Finite Group Databasebeta

← #27271465

Subgroups — #27271465

Order 13568.

302 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 43 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#20272643C212yesmaximal
2C53×#1819C211yesmaximal, characteristic
3#16810347—22no—
4C22×C22⋊C4×C53C411yescharacteristic
5C53×#379C411yescharacteristic
6C53×#379C2211yescharacteristic
7C53×#342—22no—
8C22⋊#88322—42no—
9C25×C53C2×C411yescharacteristic
10C23×C4×C53—21no—
11C2×C22⋊C4×C53—21no—
12C2×C22⋊C4×C53—21no—
13C2×C22⋊C4×C53—41no—
14C2×C22⋊C4×C53—41no—
15C2×C22⋊C4×C53—41no—
16C2×C22⋊C4×C53—41no—
17C2×C22⋊C4×C53C812yescomplemented
18C2×C22⋊C4×C53D412yes—
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—22no—
28C22×C4×C53—41no—
29C22×C4×C53—41no—
30C22×C4×C53—41no—
31C22×C4×C53—82no—
32C22⋊C4×C53—41no—
33C22⋊C4×C53—41no—
34C22⋊C4×C53—42no—
35C22⋊C4×C53—42no—
36C22⋊C4×C53—42no—
37C22⋊C4×C53—82no—
38C22⋊C4×C53—82no—
39C2×#88322—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—42no—
61C2×C4×C53—82no—
62C2×C4×C53—82no—
63C2×C4×C53—82no—
64C2×C4×C53—82no—
65C2×C4×C53—81no—
66C2×C4×C53—82no—
67#88322—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—
89C4×C53—82no—
90C4×C53—161no—
91#1819D5311yescharacteristic
92#1316—532no—
93C2×C53—21no—
94C2×C53—22no—
95C2×C53—41no—
96C2×C53—42no—
97C2×C53—41no—
98C2×C53—81no—
99C2×C53#131711yescharacteristic
100C22×C22⋊C4Dic5311yescharacteristic
101#379Dic5311yescharacteristic
102#379C2×D5311yescharacteristic
103#342—22no—
104#323—1062no—
105C53#734911yescharacteristic, complemented, Sylow
106C25C2×Dic5311yescharacteristic
107C23×C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4#8832212yescomplemented
115C2×C22⋊C4C22⋊D5312yes—
116C22⋊C8—2122no—
117C24—21no—
118C24—21no—
119C24—42no—
120C24—82no—
121C24C2×#8832211yescharacteristic
122C24#1171991711yescharacteristic
123C24C22⋊Dic5311yescharacteristic
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—82no—
130C2×C8—4242no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—82no—
137C22⋊C4—82no—
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⋊#8832211yescharacteristic
155C23#1313313211yescharacteristic
156C23#1313313411yescharacteristic
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—81no—
164C2×C4—82no—
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#1681034711yescharacteristic
184C22#1681041312yes—
185C4—82no—
186C4—82no—
187C4—161no—
188C2—21no—
189C2—22no—
190C2—41no—
191C2—42no—
192C2—41no—
193C2—81no—
194C2#2027264511yescharacteristic
subgroupG/H
centerC2#20272645
commutatorC24×C53C2×C8
Frattini#379C2×D53
FittingC53×#1819C2
socleC2×C53#1317
prime radicalC25×C53C2×C4
cosocleC53×#379C22
hypercenter#1819D53
hypocenterC53#7349
LCSG ▷ C24×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ #1819

Complements

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

NH ≅ G/Ntype
17. C2×C22⋊C4×C53C8semidirect
105. C53#7349semidirect
114. C2×C22⋊C4#88322semidirect

Maximal subgroups

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

subgroupindex
#202726432
C53×#18192
#734953