Finite Group Databasebeta

← #36361855

Subgroups — #36361855

Order 22144.

1102 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C2×#19149662C212yesmaximal, complemented
2C2×#19149648C211yesmaximal, characteristic
3#26011370C214yesmaximal, complemented
4#26011364C214yesmaximal, complemented
5#26011354C212yesmaximal, complemented
6#26011347C211yesmaximal, characteristic, complemented
7C24×C4×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9C22×C22⋊D173C2212yes—
10#19149662—28no—
11#19149648—24no—
12C2×#14583805C2214yescomplemented
13C2×C22⋊Dic173C2211yescharacteristic
14C2×C22⋊Dic173C2212yes—
15C2×C22⋊Dic173C2212yes—
16C2×C22⋊Dic173C2214yes—
17C2×#14583803C2214yescomplemented
18#19149586C2218yescomplemented
19C25×C173C2211yescharacteristic
20C23×C4×C173—24no—
21C23×C4×C173—28no—
22C23×C4×C173C2214yes—
23C23×C4×C173C2212yes—
24C23×D173C2311yescharacteristic
25C2×C22⋊D173—22no—
26C2×C22⋊D173—24no—
27C2×C22⋊D173—24no—
28C2×C22⋊D173—28no—
29C22×Dic173C2312yes—
30C22×Dic173C2314yes—
31C22×Dic173C2311yescharacteristic
32#14583805—28no—
33C22⋊Dic173—22no—
34C22⋊Dic173—24no—
35C22⋊Dic173—24no—
36C22⋊Dic173—28no—
37#14583803—28no—
38C24×C173—24no—
39C24×C173—28no—
40C24×C173D414yes—
41C24×C173C2311yescharacteristic
42C24×C173C2312yes—
43C22×C4×C173—232no—
44C22×C4×C173—28no—
45C22×C4×C173—24no—
46C22×C4×C173—24no—
47C22×C4×C173—216no—
48C22×C4×C173D418yescomplemented
49C22×C4×C173C2314yes—
50C22×D173—21no—
51C22×D173—22no—
52C22×D173—24no—
53C22⋊D173—44no—
54C22⋊D173—48no—
55C2×Dic173—21no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—24no—
59C2×Dic173—24no—
60C2×Dic173—28no—
61C23×C173—232no—
62C23×C173—28no—
63C23×C173—216no—
64C23×C173—24no—
65C23×C173—24no—
66C23×C173—24no—
67C23×C173C2×D412yes—
68C23×C173C2×D414yes—
69C23×C173C2×D414yes—
70C23×C173#4118yes—
71C23×C173C2411yescharacteristic
72C2×C4×C173—232no—
73C2×C4×C173—216no—
74C2×C4×C173—24no—
75C2×C4×C173—24no—
76C2×C4×C173C2×D418yes—
77C22×C173—232no—
78C22×C173—24no—
79C22×C173—24no—
80C22×C173—24no—
81C22×C173—28no—
82C22×C173—216no—
83C22×C173#12014yes—
84C22×C173#12318yescomplemented
85C22×C173C22×D411yescharacteristic
86C22×C173C22×D412yes—
87C22×C173C2×#4114yes—
88C2×D173—41no—
89C2×D173—42no—
90C2×D173—44no—
91C4×C173—28no—
92Dic173—41no—
93Dic173—42no—
94Dic173—44no—
95C2×C173—24no—
96C2×C173—28no—
97C2×C173C2×#12011yescharacteristic
98C2×C173C2×#12312yes—
99C2×C173#52514yes—
100D173—81no—
101C173#284911yescharacteristic, complemented, Sylow
102#2849—1731nomaximal, Sylow
103C24×C4D17311yescharacteristic, complemented
104C2×#123—1732no—
105C2×#120—1731no—
106#386—1734no—
107#385—1734no—
108#381—1732no—
109#379—1731no—
110C25C2×D17311yescharacteristic
111C22×D4—1731no—
112C22×D4—1732no—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4C2×D17314yes—
116C23×C4C2×D17312yes—
117#123—3468no—
118#120—3464no—
119C2×C4⋊C4—1734no—
120C2×C22⋊C4—1731no—
121C2×C22⋊C4—1732no—
122C2×C22⋊C4—1732no—
123C2×C22⋊C4—1734no—
124C2×C22⋊C4—1734no—
125#95—1738no—
126C24—24no—
127C24—28no—
128C24—1731no—
129C24C22⋊D17314yes—
130C24C22×D17311yescharacteristic
131C24C22×D17312yes—
132C2×D4—3462no—
133C2×D4—3464no—
134C2×D4—3464no—
135C2×D4—3468no—
136C22×C4—232no—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—216no—
141C22×C4—1732no—
142C22×C4—1734no—
143C22×C4—1731no—
144C22×C4C22⋊D17318yescomplemented
145C22×C4C22×D17314yes—
146C4⋊C4—3468no—
147C22⋊C4—3462no—
148C22⋊C4—3464no—
149C22⋊C4—3464no—
150C22⋊C4—3468no—
151C22⋊C4—3468no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—3462no—
159C23—3461no—
160C23—3464no—
161C23C2×C22⋊D17312yes—
162C23C2×C22⋊D17314yes—
163C23C2×C22⋊D17314yes—
164C23#1458382618yes—
165C23C23×D17311yescharacteristic
166D4—6924no—
167D4—6928no—
168C2×C4—232no—
169C2×C4—216no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—3464no—
173C2×C4—3461no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—3468no—
177C2×C4—3464no—
178C2×C4C2×C22⋊D17318yes—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—6921no—
186C22—6922no—
187C22—6924no—
188C22#1914964814yes—
189C22#1914966218yescomplemented
190C22C22×C22⋊D17311yescharacteristic
191C22C22×C22⋊D17312yes—
192C22C2×#1458382614yes—
193C4—28no—
194C4—6921no—
195C4—6922no—
196C4—6924no—
197C2—24no—
198C2—28no—
199C2—13841no—
200C2C2×#1914964811yescharacteristic
201C2C2×#1914966212yes—
202C2#2601200214yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
FittingC24×C4×C173C2
prime radicalC2×#19149648C2
hypercenterC24×C4D173
hypocenterC173#2849
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C2×#19149662C2semidirect
3. #26011370C2semidirect
4. #26011364C2semidirect
5. #26011354C2semidirect
6. #26011347C2semidirect
7. C24×C4×C173C2semidirect
12. C2×#14583805C22semidirect
17. C2×#14583803C22semidirect
18. #19149586C22semidirect
48. C22×C4×C173D4semidirect
84. C22×C173#123semidirect
101. C173#2849semidirect
103. C24×C4D173semidirect
144. C22×C4C22⋊D173semidirect
189. C22#19149662semidirect

Maximal subgroups

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

subgroupindex
C2×#191496622
C2×#191496482
#260113702
#260113642
#260113542
#260113472
C24×C4×C1732
#2849173