Finite Group Databasebeta

← #28298922

Subgroups — #28298922

Order 13568.

202 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289661C212yesmaximal, complemented
2#20275547C211yesmaximal, characteristic
3#16811519—22no—
4#16811173C411yescharacteristic
5#16810647C2211yescharacteristic
6C2×#13133145—22no—
7C53×#376C411yescharacteristic
8#13133291—42no—
9C2×#11719945—21no—
10C2×#11719917D412yes—
11#13133217C811yescharacteristic
12#13133214C811yescharacteristic
13#13133205—21no—
14#13133191—41no—
15C4×D212—41no—
16C4⋊#88322—41no—
17#13133145—44no—
18C2×C42×C53—21no—
19C2×C42×C53C2×C411yescharacteristic
20#95×C53—41no—
21C22×C4×C53—21no—
22C22×C4×C53—41no—
23C22×C4×C53C22⋊C411yescharacteristic
24C2×C22⋊D53—41no—
25#11719945—42no—
26#11719945—42no—
27C2×D212—21no—
28C2×D212—41no—
29C2×C4×D53—41no—
30C2×Dic106—21no—
31C42×C53—21no—
32C42×C53—22no—
33C42×C53—41no—
34C42×C53C2×C811yescharacteristic
35C42×C53M1611yescharacteristic
36#11719921—42no—
37C4⋊Dic53—41no—
38#11719917—42no—
39C2×#88322—22no—
40#11719910—82no—
41C23×C53#10011yescharacteristic
42C22×D53—41no—
43C2×C4×C53—21no—
44C2×C4×C53—21no—
45C2×C4×C53—21no—
46C2×C4×C53—21no—
47C2×C4×C53—42no—
48C2×C4×C53—81no—
49C2×C4×C53—82no—
50C2×C4×C53C22⋊C811yescharacteristic
51C2×C4×C53#9911yescharacteristic
52C22⋊D53—82no—
53C2×Dic53—41no—
54D212—42no—
55D212—82no—
56C4×D53—81no—
57C4×D53—81no—
58Dic106—42no—
59#88322—42no—
60#7356—531nomaximal, Sylow
61C22×C53—21no—
62C22×C53—41no—
63C22×C53#32311yescharacteristic
64C2×D53—82no—
65C2×D53—81no—
66C4×C53—21no—
67C4×C53—41no—
68C4×C53—41no—
69C4×C53—82no—
70C4×C53#35311yescharacteristic
71C4×C53#35511yescharacteristic
72Dic53—81no—
73#1963—531no—
74#1327—532no—
75C2×C53—21no—
76C2×C53—41no—
77C2×C53#131811yescharacteristic
78D53—161no—
79#532—531no—
80#423—531no—
81C2×#104—1062no—
82#376#615711yescharacteristic
83#349—1062no—
84C53#735611yescharacteristic, complemented, Sylow
85C2×#41—1061no—
86C2×M16—532no—
87C4⋊Q8—531no—
88#127—531no—
89#124—1061no—
90#121—2121no—
91C4×D4—2121no—
92C2×C42—21no—
93C2×C42C2×#615711yescharacteristic
94M32—2122no—
95C4⋊C8—2121no—
96#104—2124no—
97#95—41no—
98#41—2122no—
99#41—2122no—
100C2×Q8—1061no—
101C2×D4—1061no—
102C2×D4—2121no—
103C2×D4—2121no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—2121no—
107C22×C4C22⋊#615711yescharacteristic
108M16—2122no—
109C2×C8—1062no—
110C4⋊C4—2121no—
111C22⋊C4—2122no—
112C42—21no—
113C42—22no—
114C42—41no—
115C42D53⋊C811yescharacteristic
116C42#1171993611yescharacteristic
117C16—4242no—
118C23—2121no—
119C23#1313327711yescharacteristic
120Q8—2122no—
121D4—4242no—
122D4—4242no—
123D4—2122no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—81no—
130C2×C4—82no—
131C2×C4—4241no—
132C2×C4—4241no—
133C2×C4—2121no—
134C2×C4#1313327411yescharacteristic
135C2×C4#1313327611yescharacteristic
136C8—2122no—
137C22—21no—
138C22—41no—
139C22—4242no—
140C22—4241no—
141C22#1681146711yescharacteristic
142C4—21no—
143C4—41no—
144C4—41no—
145C4—82no—
146C4—4241no—
147C4#1681152811yescharacteristic
148C4#1681153211yescharacteristic
149C2—21no—
150C2—41no—
151C2—8481no—
152C2#2028964411yescharacteristic
subgroupG/H
centerC4#16811528
commutatorC42×C53C2×C8
FrattiniC2×C42C2×#6157
FittingC53×#376C4
socleC2×C53#1318
prime radical#16811173C4
cosocle#16810647C22
hypercenter#376#6157
hypocenterC53#7356
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
1. #20289661C2semidirect
84. C53#7356semidirect

Maximal subgroups

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

subgroupindex
#202896612
#202755472
#735653