Finite Group Databasebeta

← #36354091

Subgroups — #36354091

Order 22144.

344 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 93 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#26011496C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic, complemented
4#26011247C212yesmaximal, complemented
5#535×C173C211yesmaximal, characteristic
6#19149686—24no—
7C2×#14583810—22no—
8C2×#14583810C2212yes—
9#19149638C2212yes—
10#19149600—24no—
11#19149594—22no—
12C2×#41×C173C2211yescharacteristic
13C22×D4×C173C2211yescharacteristic
14#127×C173C412yescomplemented
15#124×C173C412yescomplemented
16#121×C173—22no—
17#120×C173—22no—
18#117×C173C2211yescharacteristic
19#14583810—24no—
20#14583810—22no—
21#14583810—22no—
22C2×#13063436—24no—
23C2×#13063436D414yescomplemented
24C4⋊Dic173—22no—
25C22⋊Dic173—22no—
26C4×Dic173—22no—
27C24×C173—21no—
28#41×C173—22no—
29#41×C173—22no—
30C2×Q8×C173C2×C411yescharacteristic
31C2×D4×C173—24no—
32C2×D4×C173—22no—
33C2×D4×C173—21no—
34C2×D4×C173—22no—
35C2×D4×C173C2×C412yes—
36C2×D4×C173C2×C411yescharacteristic
37C2×D4×C173D412yes—
38C22×C4×C173D412yes—
39C22×C4×C173C2311yescharacteristic
40C4⋊C4×C173—21no—
41C22⋊C4×C173—21no—
42C22⋊C4×C173—24no—
43C42×C173C2×C412yes—
44C2×Dic173—42no—
45#13063436—24no—
46C23×C173—22no—
47C23×C173—42no—
48C23×C173—82no—
49C23×C173C22⋊C412yes—
50C23×C173C2×D411yescharacteristic
51Q8×C173—22no—
52D4×C173—24no—
53D4×C173—22no—
54D4×C173—24no—
55D4×C173—41no—
56D4×C173—41no—
57D4×C173—84no—
58C2×C4×C173—22no—
59C2×C4×C173—22no—
60C2×C4×C173—22no—
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×C173—44no—
70C22×C173—42no—
71C22×C173—84no—
72C22×C173C2×C22⋊C411yescharacteristic
73C22×C173#12112yes—
74C4×C173—22no—
75C4×C173—42no—
76C4×C173C4×D412yes—
77C4×C173#12011yescharacteristic
78C4×C173#12311yescharacteristic
79Dic173—82no—
80C2×C173—22no—
81C2×C173—21no—
82C2×C173—42no—
83C2×C173—82no—
84C2×C173#38611yescharacteristic
85C173#191011yescharacteristic, complemented, Sylow
86#1910—1731nomaximal, Sylow
87#568—1731no—
88#535D17311yescharacteristic
89#418—1732no—
90#410—1731no—
91H(3,ℤ/4)—1732no—
92C2×#41C2×D17311yescharacteristic
93C22×D4C2×D17311yescharacteristic
94#131—3464no—
95C2×M16—3462no—
96C2×M16—1732no—
97#127Dic17312yescomplemented
98#124Dic17312yescomplemented
99#121—22no—
100#120—22no—
101#117—1732no—
102#117C2×D17311yescharacteristic
103#102—3464no—
104#99—3462no—
105C24—21no—
106#41—22no—
107#41—22no—
108C2×Q8C2×Dic17311yescharacteristic
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4C2×Dic17312yes—
114C2×D4C2×Dic17311yescharacteristic
115C2×D4C22⋊D17312yes—
116C22×C4C22⋊D17312yes—
117C22×C4C22×D17311yescharacteristic
118M16—3464no—
119M16—3462no—
120M16—3462no—
121C2×C8—3464no—
122C2×C8—1734no—
123C4⋊C4—21no—
124C4⋊C4—3462no—
125C22⋊C4—21no—
126C22⋊C4—24no—
127C22⋊C4—3462no—
128C42—3462no—
129C42C2×Dic17312yes—
130C23—22no—
131C23—42no—
132C23—82no—
133C23C22⋊Dic17312yes—
134C23C2×C22⋊D17311yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—24no—
139D4—41no—
140D4—41no—
141D4—84no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—6922no—
146C2×C4C22⋊Dic17312yes—
147C2×C4C22×Dic17311yescharacteristic
148C2×C4D4×D17312yes—
149C2×C4C2×C22⋊D17311yescharacteristic
150C2×C4#1458382512yes—
151C8—3464no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—84no—
158C22C2×C22⋊Dic17311yescharacteristic
159C22#1914965112yes—
160C4—22no—
161C4—42no—
162C4—13842no—
163C4D4×Dic17312yes—
164C4#1914964711yescharacteristic
165C4#1914966011yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2601137111yescharacteristic
subgroupG/H
centerC2#26011371
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radical#535×C173C2
socleC2×C173#386
cosocleC22×C4×C173C23
hypercenter#535D173
hypocenterC173#1910
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26012258C2semidirect
2. #26011496C2semidirect
3. #26011465C2semidirect
4. #26011247C2semidirect
14. #127×C173C4semidirect
15. #124×C173C4semidirect
23. C2×#13063436D4semidirect
85. C173#1910semidirect
97. #127Dic173semidirect
98. #124Dic173semidirect

Maximal subgroups

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

subgroupindex
#260122582
#260114962
#260114652
#260112472
#535×C1732
#1910173