Finite Group Databasebeta

← #28299283

Subgroups — #28299283

Order 13568.

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

normal subgroups only

This group has 286 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#20291008C212yesmaximal
2#20275542C211yesmaximal, characteristic
3#16811609—22no—
4#16811534—22no—
5#16811532C414yescomplemented
6#16811173C2211yescharacteristic
7C2×#13133143—22no—
8C42×Dic53C411yescharacteristic
9C53×#423C411yescharacteristic
10D53⋊M16—42no—
11#13133278—24no—
12#13133277—42no—
13C2×#11719945D412yes—
14#13133217D411yescharacteristic
15#13133214Q811yescharacteristic
16#13133205C2×C412yes—
17#13133191—41no—
18C4×D212—41no—
19C2×C4×Dic53—21no—
20C2×C4×Dic53—41no—
21#13133143—44no—
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×D212C4⋊C412yes—
33C2×C4×D53—22no—
34C2×Dic106C22⋊C412yes—
35#11719940—44no—
36#11719936—82no—
37D53⋊C8—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—
47C4×Dic53—44no—
48C23×C53#9911yescharacteristic
49C22×D53—22no—
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—42no—
55C2×C4×C53#9511yescharacteristic
56C2×C4×C53#9911yescharacteristic
57C22⋊D53—82no—
58C2×Dic53—22no—
59C2×Dic53—42no—
60C2×Dic53—44no—
61D212—42no—
62D212—82no—
63C4×D53—42no—
64C4×D53—42no—
65Dic106—42no—
66#88324—84no—
67C8×C53—81no—
68#7501—531nomaximal, Sylow
69C22×C53—21no—
70C22×C53—41no—
71C22×C53#34211yescharacteristic
72C2×D53—42no—
73C2×D53—82no—
74C4×C53—22no—
75C4×C53—21no—
76C4×C53—41no—
77C4×C53#35311yescharacteristic
78C4×C53#35411yescharacteristic
79Dic53—44no—
80Dic53—42no—
81#2132—532no—
82#1962—531no—
83C2×C53—21no—
84C2×C53—41no—
85C2×C53#139311yescharacteristic
86D53—82no—
87#532—531no—
88#423#615711yescharacteristic
89C2×#104—1062no—
90#413—1062no—
91C43—531no—
92#356—1062no—
93#355—534no—
94C53#750111yescharacteristic, complemented, Sylow
95C2×#41—532no—
96C2×M16—21no—
97C2×M16—2122no—
98C4⋊Q8—531no—
99#127—531no—
100#124—532no—
101#121—2121no—
102C4×D4—2121no—
103C2×C42—1061no—
104C2×C42—2121no—
105C2×C42C2×#615711yescharacteristic
106C4⋊C8—41no—
107#104—2124no—
108#101—1064no—
109#100—2122no—
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—
121M16—4242no—
122M16—2124no—
123C2×C8—41no—
124C2×C8—4242no—
125C4⋊C4—2121no—
126C22⋊C4—2122no—
127C42—21no—
128C42—1062no—
129C42—2121no—
130C42—2124no—
131C42C4×#615711yescharacteristic
132C42C4⋊#615711yescharacteristic
133C23—1062no—
134C23#1313327611yescharacteristic
135Q8—2122no—
136D4—4242no—
137D4—4242no—
138D4—2122no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—1062no—
145C2×C4—2122no—
146C2×C4—2124no—
147C2×C4—2122no—
148C2×C4—2122no—
149C2×C4(C2×#6157)⋊C411yescharacteristic
150C2×C4#1313327511yescharacteristic
151C8—81no—
152C8—4244no—
153C22—21no—
154C22—41no—
155C22—4242no—
156C22—2122no—
157C22#1681150111yescharacteristic
158C4—22no—
159C4—21no—
160C4—41no—
161C4—2124no—
162C4—2122no—
163C4#1681152711yescharacteristic
164C4#1681152911yescharacteristic
165C2—21no—
166C2—41no—
167C2—4242no—
168C2#2028981011yescharacteristic
subgroupG/H
centerC4#16811529
commutatorC42×C53C42
FrattiniC2×C42C2×#6157
FittingC53×#423C4
socleC2×C53#1393
prime radical, cosocle#16811173C22
hypercenter#423#6157
hypocenterC53#7501
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
5. #16811532C4semidirect
94. C53#7501semidirect

Maximal subgroups

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

subgroupindex
#202910082
#202755422
#750153