Finite Group Databasebeta

← #35213851

Subgroups — #35213851

Order 20864.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#23782413C211yesmaximal, characteristic
2#23782057C211yesmaximal, characteristic, complemented
3#23782027C211yesmaximal, characteristic, complemented
4#23781407C211yesmaximal, characteristic, complemented
5#23781404C211yesmaximal, characteristic, complemented
6#23781404C211yesmaximal, characteristic, complemented
7#501×C163C211yesmaximal, characteristic, complemented
8C2×#14579689—21no—
9C2×#14579686—21no—
10#19104723C2211yescharacteristic
11#19104723C2211yescharacteristic
12#19104720—21no—
13#19104720—21no—
14#19104713—21no—
15#19104713—21no—
16#19104710—21no—
17#19104710—21no—
18#19104658—21no—
19#19104655—21no—
20#19104646C2211yescharacteristic
21#19104646C2211yescharacteristic
22C4⋊Q8×C163C2211yescharacteristic
23#125×C163C2211yescharacteristic
24#107×C163—21no—
25#106×C163—21no—
26#97×C163C2211yescharacteristic
27C2×Dic326D411yescharacteristic
28C2×Dic326D411yescharacteristic
29C2×D652—21no—
30C2×D652—21no—
31C2×D652D411yescharacteristic
32C2×D652D411yescharacteristic
33C2×C4×D163—21no—
34C2×C4×D163—21no—
35#14579689—42no—
36#14579686—42no—
37C2×C163⋊C8—21no—
38C2×C163⋊C8—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41(C2×D163)⋊C4—21no—
42(C2×D163)⋊C4—21no—
43(C2×D163)⋊C4—21no—
44(C2×D163)⋊C4—21no—
45(C2×D163)⋊C4—21no—
46(C2×D163)⋊C4—21no—
47C2×Q8×C163—21no—
48C2×C8×C163—21no—
49C4⋊C4×C163—21no—
50C4⋊C4×C163—21no—
51C4⋊C4×C163—21no—
52C4⋊C4×C163—21no—
53C4⋊C4×C163D411yescharacteristic
54C4⋊C4×C163D411yescharacteristic
55C42×C163C2311yescharacteristic
56C22×D163—21no—
57C22×D163—21no—
58Dic326—41no—
59Dic326—41no—
60D652—41no—
61D652—41no—
62D652—81no—
63D652—81no—
64C2×Dic163—21no—
65C2×Dic163—21no—
66C4×D163—81no—
67C4×D163—81no—
68C163⋊C8—22no—
69C163⋊C8—41no—
70Q8×C163—41no—
71Q8×C163—41no—
72C2×C4×C163—21no—
73C2×C4×C163—21no—
74C2×C4×C163—41no—
75C2×C4×C163C2×D411yescharacteristic
76C2×C4×C163C2×D411yescharacteristic
77C2×C4×C163C2×D411yescharacteristic
78C8×C163—22no—
79C22×C163#12011yescharacteristic
80C2×D163—41no—
81C2×D163—41no—
82C2×D163—41no—
83C2×D163—41no—
84C2×D163—41no—
85C2×D163—41no—
86C4×C163—21no—
87C4×C163—21no—
88C4×C163—21no—
89C4×C163—41no—
90C4×C163—41no—
91C4×C163—81no—
92Dic163—41no—
93Dic163—41no—
94C2×C163#45311yescharacteristic
95C2×C163#45511yescharacteristic
96C2×C163#45611yescharacteristic
97D163—81no—
98D163—81no—
99C163#166111yescharacteristic, complemented, Sylow
100#1661—1631nomaximal, Sylow
101#562—1631no—
102#501D16311yescharacteristic, complemented
103#497—1631no—
104#489—1631no—
105#330—1631no—
106#329—1631no—
107#329—1631no—
108C2×Q16—3261no—
109C2×SD16—3261no—
110C4⋊Q8C2×D16311yescharacteristic
111#125C2×D16311yescharacteristic
112#124—1631no—
113#124—1631no—
114#123—3261no—
115#123—3261no—
116C22⋊Q8—3261no—
117C22⋊Q8—3261no—
118#121—3261no—
119#121—3261no—
120#107—21no—
121#106—21no—
122#103—3261no—
123#102—3261no—
124#97—1631no—
125#97—1631no—
126#97C2×D16311yescharacteristic
127C2×Q8—21no—
128C2×Q8—1631no—
129C2×Q8—1631no—
130C2×D4—3261no—
131C2×D4—3261no—
132C2×D4—1631no—
133C2×D4—1631no—
134C22×C4—3261no—
135C22×C4—3261no—
136Q16—6522no—
137SD16—6522no—
138C2×C8—21no—
139C2×C8—3261no—
140C2×C8—3261no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—3261no—
146C4⋊C4—3261no—
147C4⋊C4C22⋊D16311yescharacteristic
148C4⋊C4C22⋊D16311yescharacteristic
149C22⋊C4—3261no—
150C22⋊C4—3261no—
151C22⋊C4—3261no—
152C22⋊C4—3261no—
153C22⋊C4—3261no—
154C22⋊C4—3261no—
155C42C22×D16311yescharacteristic
156C23—3261no—
157C23—3261no—
158Q8—41no—
159Q8—41no—
160Q8—6521no—
161Q8—6521no—
162D4—13041no—
163D4—13041no—
164D4—6521no—
165D4—6521no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—13041no—
170C2×C4—13041no—
171C2×C4—3261no—
172C2×C4—3261no—
173C2×C4D4×D16311yescharacteristic
174C2×C4D4×D16311yescharacteristic
175C2×C4C2×C22⋊D16311yescharacteristic
176C8—22no—
177C8—3262no—
178C8—6521no—
179C22—6521no—
180C22—6521no—
181C22—6521no—
182C22—6521no—
183C22—6521no—
184C22—6521no—
185C22#1910470311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—81no—
192C4—6521no—
193C4—6521no—
194C2—13041no—
195C2—13041no—
196C2#2378183111yescharacteristic
197C2#2378184111yescharacteristic
198C2#2378184711yescharacteristic
subgroupG/H
centerC22#19104703
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#501×C163C2
socleC22×C163#120
prime radical#23782413C2
hypercenter#501D163
hypocenterC163#1661
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #23782057C2semidirect
3. #23782027C2semidirect
4. #23781407C2semidirect
5. #23781404C2semidirect
6. #23781404C2semidirect
7. #501×C163C2semidirect
99. C163#1661semidirect
102. #501D163semidirect

Maximal subgroups

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

subgroupindex
#237824132
#237820572
#237820272
#237814072
#237814042
#501×C1632
#1661163