Finite Group Databasebeta

← #28298905

Subgroups — #28298905

Order 13568.

302 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 38 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#20289639C212yesmaximal
2#20274789C211yesmaximal, characteristic
3#16811466—22no—
4#16810479C411yescharacteristic
5#16810479C2211yescharacteristic
6#16810390—22no—
7C22×C22⋊C4×C53C411yescharacteristic
8C22⋊#88324—42no—
9C2×C22⋊Dic53—21no—
10C2×C22⋊Dic53—41no—
11C2×C22⋊Dic53—41no—
12C2×C22⋊Dic53D412yes—
13C25×C53C2×C411yescharacteristic
14C23×C4×C53—21no—
15C2×C22⋊C4×C53—21no—
16C2×C22⋊C4×C53—41no—
17C2×C22⋊C4×C53—41no—
18C2×C22⋊C4×C53C812yescomplemented
19C24×C53—21no—
20C24×C53—21no—
21C24×C53—42no—
22C24×C53—82no—
23C24×C53C22⋊C411yescharacteristic
24C24×C53C2×C811yescharacteristic
25C24×C53M1611yescharacteristic
26C22×C4×C53—22no—
27C22×C4×C53—41no—
28C22×C4×C53—41no—
29C22×C4×C53—82no—
30C22×Dic53—22no—
31C22×Dic53—41no—
32C2×#88324—82no—
33C22⋊C4×C53—41no—
34C22⋊C4×C53—41no—
35C22⋊C4×C53—42no—
36C22⋊Dic53—42no—
37C22⋊Dic53—42no—
38C22⋊Dic53—82no—
39C22⋊Dic53—82no—
40C23×C53—21no—
41C23×C53—22no—
42C23×C53—21no—
43C23×C53—42no—
44C23×C53—41no—
45C23×C53—42no—
46C23×C53—42no—
47C23×C53—42no—
48C23×C53—41no—
49C23×C53—42no—
50C23×C53—81no—
51C23×C53—81no—
52C23×C53—81no—
53C23×C53—81no—
54C23×C53—84no—
55C23×C53—84no—
56C23×C53C22⋊C811yescharacteristic
57C23×C53#9911yescharacteristic
58C23×C53#10011yescharacteristic
59C2×C4×C53—42no—
60C2×C4×C53—82no—
61C2×C4×C53—82no—
62C2×C4×C53—82no—
63C2×Dic53—42no—
64C2×Dic53—82no—
65C2×Dic53—81no—
66C2×Dic53—82no—
67#88324—162no—
68#7349—531nomaximal, Sylow
69C22×C53—22no—
70C22×C53—21no—
71C22×C53—21no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—42no—
79C22×C53—42no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53—82no—
83C22×C53—84no—
84C22×C53—84no—
85C22×C53—82no—
86C22×C53#32311yescharacteristic
87C22×C53#35212yes—
88C4×C53—82no—
89Dic53—82no—
90Dic53—161no—
91#1819—531no—
92#1316—532no—
93C2×C53—22no—
94C2×C53—21no—
95C2×C53—41no—
96C2×C53—42no—
97C2×C53—41no—
98C2×C53—81no—
99C2×C53#131711yescharacteristic
100C22×C22⋊C4#615711yescharacteristic
101#379—531no—
102#379—531no—
103#342—1062no—
104#323—1062no—
105C53#734911yescharacteristic, complemented, Sylow
106C25C2×#615711yescharacteristic
107C23×C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—532no—
112C2×C22⋊C4—1061no—
113C2×C22⋊C4—2121no—
114C2×C22⋊C4—2121no—
115C2×C22⋊C4#8832412yescomplemented
116C22⋊C8—2122no—
117C24—21no—
118C24—21no—
119C24—42no—
120C24—82no—
121C24C2×#8832411yescharacteristic
122C24#1171994011yescharacteristic
123C24C22⋊#615711yescharacteristic
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—82no—
128C22×C4—1062no—
129C22×C4—2121no—
130C2×C8—4242no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—42no—
134C22⋊C4—4242no—
135C22⋊C4—4242no—
136C22⋊C4—2122no—
137C22⋊C4—2122no—
138C23—21no—
139C23—22no—
140C23—21no—
141C23—42no—
142C23—41no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—41no—
147C23—42no—
148C23—81no—
149C23—81no—
150C23—81no—
151C23—81no—
152C23—84no—
153C23—84no—
154C23C22⋊#8832411yescharacteristic
155C23#1313327511yescharacteristic
156C23#1313327711yescharacteristic
157C2×C4—42no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—4241no—
162C2×C4—4242no—
163C2×C4—4242no—
164C2×C4—2122no—
165C8—8482no—
166C22—22no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—82no—
180C22—84no—
181C22—84no—
182C22—82no—
183C22#1681146611yescharacteristic
184C22#1681152512yes—
185C4—82no—
186C4—8481no—
187C4—4242no—
188C2—21no—
189C2—22no—
190C2—41no—
191C2—42no—
192C2—41no—
193C2—81no—
194C2#2028964111yescharacteristic
subgroupG/H
centerC2#20289641
commutatorC24×C53C2×C8
FrattiniC25C2×#6157
FittingC22×C22⋊C4×C53C4
socleC2×C53#1317
prime radicalC25×C53C2×C4
cosocle#16810479C22
hypercenterC22×C22⋊C4#6157
hypocenterC53#7349
LCSG ▷ C24×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
18. C2×C22⋊C4×C53C8semidirect
105. C53#7349semidirect
115. C2×C22⋊C4#88324semidirect

Maximal subgroups

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

subgroupindex
#202896392
#202747892
#734953