Finite Group Databasebeta

← #35213873

Subgroups — #35213873

Order 20864.

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

normal subgroups only

This group has 184 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#23782415C211yesmaximal, characteristic
2#23782074C211yesmaximal, characteristic, complemented
3#23782034C211yesmaximal, characteristic, complemented
4#23782024C211yesmaximal, characteristic
5#23781406C211yesmaximal, characteristic, complemented
6#23781406C211yesmaximal, characteristic, complemented
7#329×C163C211yesmaximal, characteristic
8#19104735C2211yescharacteristic
9#19104733—21no—
10#19104728—21no—
11#19104728—21no—
12#19104727C2211yescharacteristic
13#19104727C2211yescharacteristic
14#19104716—21no—
15#19104712—21no—
16#19104712—21no—
17#19104669—21no—
18#19104667—21no—
19#19104660—21no—
20#19104660—21no—
21#19104659—21no—
22#19104656—21no—
23#19104646C2211yescharacteristic
24#124×C163C2211yescharacteristic
25#97×C163C2211yescharacteristic
26#97×C163C2211yescharacteristic
27C2×Dic326—21no—
28C2×Dic326—21no—
29C22×Dic163—21no—
30C2×C163⋊C8—21no—
31C4⋊Dic163—21no—
32C4⋊Dic163—21no—
33C4⋊Dic163D411yescharacteristic
34C4⋊Dic163D411yescharacteristic
35C4⋊Dic163D411yescharacteristic
36C4⋊Dic163D411yescharacteristic
37Dic163⋊C4—21no—
38Dic163⋊C4—21no—
39Dic163⋊C4—21no—
40Dic163⋊C4—21no—
41Dic163⋊C4—21no—
42Dic163⋊C4—21no—
43C22⋊Dic163—21no—
44C22⋊Dic163—21no—
45C22⋊Dic163—21no—
46C4×Dic163—21no—
47C2×Q8×C163D411yescharacteristic
48C2×D4×C163D411yescharacteristic
49C2×C8×C163—21no—
50C2×C8×C163—21no—
51C22⋊C4×C163—21no—
52C22⋊C4×C163—21no—
53C42×C163C2311yescharacteristic
54Dic326—42no—
55Dic326—81no—
56C2×Dic163—21no—
57C2×Dic163—21no—
58C2×Dic163—21no—
59C2×Dic163—21no—
60C2×Dic163—41no—
61C2×Dic163—81no—
62C163⋊C8—41no—
63C23×C163—21no—
64Q8×C163—41no—
65D4×C163—41no—
66C2×C4×C163—21no—
67C2×C4×C163C2×D411yescharacteristic
68C2×C4×C163C2×D411yescharacteristic
69C2×C4×C163C2×D411yescharacteristic
70C8×C163—22no—
71C8×C163—41no—
72C22×C163—41no—
73C22×C163—41no—
74C22×C163—41no—
75C22×C163#12011yescharacteristic
76C4×C163—21no—
77C4×C163—21no—
78C4×C163—21no—
79C4×C163—41no—
80Dic163—41no—
81Dic163—41no—
82Dic163—41no—
83Dic163—41no—
84Dic163—81no—
85C2×C163—81no—
86C2×C163#45411yescharacteristic
87C2×C163#45511yescharacteristic
88C2×C163#45611yescharacteristic
89C163#166411yescharacteristic, complemented, Sylow
90#1664—1631nomaximal, Sylow
91#563—1631no—
92#501—1631no—
93#491—1631no—
94#489—1631no—
95#330—1631no—
96#330—1631no—
97#329D16311yescharacteristic
98C4⋊Q8—3261no—
99C4⋊Q8—1631no—
100#126—3261no—
101#126—3261no—
102#125—1631no—
103#125—1631no—
104#124C2×D16311yescharacteristic
105#123—3261no—
106C22⋊Q8—3261no—
107C22⋊Q8—3261no—
108#107—3261no—
109#106—3261no—
110#103—3261no—
111#103—3261no—
112#103—3261no—
113#102—3261no—
114#97—1631no—
115#97C2×D16311yescharacteristic
116#97C2×D16311yescharacteristic
117C2×Q8—3261no—
118C2×Q8—3261no—
119C2×Q8D65211yescharacteristic
120C2×D4D65211yescharacteristic
121C22×C4—3261no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—3261no—
125C4⋊C4—3261no—
126C4⋊C4—3261no—
127C4⋊C4—3261no—
128C4⋊C4—3261no—
129C4⋊C4—3261no—
130C4⋊C4—3261no—
131C4⋊C4—3261no—
132C4⋊C4—3261no—
133C4⋊C4—1631no—
134C4⋊C4—1631no—
135C4⋊C4—1631no—
136C4⋊C4—1631no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—3261no—
140C22⋊C4—3261no—
141C22⋊C4—3261no—
142C42—3261no—
143C42C22×D16311yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—13041no—
147Q8—6522no—
148D4—41no—
149C2×C4—21no—
150C2×C4—13041no—
151C2×C4—3261no—
152C2×C4—3261no—
153C2×C4—3261no—
154C2×C4—3261no—
155C2×C4—6521no—
156C2×C4D4×D16311yescharacteristic
157C2×C4D4×D16311yescharacteristic
158C2×C4C2×D65211yescharacteristic
159C8—22no—
160C8—41no—
161C8—6521no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1910470511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—13041no—
171C4—6521no—
172C4—6521no—
173C4—6521no—
174C4—6521no—
175C2—81no—
176C2#2378183911yescharacteristic
177C2#2378184511yescharacteristic
178C2#2378184911yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#329×C163C2
socleC22×C163#120
prime radicalC2×D4×C163D4
hypercenter#329D163
hypocenterC163#1664
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #23782074C2semidirect
3. #23782034C2semidirect
5. #23781406C2semidirect
6. #23781406C2semidirect
89. C163#1664semidirect

Maximal subgroups

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

subgroupindex
#237824152
#237820742
#237820342
#237820242
#237814062
#329×C1632
#1664163