Finite Group Databasebeta

← #28303416

Subgroups — #28303416

Order 13568.

384 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289968C212yesmaximal
2#20289566C214yesmaximal, complemented
3#20277212C211yesmaximal, characteristic
4#16811643C2214yes—
5#16811469—22no—
6#16811467—22no—
7C2×#13133174C2211yescharacteristic
8#16810514—22no—
9#16810507—22no—
10C4×#11719919C2212yes—
11C4×#11719921C412yes—
12#16810469C411yescharacteristic
13C43×C53C411yescharacteristic
14#13133284—24no—
15C22⋊#88324—24no—
16#13133272—24no—
17C22×C4×D53C2×C411yescharacteristic
18#13133174—24no—
19C2×#11719919—22no—
20C2×#11719919D412yes—
21C2×#11719921—22no—
22C2×#11719921D412yes—
23C2×C4×Dic53C2×C412yes—
24C2×C4×Dic53C2311yescharacteristic
25#13133124—22no—
26#13133124—22no—
27#13133124—22no—
28C2×C42×C53—22no—
29C2×C42×C53C2×C412yes—
30C2×C42×C53C2×C411yescharacteristic
31C23×D53C22⋊C411yescharacteristic
32C22×C4×C53—22no—
33C22×C4×C53C22⋊C412yes—
34C22×C4×C53C22×C411yescharacteristic
35C22×Dic53—22no—
36C22×Dic53C22⋊C411yescharacteristic
37C22×Dic53C2×D412yes—
38C2×C4×D53—22no—
39C2×C4×D53—42no—
40C2×#88324—44no—
41C42×C53—22no—
42C42×C53—44no—
43C42×C53—42no—
44#11719921—24no—
45#11719921—44no—
46#11719919—24no—
47#11719919—44no—
48C4×Dic53—42no—
49C4×Dic53#4114yes—
50C23×C53C2×C22⋊C411yescharacteristic
51C22×D53—21no—
52C22×D53—22no—
53C22×D53—42no—
54C2×C4×C53—22no—
55C2×C4×C53—24no—
56C2×C4×C53—22no—
57C2×C4×C53—42no—
58C2×C4×C53—42no—
59C2×C4×C53—42no—
60C2×C4×C53#11712yes—
61C2×Dic53—21no—
62C2×Dic53—22no—
63C2×Dic53—42no—
64C2×Dic53—42no—
65C2×Dic53—42no—
66C2×Dic53—42no—
67C2×Dic53#12314yes—
68C4×D53—44no—
69#88324—84no—
70#8744—531nomaximal, Sylow
71C22×C53—22no—
72C22×C53#38111yescharacteristic
73C22×C53#41312yes—
74C2×D53—41no—
75C2×D53—42no—
76C2×D53—82no—
77C4×C53—22no—
78C4×C53—44no—
79C4×C53—42no—
80Dic53—24no—
81Dic53—41no—
82Dic53—82no—
83#2298—531no—
84#1464—532no—
85#1285—534no—
86C2×C53—22no—
87C2×C53#182211yescharacteristic
88C2×C53#182712yes—
89D53—81no—
90C2×#117—531no—
91#432—534no—
92#388—1062no—
93#387—1062no—
94C4×C4⋊C4—532no—
95C4×C22⋊C4—532no—
96#376—531no—
97C43#615711yescharacteristic
98#324—1062no—
99#323—1062no—
100C53#874411yescharacteristic, complemented, Sylow
101C23×C4—531no—
102#117—1064no—
103C2×C4⋊C4—532no—
104C2×C4⋊C4—1062no—
105C2×C22⋊C4—532no—
106C2×C22⋊C4—1062no—
107C2×C42—22no—
108C2×C42—531no—
109C2×C42—532no—
110C2×C42C2×#615711yescharacteristic
111C2×C42C2×#615712yes—
112C4⋊C8—1064no—
113C22⋊C8—1064no—
114#97—1064no—
115#95—1062no—
116#95—1062no—
117#95—1062no—
118C24—531no—
119C22×C4—22no—
120C22×C4—532no—
121C22×C4—531no—
122C22×C4—1062no—
123C22×C4—1062no—
124C22×C4—2122no—
125C22×C4C22⋊#615712yes—
126C22×C4C22×#615711yescharacteristic
127C2×C8—2124no—
128C4⋊C4—1064no—
129C4⋊C4—2124no—
130C22⋊C4—1064no—
131C22⋊C4—2124no—
132C42—22no—
133C42—42no—
134C42—44no—
135C42—534no—
136C42—2122no—
137C23—1062no—
138C23—1061no—
139C23—2122no—
140C23C2×C22⋊#615711yescharacteristic
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—534no—
148C2×C4—1061no—
149C2×C4—1062no—
150C2×C4—2124no—
151C2×C4—2122no—
152C2×C4—2122no—
153C2×C4—2122no—
154C2×C4—2122no—
155C2×C4#1313329612yes—
156C8—4244no—
157C22—22no—
158C22—4242no—
159C22—2121no—
160C22—2122no—
161C22#1681156611yescharacteristic
162C22#1681160912yes—
163C4—22no—
164C4—44no—
165C4—42no—
166C4—1064no—
167C4—4242no—
168C4—2121no—
169C2—22no—
170C2—4241no—
171C2#2029061611yescharacteristic
172C2#2029062412yes—
subgroupG/H
centerC22#16811566
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC43×C53C4
socleC22×C53#381
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC43#6157
hypocenterC53#8744
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C43

Complements

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

NH ≅ G/Ntype
2. #20289566C2semidirect
100. C53#8744semidirect

Maximal subgroups

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

subgroupindex
#202899682
#202895662
#202772122
#874453