Finite Group Databasebeta

← #36352072

Subgroups — #36352072

Order 22144.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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#26012041C211yesmaximal, characteristic
2#26011337C211yesmaximal, characteristic, complemented
3#26011226C214yesmaximal, complemented
4#432×C173C211yesmaximal, characteristic, complemented
5C2×#14583826—21no—
6#19149679C411yescharacteristic, complemented
7#19149676C411yescharacteristic, complemented
8#19149667—21no—
9#19149653—21no—
10#19149653C2212yes—
11C4×D692—21no—
12C4×D692C412yescomplemented
13C2×C4×Dic173—21no—
14#19149586—21no—
15#19149586C2212yes—
16C2×C42×C173C2211yescharacteristic
17C4⋊C8×C173—21no—
18C22⋊C8×C173C2212yes—
19#97×C173—21no—
20#14583826—44no—
21C2×Dic346—21no—
22C2×D692—21no—
23C2×D692—21no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—22no—
26C22×Dic173—22no—
27C2×C4×D173—22no—
28C4⋊Dic173—21no—
29C4⋊Dic173C2×C412yes—
30#14583803—22no—
31#14583803—22no—
32C4×Dic173—24no—
33C22×C4×C173D411yescharacteristic
34C22×C4×C173D411yescharacteristic
35C22×C4×C173C2311yescharacteristic
36C2×C8×C173—22no—
37C42×C173C2×C411yescharacteristic
38C42×C173C2×C411yescharacteristic
39C42×C173D411yescharacteristic
40C42×C173D411yescharacteristic
41C22×D173—22no—
42Dic346—42no—
43D692—42no—
44D692—42no—
45D692—42no—
46C22⋊D173—44no—
47C2×Dic173—24no—
48C2×Dic173—22no—
49C2×Dic173—42no—
50C2×Dic173—42no—
51C4×D173—44no—
52C23×C173C2×D411yescharacteristic
53C2×C4×C173—22no—
54C2×C4×C173—22no—
55C2×C4×C173—41no—
56C2×C4×C173C22⋊C411yescharacteristic
57C2×C4×C173C22⋊C411yescharacteristic
58C2×C4×C173C22⋊C411yescharacteristic
59C2×C4×C173C22⋊C411yescharacteristic
60C2×C4×C173C22×C411yescharacteristic
61C2×C4×C173C2×D411yescharacteristic
62C8×C173—42no—
63C22×C173—21no—
64C22×C173—21no—
65C22×C173C2×C22⋊C411yescharacteristic
66C22×C173AGL(1,ℤ/8)12yes—
67C2×D173—42no—
68C2×D173—42no—
69C2×D173—42no—
70C4×C173—21no—
71C4×C173—21no—
72C4×C173—21no—
73C4×C173—21no—
74C4×C173—21no—
75C4×C173#10412yes—
76Dic173—44no—
77Dic173—42no—
78C2×C173—22no—
79C2×C173#41011yescharacteristic
80C2×C173#41811yescharacteristic
81C2×C173C2×#10411yescharacteristic
82D173—82no—
83C173#150511yescharacteristic, complemented, Sylow
84#1505—1731nomaximal, Sylow
85#532—1731no—
86#432D17311yescharacteristic, complemented
87#376—1731no—
88#327—1734no—
89C2×#41—3461no—
90C4⋊Q8—1731no—
91#127—1731no—
92#124—3461no—
93#121—3461no—
94#121—1732no—
95C4×D4—3461no—
96C4×D4—1732no—
97C2×C42—3461no—
98C2×C42C2×D17311yescharacteristic
99C4⋊C8—21no—
100C22⋊C8C2×D17312yes—
101#97—21no—
102#95—3461no—
103#95—1732no—
104#41—6924no—
105C2×Q8—3461no—
106C2×D4—3462no—
107C2×D4—3461no—
108C2×D4—3461no—
109C2×D4—1732no—
110C22×C4—3462no—
111C22×C4—3462no—
112C22×C4D69211yescharacteristic
113C22×C4D69211yescharacteristic
114C22×C4C22×D17311yescharacteristic
115C2×C8—22no—
116C4⋊C4—3461no—
117C4⋊C4—1732no—
118C22⋊C4—3462no—
119C22⋊C4—3462no—
120C42—3464no—
121C42C4×D17311yescharacteristic
122C42C4×D17311yescharacteristic
123C42C22⋊D17311yescharacteristic
124C42C22⋊D17311yescharacteristic
125C23—3462no—
126C23C2×D69211yescharacteristic
127Q8—6922no—
128D4—6922no—
129D4—6922no—
130D4—6924no—
131D4—6922no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—3464no—
136C2×C4—3462no—
137C2×C4—6924no—
138C2×C4—6922no—
139C2×C4—6922no—
140C2×C4#1458380311yescharacteristic
141C2×C4#1458380311yescharacteristic
142C2×C4#1458380311yescharacteristic
143C2×C4#1458380311yescharacteristic
144C2×C4C2×C4×D17311yescharacteristic
145C2×C4C2×C22⋊D17311yescharacteristic
146C8—42no—
147C22—21no—
148C22—21no—
149C22—6922no—
150C22—6922no—
151C22—6922no—
152C22C2×#1458380311yescharacteristic
153C22#1914970712yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—6924no—
160C4—6922no—
161C4#1914960512yes—
162C2—22no—
163C2—13842no—
164C2#2601146411yescharacteristic
165C2#2601149711yescharacteristic
166C2C2×#1914960511yescharacteristic
subgroupG/H
centerC22C2×#14583803
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#432×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012041C2
cosocleC22×C4×C173C23
hypercenter#432D173
hypocenterC173#1505
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #26011337C2semidirect
3. #26011226C2semidirect
4. #432×C173C2semidirect
6. #19149679C4semidirect
7. #19149676C4semidirect
12. C4×D692C4semidirect
83. C173#1505semidirect
86. #432D173semidirect

Maximal subgroups

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

subgroupindex
#260120412
#260113372
#260112262
#432×C1732
#1505173