Finite Group Databasebeta

← #28298920

Subgroups — #28298920

Order 13568.

216 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 40 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#20289657C212yesmaximal
2#20275517C211yesmaximal, characteristic
3#16811468—22no—
4C2×#13133217C411yescharacteristic
5C2×#13133142—22no—
6#16810493C2211yescharacteristic
7C2×C4×C8×C53C411yescharacteristic
8C22⋊#88324—42no—
9C22×Dic106—21no—
10#13133217—21no—
11#13133217—41no—
12#13133217C812yescomplemented
13C2×C4⋊Dic53—41no—
14C2×C4⋊Dic53—41no—
15C2×C4⋊Dic53D412yes—
16#13133142—44no—
17C22×C8×C53—21no—
18C2×C42×C53C2×C411yescharacteristic
19C4×C8×C53—41no—
20C22×C4×C53—21no—
21C22×C4×C53C22⋊C411yescharacteristic
22C22×Dic53—22no—
23C22×Dic53—41no—
24C2×Dic106—22no—
25C2×Dic106—42no—
26C2×Dic106—42no—
27C2×Dic106—41no—
28C2×#88324—82no—
29C2×C8×C53—22no—
30C2×C8×C53—41no—
31C2×C8×C53—41no—
32C42×C53—21no—
33C42×C53C2×C811yescharacteristic
34C42×C53M1611yescharacteristic
35C4⋊Dic53—42no—
36C4⋊Dic53—42no—
37C4⋊Dic53—42no—
38C4⋊Dic53—82no—
39C23×C53#9911yescharacteristic
40C2×C4×C53—22no—
41C2×C4×C53—22no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53—21no—
45C2×C4×C53—21no—
46C2×C4×C53C22⋊C811yescharacteristic
47C2×C4×C53#10011yescharacteristic
48C2×Dic53—42no—
49C2×Dic53—42no—
50C2×Dic53—81no—
51C2×Dic53—81no—
52C2×Dic53—82no—
53Dic106—44no—
54Dic106—82no—
55#88324—162no—
56C8×C53—42no—
57#7355—531nomaximal, Sylow
58C22×C53—21no—
59C22×C53—21no—
60C22×C53#32311yescharacteristic
61C22×C53#35311yescharacteristic
62C22×C53#35511yescharacteristic
63C4×C53—22no—
64C4×C53—22no—
65C4×C53—21no—
66C4×C53—21no—
67Dic53—82no—
68Dic53—82no—
69#1956—531no—
70#1325—532no—
71C2×C53—21no—
72C2×C53—21no—
73C2×C53#131811yescharacteristic
74C2×C53#140911yescharacteristic
75C2×C53#141111yescharacteristic
76C2×C4⋊Q8—531no—
77C2×#103—1062no—
78C2×C4×C8#615711yescharacteristic
79#384—531no—
80#324—1062no—
81C53#735511yescharacteristic, complemented, Sylow
82C22×Q8—1061no—
83C22×C8—21no—
84C4⋊Q8—532no—
85C4⋊Q8—1061no—
86C4⋊Q8—2121no—
87C2×C4⋊C4—532no—
88C2×C4⋊C4—2121no—
89C2×C4⋊C4—2121no—
90C2×C42C2×#615711yescharacteristic
91#103—2124no—
92C22⋊C8—2122no—
93C4×C8—41no—
94C2×Q8—1062no—
95C2×Q8—2122no—
96C2×Q8—2122no—
97C2×Q8—2121no—
98C22×C4—21no—
99C22×C4—1062no—
100C22×C4—2121no—
101C22×C4C22⋊#615711yescharacteristic
102C2×C8—22no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—4242no—
106C4⋊C4—4242no—
107C4⋊C4—2122no—
108C4⋊C4—2122no—
109C4⋊C4—2122no—
110C42—21no—
111C42D53⋊C811yescharacteristic
112C42#1171993611yescharacteristic
113C23#1313327611yescharacteristic
114Q8—4242no—
115Q8—2124no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—4241no—
123C2×C4—4241no—
124C2×C4—4242no—
125C2×C4—2122no—
126C2×C4—2122no—
127C2×C4#1313327411yescharacteristic
128C2×C4#1313327711yescharacteristic
129C8—42no—
130C8—8482no—
131C22—21no—
132C22—21no—
133C22#1681146711yescharacteristic
134C22#1681152811yescharacteristic
135C22#1681153211yescharacteristic
136C4—22no—
137C4—22no—
138C4—21no—
139C4—21no—
140C4—4242no—
141C4—4242no—
142C2—21no—
143C2—21no—
144C2#2028964411yescharacteristic
145C2#2028984911yescharacteristic
146C2#2028985311yescharacteristic
subgroupG/H
centerC22#16811528
commutatorC42×C53C2×C8
FrattiniC2×C42C2×#6157
FittingC2×C4×C8×C53C4
socleC22×C53#353
prime radicalC23×C53#99
cosocle#16810493C22
hypercenterC2×C4×C8#6157
hypocenterC53#7355
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
12. #13133217C8semidirect
81. C53#7355semidirect

Maximal subgroups

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

subgroupindex
#202896572
#202755172
#735553