Finite Group Databasebeta

← #35216488

Subgroups — #35216488

Order 20864.

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

normal subgroups only

This group has 216 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
2#23782108C211yesmaximal, characteristic, complemented
3#23782105C211yesmaximal, characteristic, complemented
4#23781924C211yesmaximal, 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×Dic652C2211yescharacteristic, complemented
16C2×#14579682C2211yescharacteristic
17#19104715—21no—
18Dic1304—21no—
19#19104681—21no—
20#19104680—21no—
21#19104679—21no—
22#19104675C2211yescharacteristic, complemented
23#19104658—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—
32C2×Dic326—21no—
33Q8×D163—21no—
34D4×D163—21no—
35D4×D163—41no—
36C2×C4×D163—41no—
37C2×C4×D163D411yescharacteristic
38Dic652—21no—
39Dic652—21no—
40Dic652D411yescharacteristic
41#14579689—21no—
42#14579688—21no—
43#14579687—21no—
44#14579686—21no—
45D1304D411yescharacteristic
46#14579684—21no—
47#14579682—21no—
48#14579682—21no—
49C2×C163⋊C8D411yescharacteristic
50C4⋊Dic163—41no—
51(C2×D163)⋊C4—41no—
52C163⋊C16—21no—
53#41×C163—21no—
54Q16×C163D411yescharacteristic
55SD16×C163—21no—
56D8×C163D411yescharacteristic
57C2×C8×C163C2311yescharacteristic
58C4⋊C4×C163—21no—
59C16×C163—21no—
60C22×D163—41no—
61C22×D163D811yescharacteristic
62Dic326—21no—
63Dic326—41no—
64Dic326—41no—
65Dic326—41no—
66D652—21no—
67D652—41no—
68C22⋊D163—41no—
69C22⋊D163—41no—
70C22⋊D163—41no—
71C2×Dic163—41no—
72C2×Dic163—41no—
73C2×Dic163D811yescharacteristic
74C4×D163—21no—
75C4×D163—21no—
76C4×D163—41no—
77C4×D163—41no—
78C4×D163—41no—
79C163⋊C8—21no—
80Q8×C163—21no—
81D4×C163—21no—
82D4×C163—41no—
83C2×C4×C163—41no—
84C2×C4×C163—41no—
85C2×C4×C163C2×D411yescharacteristic
86C8×C163C2×D411yescharacteristic
87C8×C163C2×D411yescharacteristic
88C22×C163—41no—
89C22×C163C2×D811yescharacteristic
90C2×D163—21no—
91C2×D163—41no—
92C2×D163—41no—
93C2×D163—82no—
94C4×C163—41no—
95C4×C163—81no—
96C4×C163#12011yescharacteristic
97C4×C163AGL(1,ℤ/8)11yescharacteristic
98Dic163—21no—
99Dic163—41no—
100Dic163—81no—
101C2×C163—21no—
102C2×C163—81no—
103C2×C163#44711yescharacteristic
104D163—41no—
105D163—81no—
106C163#219111yescharacteristic, complemented, Sylow
107#2191—1631nomaximal, Sylow
108#575—1631no—
109#510—1631no—
110#510—1631no—
111#470—1631no—
112#360—1631no—
113#360D16311yescharacteristic, complemented
114#349—1631no—
115C2×#41—3261no—
116#137—3261no—
117#137—3261no—
118AGL(1,ℤ/8)—3261no—
119AGL(1,ℤ/8)—3261no—
120#135—3261no—
121#135—1631no—
122#135C2×D16311yescharacteristic
123C2×Q16—1631no—
124C2×M16—1631no—
125C22⋊Q8—3261no—
126Q32—3261no—
127Q32—3261no—
128SD32—3261no—
129SD32—3261no—
130M32—1631no—
131M32C2×D16311yescharacteristic, complemented
132#106C2×D16311yescharacteristic, complemented
133#103—3261no—
134#41—21no—
135#41—3261no—
136#41—3261no—
137#41—3261no—
138#41—6521no—
139#41—6521no—
140C2×Q8—3261no—
141C2×Q8—3261no—
142C2×D4—3261no—
143C2×D4—6521no—
144C22×C4—6521no—
145C22×C4—1631no—
146Q16—3261no—
147Q16—3261no—
148Q16—3261no—
149Q16—1631no—
150Q16D65211yescharacteristic
151SD16—21no—
152SD16—3261no—
153SD16—3261no—
154SD16—3261no—
155D8—3261no—
156D8—1631no—
157D8D65211yescharacteristic
158M16—3261no—
159M16—3261no—
160C2×C8—1631no—
161C2×C8C22×D16311yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—6521no—
164C22⋊C4—6521no—
165C16—21no—
166C16—3261no—
167C23—6521no—
168C23—1631no—
169Q8—21no—
170Q8—3261no—
171Q8—6521no—
172Q8—6521no—
173Q8—6521no—
174D4—21no—
175D4—41no—
176D4—3261no—
177D4—6521no—
178D4—6521no—
179D4—6521no—
180D4—6521no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3261no—
184C2×C4—3261no—
185C2×C4—6521no—
186C2×C4—6521no—
187C2×C4—6521no—
188C2×C4—6521no—
189C2×C4—6521no—
190C2×C4—1631no—
191C2×C4D4×D16311yescharacteristic
192C8—3261no—
193C8D4×D16311yescharacteristic
194C8C2×D65211yescharacteristic
195C22—41no—
196C22—13042no—
197C22—3261no—
198C22—6521no—
199C22—6521no—
200C22D8×D16311yescharacteristic
201C4—41no—
202C4—81no—
203C4—13041no—
204C4—3261no—
205C4—6521no—
206C4#1910470511yescharacteristic
207C4#1910476211yescharacteristic
208C2—21no—
209C2—81no—
210C2—13041no—
211C2—6521no—
212C2#2378180011yescharacteristic
subgroupG/H
centerC2#23781800
commutator, cosocleC2×C8×C163C23
FrattiniC2×C8C22×D163
Fitting#360×C163C2
socleC2×C163#447
prime radical#23782462C2
hypercenter#360D163
hypocenterC163#2191
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
2. #23782108C2semidirect
3. #23782105C2semidirect
4. #23781924C2semidirect
5. #23781474C2semidirect
6. #23781446C2semidirect
7. #360×C163C2semidirect
15. C2×Dic652C22semidirect
22. #19104675C22semidirect
25. M32×C163C22semidirect
26. #106×C163C22semidirect
106. C163#2191semidirect
113. #360D163semidirect
131. M32C2×D163semidirect
132. #106C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824622
#237821082
#237821052
#237819242
#237814742
#237814462
#360×C1632
#2191163