Finite Group Databasebeta

← #28299300

Subgroups — #28299300

Order 13568.

274 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 46 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#20290988C212yesmaximal
2#20275546C211yesmaximal, characteristic
3#16811602—22no—
4#16811530C414yescomplemented
5#16811528—22no—
6#16811173C2211yescharacteristic
7C2×#13133143—22no—
8#16810469C411yescharacteristic, complemented
9C53×#423C411yescharacteristic, complemented
10#13133296—42no—
11#13133276—24no—
12#13133275—42no—
13C2×#11719945D412yes—
14#13133217Q811yescharacteristic
15#13133214D411yescharacteristic
16#13133205C2×C412yes—
17#13133191—41no—
18C4×D212—41no—
19C2×C4×Dic53—21no—
20#13133143—44no—
21#13133124—41no—
22C2×M16×C53—21no—
23C2×C42×C53C2×C411yescharacteristic
24C4⋊C8×C53—41no—
25C22×C4×C53—21no—
26C22×C4×C53C22⋊C411yescharacteristic
27C2×C22⋊D53—22no—
28C22×Dic53—41no—
29#11719945—42no—
30#11719945—42no—
31C2×D212—41no—
32C2×D212C22⋊C412yes—
33C2×C4×D53—22no—
34C2×Dic106C4⋊C412yes—
35C22⋊#6157—44no—
36C4⋊#6157—82no—
37C4×#6157—82no—
38M16×C53—42no—
39C2×C8×C53—41no—
40C42×C53—21no—
41C42×C53C4211yescharacteristic
42C42×C53C4⋊C411yescharacteristic
43#11719921—42no—
44C4⋊Dic53—41no—
45C4×Dic53—22no—
46C4×Dic53—41no—
47C23×C53#9911yescharacteristic
48C22×D53—22no—
49C2×#6157—84no—
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—81no—
55C2×C4×C53#9511yescharacteristic
56C2×C4×C53#9911yescharacteristic
57C22⋊D53—82no—
58C2×Dic53—22no—
59C2×Dic53—42no—
60C2×Dic53—82no—
61D212—42no—
62D212—82no—
63C4×D53—42no—
64C4×D53—42no—
65Dic106—42no—
66C8×C53—81no—
67#7506—531nomaximal, Sylow
68C22×C53—21no—
69C22×C53—41no—
70C22×C53#34211yescharacteristic
71C2×D53—42no—
72C2×D53—82no—
73#6157—164no—
74C4×C53—21no—
75C4×C53—41no—
76C4×C53—41no—
77C4×C53#35311yescharacteristic
78C4×C53#35411yescharacteristic
79Dic53—42no—
80Dic53—82no—
81#2126—532no—
82#1963—531no—
83C2×C53—21no—
84C2×C53—41no—
85C2×C53#139311yescharacteristic
86D53—82no—
87#532—531no—
88#423#615711yescharacteristic, complemented
89C2×#104—1062no—
90#410—1062no—
91#376—531no—
92#354—534no—
93#353—1062no—
94C53#750611yescharacteristic, complemented, Sylow
95C2×#41—532no—
96C2×M16—21no—
97C4⋊Q8—531no—
98#127—531no—
99#124—532no—
100#121—2121no—
101C4×D4—2121no—
102#117—2122no—
103C2×C42—1061no—
104C2×C42C2×#615711yescharacteristic
105C4⋊C8—41no—
106#104—2124no—
107#99—1064no—
108#99—2122no—
109#95—2121no—
110#41—2122no—
111#41—2122no—
112C2×Q8—532no—
113C2×D4—532no—
114C2×D4—1062no—
115C2×D4—2121no—
116C22×C4—21no—
117C22×C4—1062no—
118C22×C4—2121no—
119C22×C4C22⋊#615711yescharacteristic
120M16—42no—
121C2×C8—41no—
122C4⋊C4—4242no—
123C4⋊C4—2121no—
124C22⋊C4—2122no—
125C22⋊C4—2124no—
126C42—21no—
127C42—1062no—
128C42—4242no—
129C42—2121no—
130C42C4×#615711yescharacteristic
131C42C4⋊#615711yescharacteristic
132C23—1062no—
133C23#1313327611yescharacteristic
134Q8—2122no—
135D4—4242no—
136D4—4242no—
137D4—2122no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—81no—
143C2×C4—1062no—
144C2×C4—4244no—
145C2×C4—4242no—
146C2×C4—2122no—
147C2×C4—2122no—
148C2×C4—2122no—
149C2×C4(C2×#6157)⋊C411yescharacteristic
150C2×C4#1313327511yescharacteristic
151C8—81no—
152C22—21no—
153C22—41no—
154C22—4242no—
155C22—2122no—
156C22#1681150111yescharacteristic
157C4—21no—
158C4—41no—
159C4—41no—
160C4—8484no—
161C4—4242no—
162C4—2122no—
163C4#1681152711yescharacteristic
164C4#1681152911yescharacteristic
165C2—21no—
166C2—41no—
167C2—4242no—
168C2#2028981011yescharacteristic
subgroupG/H
centerC4#16811527
commutatorC42×C53C42
FrattiniC2×C42C2×#6157
FittingC53×#423C4
socleC2×C53#1393
prime radical, cosocle#16811173C22
hypercenter#423#6157
hypocenterC53#7506
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
4. #16811530C4semidirect
8. #16810469C4semidirect
9. C53×#423C4semidirect
88. #423#6157semidirect
94. C53#7506semidirect

Maximal subgroups

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

subgroupindex
#202909882
#202755462
#750653