Finite Group Databasebeta

← #36361886

Subgroups — #36361886

Order 22144.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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
2C22×C4×Dic173C211yesmaximal, characteristic, complemented
3#26011383C214yesmaximal, complemented
4#26011376C214yesmaximal, complemented
5#26011368C212yesmaximal
6#26011366C212yesmaximal, complemented
7#381×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9C23×Dic173C2212yes—
10#19149662—24no—
11C2×C4⋊Dic173C2214yes—
12C2×#14583805C2212yes—
13C2×C22⋊Dic173C2211yescharacteristic
14C2×#14583803C2212yes—
15C2×#14583803C2214yes—
16C2×C4×Dic173—24no—
17C2×C4×Dic173C2214yes—
18#19149586C2214yes—
19#19149586C2214yes—
20C23×C4×C173C2211yescharacteristic
21C2×C22⋊C4×C173C2212yes—
22#95×C173C2214yes—
23C23×D173C2311yescharacteristic
24C2×C22⋊D173—22no—
25C2×C22⋊D173—24no—
26C22×Dic173—28no—
27C22×Dic173—24no—
28C22×Dic173C2311yescharacteristic
29C22×Dic173C2312yes—
30C22×Dic173C2314yes—
31C4⋊Dic173—28no—
32#14583805—24no—
33C22⋊Dic173—22no—
34#14583803—28no—
35#14583803—24no—
36C4×Dic173—28no—
37C24×C173C2311yescharacteristic
38C22×C4×C173—24no—
39C22×C4×C173D414yes—
40C22×C4×C173C2314yes—
41C22×C4×C173C2312yes—
42C22⋊C4×C173—24no—
43C22×D173—21no—
44C22×D173—22no—
45C22×D173—24no—
46C22⋊D173—44no—
47C2×Dic173—216no—
48C2×Dic173—24no—
49C2×Dic173—24no—
50C2×Dic173—21no—
51C2×Dic173—22no—
52C2×Dic173—22no—
53C2×Dic173—24no—
54C2×Dic173#4118yes—
55C23×C173—24no—
56C23×C173C2×D412yes—
57C23×C173#4114yes—
58C23×C173C2411yescharacteristic
59C2×C4×C173—28no—
60C2×C4×C173—24no—
61C2×C4×C173—24no—
62C2×C4×C173—24no—
63C2×C4×C173—24no—
64C2×C4×C173C2×D414yes—
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—44no—
76Dic173—28no—
77Dic173—41no—
78Dic173—42no—
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×C42—1731no—
88#388—1734no—
89#387—1734no—
90#386—1732no—
91#385—1732no—
92#381D17311yescharacteristic, complemented
93C22×D4—1731no—
94C23×C4—1732no—
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—1734no—
102C2×C22⋊C4C2×D17312yes—
103C2×C42—3464no—
104C2×C42—1734no—
105#95—1734no—
106#95—1734no—
107#95C2×D17314yes—
108C24—1731no—
109C24C22×D17311yescharacteristic
110C2×D4—3462no—
111C2×D4—3464no—
112C22×C4—24no—
113C22×C4—3464no—
114C22×C4—3468no—
115C22×C4—1731no—
116C22×C4—1732no—
117C22×C4—1734no—
118C22×C4D69214yes—
119C22×C4C22×D17314yes—
120C22×C4C22×D17312yes—
121C4⋊C4—3468no—
122C4⋊C4—3464no—
123C22⋊C4—24no—
124C22⋊C4—3468no—
125C22⋊C4—3462no—
126C22⋊C4—3464no—
127C42—3468no—
128C23—24no—
129C23—3461no—
130C23—3464no—
131C23—3462no—
132C23C2×D69212yes—
133C23#1458382514yes—
134C23C23×D17311yescharacteristic
135D4—6924no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—34616no—
142C2×C4—3464no—
143C2×C4—3464no—
144C2×C4—3462no—
145C2×C4—3461no—
146C2×C4—3464no—
147C2×C4—3462no—
148C2×C4—1738no—
149C2×C4C2×D69214yes—
150C22—24no—
151C22—28no—
152C22—6921no—
153C22—6922no—
154C22—6924no—
155C22#1914966314yes—
156C22C22×D69211yescharacteristic
157C22#41×D17314yes—
158C22C2×#1458382512yes—
159C4—24no—
160C4—44no—
161C4—3468no—
162C4—6921no—
163C4—6922no—
164C2—24no—
165C2—13841no—
166C2C2×#1914966311yescharacteristic
167C2#2601200312yes—
168C2#2601201814yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#381×C173C2
prime radicalC22×C22⋊D173C22
hypercenter#381D173
hypocenterC173#2853
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic173C2semidirect
3. #26011383C2semidirect
4. #26011376C2semidirect
6. #26011366C2semidirect
7. #381×C173C2semidirect
84. C173#2853semidirect
92. #381D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496622
C22×C4×Dic1732
#260113832
#260113762
#260113682
#260113662
#381×C1732
#2853173