Finite Group Databasebeta

← #35216506

Subgroups — #35216506

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#23782466C211yesmaximal, characteristic
2#23782106C211yesmaximal, characteristic, complemented
3#23781937C211yesmaximal, characteristic, complemented
4#23781478C211yesmaximal, characteristic, complemented
5#23781473C211yesmaximal, characteristic, complemented
6#23781448C211yesmaximal, characteristic, complemented
7C2×SD32×C163C211yesmaximal, characteristic, complemented
8#19104780—21no—
9#19104766—21no—
10#19104761—21no—
11#19104759—22no—
12#19104757C2211yescharacteristic
13#19104754—21no—
14D8×D163—21no—
15#19104741C2211yescharacteristic
16#19104681—21no—
17#19104678—21no—
18#19104675C2211yescharacteristic, complemented
19#19104671C2211yescharacteristic, complemented
20#19104654—21no—
21C2×Q16×C163C2211yescharacteristic, complemented
22C2×D8×C163C2211yescharacteristic
23SD32×C163—22no—
24C2×C16×C163C2211yescharacteristic, complemented
25#14579699—21no—
26#14579699D411yescharacteristic
27#14579698—22no—
28#14579698—41no—
29C2×C22⋊D163—42no—
30D4×D163—22no—
31D4×D163—41no—
32Dic652D411yescharacteristic
33#14579689—41no—
34#14579688—21no—
35#14579687—22no—
36#14579686—41no—
37D1304D411yescharacteristic
38#14579684—22no—
39#14579683—21no—
40#14579683D411yescharacteristic
41#14579682—21no—
42C8×D163—21no—
43C163⋊C16—21no—
44C2×Q8×C163—21no—
45C2×D4×C163—21no—
46Q16×C163—21no—
47Q16×C163—21no—
48D8×C163—21no—
49D8×C163D412yes—
50C2×C8×C163C2311yescharacteristic
51C16×C163—21no—
52C22×D163—42no—
53Dic326—21no—
54Dic326D811yescharacteristic
55D652—21no—
56D652D811yescharacteristic
57C22⋊D163—21no—
58C22⋊D163—42no—
59C22⋊D163—42no—
60C22⋊D163—41no—
61C2×Dic163—42no—
62C4×D163—21no—
63C4×D163—41no—
64C163⋊C8—21no—
65C163⋊C8—41no—
66C23×C163—41no—
67Q8×C163—41no—
68Q8×C163—41no—
69D4×C163—22no—
70D4×C163—41no—
71C2×C4×C163—41no—
72C2×C4×C163C2×D411yescharacteristic
73C8×C163C2×D411yescharacteristic
74C8×C163C2×D411yescharacteristic
75C22×C163—42no—
76C22×C163—161no—
77C22×C163AGL(1,ℤ/8)11yescharacteristic
78C2×D163—21no—
79C2×D163—41no—
80C2×D163—162no—
81C4×C163—81no—
82C4×C163#12011yescharacteristic
83C4×C163C2×D811yescharacteristic
84Dic163—21no—
85Dic163—41no—
86C2×C163—21no—
87C2×C163—82no—
88C2×C163#44711yescharacteristic
89D163—41no—
90D163—81no—
91C163#219411yescharacteristic, complemented, Sylow
92#2194—1631nomaximal, Sylow
93#576—1631no—
94#510—1631no—
95C2×SD32D16311yescharacteristic, complemented
96#473—1631no—
97#361—1631no—
98#359—1631no—
99#350—1631no—
100H(5,𝔽2)—3261no—
101#137—3261no—
102AGL(1,ℤ/8)—3262no—
103AGL(1,ℤ/8)—3261no—
104#135—3261no—
105#135—1631no—
106C2×Q16C2×D16311yescharacteristic, complemented
107C2×D8—3261no—
108C2×D8C2×D16311yescharacteristic
109#131—1631no—
110Q32—3261no—
111SD32—22no—
112SD32—3261no—
113M32—1631no—
114C2×C16C2×D16311yescharacteristic, complemented
115#108—1631no—
116#101—3261no—
117#41—3262no—
118#41—3261no—
119#41—6521no—
120#41—1631no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—3262no—
124C2×D4—6521no—
125C2×D4—6522no—
126Q16—21no—
127Q16—21no—
128Q16—6521no—
129Q16—1631no—
130SD16—3262no—
131SD16—3261no—
132SD16—6521no—
133D8—21no—
134D8—3262no—
135D8—1631no—
136D8C22⋊D16312yes—
137M16—3261no—
138M16—3261no—
139M16—1631no—
140C2×C8—3261no—
141C2×C8C22×D16311yescharacteristic
142C16—21no—
143C16—3261no—
144C23—41no—
145C23—6522no—
146Q8—41no—
147Q8—41no—
148Q8—3261no—
149Q8—1631no—
150D4—22no—
151D4—41no—
152D4—3261no—
153D4—3261no—
154D4—6522no—
155D4—6522no—
156D4—6521no—
157D4—1631no—
158C2×C4—41no—
159C2×C4—3261no—
160C2×C4—6522no—
161C2×C4—6521no—
162C2×C4D4×D16311yescharacteristic
163C8—3261no—
164C8—6521no—
165C8D4×D16311yescharacteristic
166C8C2×C22⋊D16311yescharacteristic
167C22—42no—
168C22—161no—
169C22—26082no—
170C22—3261no—
171C22—6521no—
172C22#1910475911yescharacteristic
173C4—81no—
174C4—3261no—
175C4—6521no—
176C4#1910470311yescharacteristic
177C4D8×D16311yescharacteristic
178C2—21no—
179C2—82no—
180C2—13041no—
181C2—6521no—
182C2#2378179711yescharacteristic
subgroupG/H
centerC2#23781797
commutator, cosocleC2×C8×C163C23
FrattiniC2×C8C22×D163
FittingC2×SD32×C163C2
socleC2×C163#447
prime radical#23782466C2
hypercenterC2×SD32D163
hypocenterC163#2194
LCSG ▷ C2×C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #23782106C2semidirect
3. #23781937C2semidirect
4. #23781478C2semidirect
5. #23781473C2semidirect
6. #23781448C2semidirect
7. C2×SD32×C163C2semidirect
18. #19104675C22semidirect
19. #19104671C22semidirect
21. C2×Q16×C163C22semidirect
24. C2×C16×C163C22semidirect
91. C163#2194semidirect
95. C2×SD32D163semidirect
106. C2×Q16C2×D163semidirect
114. C2×C16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824662
#237821062
#237819372
#237814782
#237814732
#237814482
C2×SD32×C1632
#2194163