Finite Group Databasebeta

← #36361885

Subgroups — #36361885

Order 22144.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19149662C211yesmaximal, characteristic
2#26011382C214yesmaximal, complemented
3#26011375C214yesmaximal, complemented
4#26011370C212yesmaximal
5#26011364C212yesmaximal, complemented
6#26011354C211yesmaximal, characteristic, complemented
7C22×C42×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9#19149662—24no—
10C2×#14583805C2212yes—
11C2×#14583805C2214yes—
12C2×C22⋊Dic173C2211yescharacteristic
13C2×C22⋊Dic173C2212yes—
14C2×#14583803C2212yes—
15C2×#14583803C2214yes—
16#19149586C2214yes—
17#19149586C2214yes—
18#19149586C2214yes—
19C23×C4×C173C2212yes—
20C23×C4×C173C2211yescharacteristic
21C2×C42×C173—24no—
22C2×C42×C173C2214yes—
23C23×D173C2311yescharacteristic
24C2×C22⋊D173—22no—
25C2×C22⋊D173—24no—
26C22×Dic173C2314yes—
27C22×Dic173C2311yescharacteristic
28C22×Dic173C2312yes—
29#14583805—24no—
30#14583805—28no—
31C22⋊Dic173—22no—
32C22⋊Dic173—24no—
33#14583803—24no—
34#14583803—28no—
35C24×C173C2311yescharacteristic
36C22×C4×C173—24no—
37C22×C4×C173—24no—
38C22×C4×C173—28no—
39C22×C4×C173D414yes—
40C22×C4×C173C2312yes—
41C22×C4×C173C2314yes—
42C42×C173—28no—
43C22×D173—21no—
44C22×D173—22no—
45C22×D173—24no—
46C22⋊D173—44no—
47C2×Dic173—21no—
48C2×Dic173—22no—
49C2×Dic173—22no—
50C2×Dic173—24no—
51C2×Dic173—24no—
52C2×Dic173—24no—
53C2×Dic173—24no—
54C23×C173—24no—
55C23×C173C2×D412yes—
56C23×C173#4114yes—
57C23×C173C2411yescharacteristic
58C2×C4×C173—216no—
59C2×C4×C173—28no—
60C2×C4×C173—24no—
61C2×C4×C173—24no—
62C2×C4×C173—24no—
63C2×C4×C173C2×D414yes—
64C2×C4×C173#4118yes—
65C22×C173—24no—
66C22×C173—28no—
67C22×C173#12314yes—
68C22×C173C22×D411yescharacteristic
69C22×C173C2×#4112yes—
70C22×C173C2×#4114yes—
71C2×D173—41no—
72C2×D173—42no—
73C2×D173—44no—
74C4×C173—24no—
75C4×C173—28no—
76Dic173—41no—
77Dic173—42no—
78Dic173—44no—
79C2×C173—24no—
80C2×C173C2×#12311yescharacteristic
81C2×C173#52512yes—
82C2×C173#52914yes—
83D173—81no—
84C173#285311yescharacteristic, complemented, Sylow
85#2853—1731nomaximal, Sylow
86C2×#123—1731no—
87C22×C42D17311yescharacteristic, complemented
88#388—1734no—
89#387—1734no—
90#386—1732no—
91#385—1732no—
92#381—1731no—
93C22×D4—1731no—
94C23×C4C2×D17312yes—
95C23×C4C2×D17311yescharacteristic
96#123—3464no—
97C2×C4⋊C4—1732no—
98C2×C4⋊C4—1734no—
99C2×C22⋊C4—1731no—
100C2×C22⋊C4—1732no—
101C2×C22⋊C4—1732no—
102C2×C22⋊C4—1734no—
103C2×C42—24no—
104C2×C42C2×D17314yes—
105#95—1734no—
106#95—1734no—
107#95—1734no—
108C24—1731no—
109C24C22×D17311yescharacteristic
110C2×D4—3462no—
111C2×D4—3464no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—1734no—
116C22×C4—1731no—
117C22×C4—1732no—
118C22×C4C22⋊D17314yes—
119C22×C4C22×D17312yes—
120C22×C4C22×D17314yes—
121C4⋊C4—3464no—
122C4⋊C4—3468no—
123C22⋊C4—3462no—
124C22⋊C4—3464no—
125C22⋊C4—3464no—
126C22⋊C4—3468no—
127C42—28no—
128C23—24no—
129C23—3461no—
130C23—3464no—
131C23—3462no—
132C23C2×C22⋊D17312yes—
133C23#1458382614yes—
134C23C23×D17311yescharacteristic
135D4—6924no—
136C2×C4—216no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—3462no—
142C2×C4—3461no—
143C2×C4—3464no—
144C2×C4—3464no—
145C2×C4—3464no—
146C2×C4—3464no—
147C2×C4—3462no—
148C2×C4C2×C22⋊D17314yes—
149C2×C4#1458382618yes—
150C22—24no—
151C22—28no—
152C22—6921no—
153C22—6922no—
154C22—6924no—
155C22#1914966214yes—
156C22C22×C22⋊D17311yescharacteristic
157C22C2×#1458382612yes—
158C22C2×#1458382614yes—
159C4—24no—
160C4—28no—
161C4—6921no—
162C4—6922no—
163C4—6924no—
164C2—24no—
165C2—13841no—
166C2C2×#1914966211yescharacteristic
167C2#2601200212yes—
168C2#2601202514yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
FittingC22×C42×C173C2
prime radicalC22×C22⋊D173C22
hypercenterC22×C42D173
hypocenterC173#2853
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #26011382C2semidirect
3. #26011375C2semidirect
5. #26011364C2semidirect
6. #26011354C2semidirect
7. C22×C42×C173C2semidirect
84. C173#2853semidirect
87. C22×C42D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496622
#260113822
#260113752
#260113702
#260113642
#260113542
C22×C42×C1732
#2853173