Finite Group Databasebeta

← #36354090

Subgroups — #36354090

Order 22144.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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#26012055C211yesmaximal, characteristic
2#26011497C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic, complemented
4#26011248C212yesmaximal, complemented
5#568×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7#19149675C412yescomplemented
8#19149668C412yescomplemented
9#19149653—22no—
10#19149647—22no—
11#19149638C2212yes—
12#19149638C2211yescharacteristic
13#19149601—24no—
14#19149594—22no—
15C2×#41×C173C2211yescharacteristic
16#131×C173—24no—
17C2×M16×C173—22no—
18C2×M16×C173C2212yes—
19C23×D173—21no—
20C2×D692—22no—
21C2×D692—21no—
22C2×D692—22no—
23C2×D692C2×C412yes—
24C2×C22⋊D173—24no—
25C4⋊Dic173—21no—
26C4⋊Dic173—22no—
27C22⋊Dic173—21no—
28C22⋊Dic173—22no—
29#14583803—24no—
30C4×Dic173—22no—
31C4×Dic173C2×C412yes—
32#41×C173—22no—
33#41×C173—22no—
34C2×Q8×C173C2×C411yescharacteristic
35C2×D4×C173C2×C411yescharacteristic
36C2×D4×C173D412yes—
37C22×C4×C173D412yes—
38C22×C4×C173C2311yescharacteristic
39M16×C173—24no—
40M16×C173—22no—
41M16×C173—22no—
42C2×C8×C173—24no—
43C2×C8×C173D414yescomplemented
44C22×D173—22no—
45C22×D173—42no—
46C22×D173—82no—
47D692—24no—
48D692—41no—
49D692—41no—
50C22⋊D173—84no—
51C2×Dic173—22no—
52C2×Dic173—42no—
53C23×C173C22⋊C412yes—
54C23×C173C2×D411yescharacteristic
55Q8×C173—22no—
56D4×C173—24no—
57D4×C173—22no—
58C2×C4×C173—22no—
59C2×C4×C173—22no—
60C2×C4×C173C22⋊C412yes—
61C2×C4×C173C22×C411yescharacteristic
62C2×C4×C173C2×D412yes—
63C2×C4×C173C2×D411yescharacteristic
64C2×C4×C173#4112yes—
65C8×C173—24no—
66C22×C173—22no—
67C22×C173—41no—
68C22×C173—42no—
69C22×C173C2×C22⋊C411yescharacteristic
70C22×C173#12112yes—
71C2×D173—44no—
72C2×D173—42no—
73C2×D173—84no—
74C4×C173—22no—
75C4×C173C4×D412yes—
76C4×C173#12011yescharacteristic
77C4×C173#12311yescharacteristic
78Dic173—42no—
79Dic173—82no—
80C2×C173—21no—
81C2×C173—22no—
82C2×C173—42no—
83C2×C173#38611yescharacteristic
84D173—82no—
85C173#191011yescharacteristic, complemented, Sylow
86#1910—1731nomaximal, Sylow
87#568D17311yescharacteristic, complemented
88#535—1731no—
89#418—1732no—
90#410—1731no—
91H(3,ℤ/4)—1732no—
92C2×#41C2×D17311yescharacteristic
93C22×D4—1731no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D17312yes—
97#127—1732no—
98#124—1732no—
99#121—3462no—
100#120—3462no—
101#117—1732no—
102#117—1731no—
103#102—3464no—
104#99—3462no—
105C24—3461no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D17311yescharacteristic
109C2×D4—3464no—
110C2×D4—3462no—
111C2×D4—3461no—
112C2×D4—3462no—
113C2×D4—1732no—
114C2×D4C4×D17311yescharacteristic
115C2×D4D69212yes—
116C22×C4C22⋊D17312yes—
117C22×C4C22×D17311yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D17314yescomplemented
123C4⋊C4—3461no—
124C4⋊C4—3462no—
125C22⋊C4—3461no—
126C22⋊C4—3464no—
127C22⋊C4—3462no—
128C42—3462no—
129C42—1732no—
130C23—3462no—
131C23—13842no—
132C23—6922no—
133C23#1458380312yes—
134C23C2×D69211yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—3464no—
139D4—13844no—
140D4—6921no—
141D4—6921no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—3462no—
145C2×C4—6922no—
146C2×C4#1458380312yes—
147C2×C4C2×C4×D17311yescharacteristic
148C2×C4C2×C22⋊D17312yes—
149C2×C4C2×C22⋊D17311yescharacteristic
150C2×C4#1458382612yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—13844no—
156C22—6924no—
157C22—6922no—
158C22C2×#1458380311yescharacteristic
159C22#1914965312yes—
160C4—22no—
161C4—13842no—
162C4—6922no—
163C4C4×C22⋊D17312yes—
164C4#1914964811yescharacteristic
165C4#1914966211yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—13842no—
170C2#2601137011yescharacteristic
subgroupG/H
centerC2#26011370
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#568×C173C2
socleC2×C173#386
prime radical#26012055C2
cosocleC22×C4×C173C23
hypercenter#568D173
hypocenterC173#1910
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26011497C2semidirect
3. #26011465C2semidirect
4. #26011248C2semidirect
5. #568×C173C2semidirect
7. #19149675C4semidirect
8. #19149668C4semidirect
43. C2×C8×C173D4semidirect
85. C173#1910semidirect
87. #568D173semidirect
122. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120552
#260114972
#260114652
#260112482
#568×C1732
#1910173