Finite Group Databasebeta

← #35216483

Subgroups — #35216483

Order 20864.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#23782462C211yesmaximal, characteristic, complemented
2#23782103C211yesmaximal, characteristic, complemented
3#23782100C211yesmaximal, characteristic, complemented
4#23781919C211yesmaximal, characteristic, complemented
5#23781474C211yesmaximal, characteristic, complemented
6#23781446C211yesmaximal, characteristic, complemented
7#360×C163C211yesmaximal, characteristic, complemented
8#41×D163—21no—
9#19104767—21no—
10#19104764—21no—
11#19104762—21no—
12#19104759—21no—
13#19104757C2211yescharacteristic
14#19104755—21no—
15C2×D1304C2211yescharacteristic, complemented
16C2×#14579682C2211yescharacteristic
17#19104710—21no—
18#19104680—21no—
19#19104678—21no—
20D2608—21no—
21#19104676—21no—
22#19104675C2211yescharacteristic, complemented
23#19104655—21no—
24#135×C163C2211yescharacteristic
25M32×C163C2211yescharacteristic, complemented
26#106×C163C2211yescharacteristic, complemented
27#14579700—21no—
28#14579699—21no—
29#14579699—41no—
30#14579698—21no—
31#14579698—41no—
32Q8×D163—21no—
33C2×D652—21no—
34C2×D652—41no—
35D4×D163—21no—
36D4×D163—41no—
37C2×C4×D163—41no—
38C2×C4×D163D411yescharacteristic
39Dic652D411yescharacteristic, complemented
40#14579689—21no—
41#14579688—21no—
42#14579687—21no—
43#14579686—21no—
44D1304—21no—
45D1304—21no—
46D1304D411yescharacteristic, complemented
47#14579684—21no—
48#14579682—21no—
49#14579682—21no—
50C2×C163⋊C8D411yescharacteristic
51(C2×D163)⋊C4—41no—
52C163⋊C16—21no—
53#41×C163—21no—
54Q16×C163D411yescharacteristic, complemented
55SD16×C163—21no—
56D8×C163D411yescharacteristic, complemented
57C2×C8×C163C2311yescharacteristic
58C4⋊C4×C163—21no—
59C16×C163—21no—
60C22×D163—41no—
61C22×D163—41no—
62C22×D163D811yescharacteristic
63Dic326—21no—
64Dic326—41no—
65D652—21no—
66D652—41no—
67D652—41no—
68D652—41no—
69D652—82no—
70C22⋊D163—41no—
71C22⋊D163—41no—
72C22⋊D163—41no—
73C2×Dic163—41no—
74C2×Dic163D811yescharacteristic
75C4×D163—21no—
76C4×D163—21no—
77C4×D163—41no—
78C4×D163—41no—
79C4×D163—41no—
80C163⋊C8—21no—
81Q8×C163—21no—
82D4×C163—21no—
83D4×C163—41no—
84C2×C4×C163—41no—
85C2×C4×C163—41no—
86C2×C4×C163C2×D411yescharacteristic
87C8×C163C2×D411yescharacteristic
88C8×C163C2×D411yescharacteristic
89C22×C163—41no—
90C22×C163C2×D811yescharacteristic
91C2×D163—21no—
92C2×D163—41no—
93C2×D163—41no—
94C2×D163—82no—
95C2×D163—81no—
96C2×D163—81no—
97C2×D163—81no—
98C4×C163—41no—
99C4×C163—81no—
100C4×C163#12011yescharacteristic
101C4×C163AGL(1,ℤ/8)11yescharacteristic
102Dic163—21no—
103Dic163—41no—
104C2×C163—21no—
105C2×C163—81no—
106C2×C163#44711yescharacteristic
107D163—41no—
108D163—81no—
109D163—161no—
110C163#219011yescharacteristic, complemented, Sylow
111#2190—1631nomaximal, Sylow
112#575—1631no—
113#509—1631no—
114#509—1631no—
115#469—1631no—
