Finite Group Databasebeta

← #35227669

Subgroups — #35227669

Order 20864.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D163C212yesmaximal, complemented
2#23782107C212yesmaximal, complemented
3#23782104C212yesmaximal, complemented
4#23782102C212yesmaximal, complemented
5#23782099C212yesmaximal, complemented
6#23782097C211yesmaximal, characteristic, complemented
7#23782095C212yesmaximal, complemented
8C2×#19104674C211yesmaximal, characteristic, complemented
9#508×C163C211yesmaximal, characteristic, complemented
10#41×D163—22no—
11#19104758C2212yes—
12#19104757C2212yescomplemented
13#19104756—22no—
14#19104755—22no—
15#19104754C2212yes—
16Q16×D163C2212yescomplemented
17SD16×D163—22no—
18D8×D163C2212yescomplemented
19C2×C8×D163C2211yescharacteristic
20Dic1304C2211yescharacteristic, complemented
21#19104681C2212yescomplemented
22#19104680C2212yescomplemented
23#19104679C2212yescomplemented
24#19104678C2212yescomplemented
25D2608C2211yescharacteristic, complemented
26#19104676C2212yescomplemented
27#19104674C2212yescomplemented
28#19104674C2212yescomplemented
29C2×C163⋊C16C2211yescharacteristic, complemented
30#135×C163C2212yescomplemented
31Q32×C163C2211yescharacteristic, complemented
32SD32×C163C2212yescomplemented
33D16×C163C2211yescharacteristic, complemented
34C2×C16×C163C2211yescharacteristic, complemented
35#14579700—22no—
36#14579699—22no—
37#14579699—42no—
38#14579698—22no—
39#14579698—42no—
40Q8×D163—22no—
41D4×D163—22no—
42D4×D163—42no—
43C2×C4×D163—42no—
44C2×C4×D163D411yescharacteristic
45Dic652C2312yes—
46#14579689—22no—
47#14579688—22no—
48#14579687—22no—
49#14579686—22no—
50D1304C2312yes—
51#14579684—22no—
52C2×C163⋊C8D411yescharacteristic
53C8×D163D412yes—
54C8×D163C2312yes—
55C163⋊C16C2312yes—
56#41×C163—22no—
57Q16×C163C2312yes—
58SD16×C163—22no—
59D8×C163C2312yes—
60C2×C8×C163C2311yescharacteristic
61C16×C163C2311yescharacteristic
62C16×C163C2311yescharacteristic
63C22×D163—42no—
64C22×D163D811yescharacteristic
65Dic326—22no—
66Dic326—42no—
67D652—22no—
68D652—42no—
69C22⋊D163—42no—
70C22⋊D163—42no—
71C22⋊D163—42no—
72C2×Dic163—42no—
73C2×Dic163D811yescharacteristic
74C4×D163—42no—
75C4×D163—42no—
76C4×D163—42no—
77C4×D163D812yes—
78C4×D163C2×D412yes—
79C163⋊C8C2×D412yes—
80Q8×C163—22no—
81D4×C163—22no—
82D4×C163—42no—
83C2×C4×C163—42no—
84C2×C4×C163C2×D411yescharacteristic
85C8×C163C2×D411yescharacteristic
86C8×C163C2411yescharacteristic
87C22×C163—42no—
88C22×C163C2×D811yescharacteristic
89C2×D163—41no—
90C2×D163—42no—
91C2×D163—84no—
92C2×D163C2×D812yes—
93C4×C163—42no—
94C4×C163C2×D811yescharacteristic
95C4×C163C22×D411yescharacteristic
96Dic163—42no—
97Dic163C2×D812yes—
98C2×C163—21no—
99C2×C163—82no—
100C2×C163C22×D811yescharacteristic
101D163—22no—
102D163—82no—
103C163#341411yescharacteristic, complemented, Sylow
104#3414—1631nomaximal, Sylow
105C2×#135—1632no—
106#510—1632no—
107#510—1632no—
108#509—1632no—
109#509—1632no—
110#508—1632no—
111#508—1631no—
112#508D16311yescharacteristic, complemented
