×D4—16312no—
135#117—1636no—
136C2×C4⋊C4—1633no—
137C2×C22⋊C4—1633no—
138C2×C22⋊C4C2×D16313yes—
139C2×C42—1633no—
140C24—1633no—
141C24C22×D16311yescharacteristic
142#41—6524no—
143C2×Q8C22×D16312yes—
144C2×Q8C22×D16312yes—
145C2×D4—32612no—
146C2×D4—16312no—
147C2×D4—1636no—
148C2×D4—1636no—
149C22×C4—6526no—
150C22×C4—6522no—
151C22×C4—1633no—
152C22×C4—1633no—
153C22×C4—1636no—
154C22×C4—1633no—
155C22×C4—1636no—
156C22×C4C22×D16313yes—
157C22×C4C22×D16313yes—
158C4⋊C4—16312no—
159C4⋊C4—1636no—
160C4⋊C4—1636no—
161C4⋊C4C22×D16316yes—
162C4⋊C4C22×D16316yes—
163C22⋊C4—16312no—
164C22⋊C4—16312no—
165C22⋊C4—16312no—
166C22⋊C4C22×D163112yes—
167C42—1636no—
168C42—1636no—
169C23—41no—
170C23—41no—
171C23—3266no—
172C23—6523no—
173C23—6523no—
174C23—1636no—
175C23#1457970014yescomplemented
176C23C23×D16313yes—
177Q8—42no—
178Q8—42no—
179D4—32624no—
180D4—6526no—
181D4—6526no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—32612no—
185C2×C4—3266no—
186C2×C4—3266no—
187C2×C4—6526no—
188C2×C4—6523no—
189C2×C4—1636no—
190C2×C4—1633no—
191C2×C4—1631no—
192C2×C4C23×D16316yes—
193C2×C4C23×D16316yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—3266no—
198C22—3266no—
199C22—3266no—
200C22—65212no—
201C22C2×#1457970016yes—
202C22C24×D16311yescharacteristic
203C4—26no—
204C4—26no—
205C4—3266no—
206C4—3263no—
207C4—1632no—
208C2—26no—
209C2—6526no—
210C2C22×#1457970011yescharacteristic
211C2H(5,𝔽2)×D16311yescharacteristic
212C2#2378250011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#543×C163C2
hypercenter#543D163
hypocenterC163#3459
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #23782264C2semidirect
2. #23782255C2semidirect
3. #23782227C2semidirect
4. #23782201C2semidirect
5. #23782178C2semidirect
6. C2×(C22⋊D163)⋊C4C2semidirect
7. #543×C163C2semidirect
8. C2×#14579700C22semidirect
11. C22×C4×D163C22semidirect
12. #19104730C22semidirect
13. #19104726C22semidirect
14. #19104724C22semidirect
15. #19104720C22semidirect
16. #19104719C22semidirect
17. #19104711C22semidirect
18. #19104710C22semidirect
21. #19104705C22semidirect
22. Q8×Dic163C22semidirect
23. #19104697C22semidirect
24. (C22⋊D163)⋊C4C22semidirect
25. (C22⋊D163)⋊C4C22semidirect
26. #19104693C22semidirect
27. C2×Dic163⋊C4C22semidirect
28. C2×(C2×D163)⋊C4C22semidirect
29. C2×C4×Dic163C22semidirect
30. C22⋊Q8×C163C22semidirect
31. C22⋊Q8×C163C22semidirect
79. C23×C163#41semidirect
106. C163#3459semidirect
108. #543D163semidirect
125. C22⋊Q8C2×D163semidirect
126. C22⋊Q8C2×D163semidirect
175. C23#14579700semidirect

Maximal subgroups

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

subgroupindex
#237822642
#237822552
#237822272
#237822012
#237821782
C2×(C22⋊D163)⋊C42
#543×C1632
#3459163