Finite Group Databasebeta

← #36354088

Subgroups — #36354088

Order 22144.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#26012258C211yesmaximal, characteristic, complemented
2#26012055C211yesmaximal, characteristic
3#26011495C212yesmaximal, complemented
4#26011247C212yesmaximal, complemented
5#410×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7#19149686—24no—
8C2×#14583810—22no—
9C2×#14583810C2212yes—
10#19149675C412yescomplemented
11#19149668C412yescomplemented
12#19149653—22no—
13#19149647—22no—
14#19149638C2211yescharacteristic
15#19149599—24no—
16C2×#41×C173C2211yescharacteristic
17#117×C173C2212yes—
18#99×C173—22no—
19C23×D173—21no—
20C2×D692—22no—
21C2×D692—21no—
22C2×D692—22no—
23C2×D692C2×C412yes—
24C2×C22⋊D173—24no—
25#14583810—24no—
26#14583810—22no—
27#14583810—22no—
28C2×#13063436—24no—
29C2×#13063436D414yescomplemented
30C4⋊Dic173—21no—
31C22⋊Dic173—21no—
32#14583803—24no—
33C4×Dic173C2×C412yes—
34#41×C173—22no—
35#41×C173—22no—
36C2×Q8×C173C2×C411yescharacteristic
37C2×D4×C173C2×C411yescharacteristic
38C2×D4×C173D412yes—
39C22×C4×C173D412yes—
40C22×C4×C173C2311yescharacteristic
41C4⋊C4×C173—22no—
42C22⋊C4×C173—22no—
43C42×C173—22no—
44C22×D173—22no—
45C22×D173—42no—
46C22×D173—82no—
47D692—24no—
48D692—41no—
49D692—41no—
50C22⋊D173—84no—
51C2×Dic173—22no—
52#13063436—24no—
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×C173—42no—
61C2×C4×C173C22⋊C412yes—
62C2×C4×C173C22×C411yescharacteristic
63C2×C4×C173C2×D412yes—
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173#4112yes—
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×C173—82no—
76C4×C173C4×D412yes—
77C4×C173#12011yescharacteristic
78C4×C173#12311yescharacteristic
79Dic173—42no—
80C2×C173—21no—
81C2×C173—22no—
82C2×C173—42no—
83C2×C173#38611yescharacteristic
84D173—82no—
85C173#191011yescharacteristic, complemented, Sylow
86#1910—1731nomaximal, Sylow
87#568—1731no—
88#535—1731no—
89#418—1732no—
90#410D17311yescharacteristic, complemented
91H(3,ℤ/4)—1732no—
92C2×#41C2×D17311yescharacteristic
93C22×D4—1731no—
94#131—3464no—
95C2×M16—3462no—
96C2×M16—1732no—
97#127—1732no—
98#124—1732no—
99#121—3462no—
100#120—3462no—
101#117—1731no—
102#117C2×D17312yes—
103#102—3464no—
104#99—22no—
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×D4C22⋊D17312yes—
116C22×C4D69212yes—
117C22×C4C22×D17311yescharacteristic
118M16—3464no—
119M16—3462no—
120M16—3462no—
121C2×C8—3464no—
122C2×C8—1734no—
123C4⋊C4—22no—
124C4⋊C4—3461no—
125C22⋊C4—22no—
126C22⋊C4—3461no—
127C22⋊C4—3464no—
128C42—22no—
129C42—1732no—
130C23—3462no—
131C23—13842no—
132C23—6922no—
133C23#1458380312yes—
134C23C2×C22⋊D17311yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—3464no—
139D4—13844no—
140D4—6921no—
141D4—6921no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—3462no—
146C2×C4#1458380312yes—
147C2×C4C2×C4×D17311yescharacteristic
148C2×C4D4×D17312yes—
149C2×C4C2×D69211yescharacteristic
150C2×C4#1458382512yes—
151C8—3464no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—13844no—
156C22—6924no—
157C22—6922no—
158C22C2×#1458380311yescharacteristic
159C22#1914965112yes—
160C4—22no—
161C4—82no—
162C4—6922no—
163C4#1914963912yes—
164C4#1914964911yescharacteristic
165C4#1914966311yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—13842no—
170C2#2601136811yescharacteristic
subgroupG/H
centerC2#26011368
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#410×C173C2
socleC2×C173#386
prime radical#26012055C2
cosocleC22×C4×C173C23
hypercenter#410D173
hypocenterC173#1910
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #26012258C2semidirect
3. #26011495C2semidirect
4. #26011247C2semidirect
5. #410×C173C2semidirect
10. #19149675C4semidirect
11. #19149668C4semidirect
29. C2×#13063436D4semidirect
85. C173#1910semidirect
90. #410D173semidirect

Maximal subgroups

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

subgroupindex
#260122582
#260120552
#260114952
#260112472
#410×C1732
#1910173