116#360—1631no—
117#360D16311yescharacteristic, complemented
118#349—1631no—
119C2×#41—3261no—
120#137—3261no—
121#137—3261no—
122AGL(1,ℤ/8)—3261no—
123AGL(1,ℤ/8)—3261no—
124#135—3261no—
125#135—1631no—
126#135C2×D16311yescharacteristic
127C2×D8—1631no—
128C2×M16—1631no—
129#121—3261no—
130SD32—3261no—
131SD32—3261no—
132D16—3261no—
133D16—3261no—
134M32—1631no—
135M32C2×D16311yescharacteristic, complemented
136#106C2×D16311yescharacteristic, complemented
137#102—3261no—
138#41—21no—
139#41—3261no—
140#41—3261no—
141#41—3261no—
142#41—6521no—
143#41—6521no—
144C2×Q8—3261no—
145C2×D4—3261no—
146C2×D4—3261no—
147C2×D4—6521no—
148C2×D4—6521no—
149C22×C4—6521no—
150C22×C4—1631no—
151Q16—3261no—
152Q16—1631no—
153Q16D65211yescharacteristic, complemented
154SD16—21no—
155SD16—3261no—
156SD16—3261no—
157SD16—3261no—
158D8—3261no—
159D8—3261no—
160D8—3261no—
161D8—1631no—
162D8D65211yescharacteristic, complemented
163M16—3261no—
164M16—3261no—
165C2×C8—1631no—
166C2×C8C22×D16311yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—6521no—
169C16—21no—
170C16—3261no—
171C23—6521no—
172C23—6521no—
173C23—1631no—
174Q8—21no—
175Q8—3261no—
176Q8—6521no—
177D4—21no—
178D4—41no—
179D4—13042no—
180D4—3261no—
181D4—6521no—
182D4—6521no—
183D4—6521no—
184D4—6521no—
185D4—6521no—
186D4—6521no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3261no—
190C2×C4—3261no—
191C2×C4—6521no—
192C2×C4—6521no—
193C2×C4—6521no—
194C2×C4—6521no—
195C2×C4—1631no—
196C2×C4D4×D16311yescharacteristic
197C8—3261no—
198C8D4×D16311yescharacteristic
199C8C2×D65211yescharacteristic
200C22—41no—
201C22—13041no—
202C22—13041no—
203C22—13042no—
204C22—13041no—
205C22—3261no—
206C22—6521no—
207C22—6521no—
208C22D8×D16311yescharacteristic
209C4—41no—
210C4—81no—
211C4—3261no—
212C4—6521no—
213C4#1910470511yescharacteristic
214C4#1910476211yescharacteristic
215C2—21no—
216C2—81no—
217C2—13041no—
218C2—26081no—
219C2—6521no—
220C2#2378180011yescharacteristic
subgroupG/H
centerC2#23781800
commutator, cosocleC2×C8×C163C23
FrattiniC2×C8C22×D163
Fitting#360×C163C2
socleC2×C163#447
hypercenter#360D163
hypocenterC163#2190
LCSG ▷ C2×C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #23782462C2semidirect
2. #23782103C2semidirect
3. #23782100C2semidirect
4. #23781919C2semidirect
5. #23781474C2semidirect
6. #23781446C2semidirect
7. #360×C163C2semidirect
15. C2×D1304C22semidirect
22. #19104675C22semidirect
25. M32×C163C22semidirect
26. #106×C163C22semidirect
39. Dic652D4semidirect
46. D1304D4semidirect
54. Q16×C163D4semidirect
56. D8×C163D4semidirect
110. C163#2190semidirect
117. #360D163semidirect
135. M32C2×D163semidirect
136. #106C2×D163semidirect
153. Q16D652semidirect
162. D8D652semidirect

Maximal subgroups

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

subgroupindex
#237824622
#237821032
#237821002
#237819192
#237814742
#237814462
#360×C1632
#2190163