113C2×M32—1631no—
114C2×#41—3262no—
115#135—3262no—
116#135—3262no—
117#135—1632no—
118#135—1632no—
119#135—1632no—
120#135C2×D16312yescomplemented
121C2×Q16—1632no—
122C2×SD16—3262no—
123C2×D8—1632no—
124C22×C8—1631no—
125Q32—1632no—
126Q32—1631no—
127Q32C2×D16311yescharacteristic, complemented
128SD32—1632no—
129SD32—1632no—
130SD32—1632no—
131SD32C2×D16312yescomplemented
132D16—1632no—
133D16—1631no—
134D16C2×D16311yescharacteristic, complemented
135M32—1632no—
136M32—1632no—
137C2×C16—1631no—
138C2×C16C2×D16311yescharacteristic, complemented
139#41—22no—
140#41—3262no—
141#41—3262no—
142#41—3262no—
143#41—6522no—
144#41—6522no—
145C2×Q8—3262no—
146C2×D4—3262no—
147C2×D4—6522no—
148C22×C4—6522no—
149C22×C4—1631no—
150Q16—3262no—
151Q16—1632no—
152Q16C22×D16312yes—
153SD16—22no—
154SD16—3262no—
155SD16—3262no—
156SD16—3262no—
157D8—3262no—
158D8—1632no—
159D8C22×D16312yes—
160C2×C8—1631no—
161C2×C8—1632no—
162C2×C8—1632no—
163C2×C8C22×D16311yescharacteristic
164C16—1632no—
165C16C22×D16311yescharacteristic
166C16C22×D16311yescharacteristic
167C23—6522no—
168C23—1631no—
169Q8—22no—
170Q8—3262no—
171Q8—6522no—
172D4—22no—
173D4—42no—
174D4—3262no—
175D4—6522no—
176D4—6522no—
177D4—6522no—
178D4—6522no—
179C2×C4—42no—
180C2×C4—6522no—
181C2×C4—6522no—
182C2×C4—6522no—
183C2×C4—6522no—
184C2×C4—1631no—
185C2×C4—1632no—
186C2×C4—1632no—
187C2×C4D4×D16311yescharacteristic
188C8—1632no—
189C8D4×D16311yescharacteristic
190C8C23×D16311yescharacteristic
191C22—42no—
192C22—13044no—
193C22—6521no—
194C22—6522no—
195C22—1632no—
196C22D8×D16311yescharacteristic
197C4—42no—
198C4—6522no—
199C4—1632no—
200C4D8×D16311yescharacteristic
201C4C2×D4×D16311yescharacteristic
202C2—21no—
203C2—82no—
204C2—13042no—
205C2—3262no—
206C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC4D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#508×C163C2
socleC2×C163C22×D8
hypercenter#508D163
hypocenterC163#3414
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D163C2semidirect
2. #23782107C2semidirect
3. #23782104C2semidirect
4. #23782102C2semidirect
5. #23782099C2semidirect
6. #23782097C2semidirect
7. #23782095C2semidirect
8. C2×#19104674C2semidirect
9. #508×C163C2semidirect
12. #19104757C22semidirect
16. Q16×D163C22semidirect
18. D8×D163C22semidirect
20. Dic1304C22semidirect
21. #19104681C22semidirect
22. #19104680C22semidirect
23. #19104679C22semidirect
24. #19104678C22semidirect
25. D2608C22semidirect
26. #19104676C22semidirect
27. #19104674C22semidirect
28. #19104674C22semidirect
29. C2×C163⋊C16C22semidirect
30. #135×C163C22semidirect
31. Q32×C163C22semidirect
32. SD32×C163C22semidirect
33. D16×C163C22semidirect
34. C2×C16×C163C22semidirect
103. C163#3414semidirect
112. #508D163semidirect
120. #135C2×D163semidirect
127. Q32C2×D163semidirect
131. SD32C2×D163semidirect
134. D16C2×D163semidirect
138. C2×C16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#135×D1632
#237821072
#237821042
#237821022
#237820992
#237820972
#237820952
C2×#191046742
#508×C1632
#3414163