Finite Group Databasebeta

← #27277604

Subgroups — #27277604

Order 13568.

538 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 99 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#20274789C212yesmaximal
2#20272619C214yesmaximal
3C53×#2312C211yesmaximal, characteristic
4#16810479C2214yes—
5#16810390—24no—
6C22×C22⋊C4×C53C2211yescharacteristic
7C53×#388—24no—
8C53×#380C412yescomplemented
9C53×#379C2212yes—
10C4×C22⋊C4×C53C412yescomplemented
11C2×C22⋊Dic53—24no—
12C2×C22⋊Dic53—24no—
13C2×C22⋊Dic53—24no—
14C25×C53C2311yescharacteristic
15C23×C4×C53C2×C411yescharacteristic
16C23×C4×C53C2×C411yescharacteristic
17C2×C4⋊C4×C53—22no—
18C2×C22⋊C4×C53—22no—
19C2×C22⋊C4×C53—22no—
20C2×C22⋊C4×C53—24no—
21C2×C22⋊C4×C53C2×C414yes—
22C2×C22⋊C4×C53D414yes—
23C2×C42×C53—22no—
24#95×C53—22no—
25#95×C53—22no—
26C24×C53—42no—
27C24×C53—81no—
28C24×C53—81no—
29C24×C53C22×C411yescharacteristic
30C24×C53C2×D412yes—
31C24×C53#4114yes—
32C22×C4×C53—22no—
33C22×C4×C53—22no—
34C22×C4×C53—24no—
35C22×C4×C53—81no—
36C22×C4×C53—81no—
37C22×C4×C53C22⋊C412yes—
38C22×C4×C53C22⋊C412yes—
39C22×Dic53—44no—
40C4⋊C4×C53—82no—
41C22⋊C4×C53—28no—
42C22⋊C4×C53—44no—
43C22⋊C4×C53—82no—
44C22⋊C4×C53—82no—
45C42×C53—82no—
46C22⋊Dic53—48no—
47C22⋊Dic53—84no—
48C22⋊Dic53—84no—
49C23×C53—22no—
50C23×C53—24no—
51C23×C53—41no—
52C23×C53—41no—
53C23×C53—44no—
54C23×C53—42no—
55C23×C53—42no—
56C23×C53—44no—
57C23×C53—44no—
58C23×C53—88no—
59C23×C53C2×C22⋊C411yescharacteristic
60C23×C53#11712yes—
61C23×C53#12314yes—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53—44no—
67C2×C4×C53—44no—
68C2×C4×C53—44no—
69C2×C4×C53—82no—
70C2×C4×C53—82no—
71C2×Dic53—84no—
72C2×Dic53—84no—
73C2×Dic53—84no—
74#8750—531nomaximal, Sylow
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—24no—
78C22×C53—22no—
79C22×C53—24no—
80C22×C53—24no—
81C22×C53—22no—
82C22×C53—22no—
83C22×C53—44no—
84C22×C53—84no—
85C22×C53—88no—
86C22×C53#38111yescharacteristic
87C22×C53#41012yes—
88C4×C53—42no—
89C4×C53—42no—
90C4×C53—84no—
91Dic53—164no—
92#2312D5311yescharacteristic
93#1819—532no—
94#1304—534no—
95C2×C53—22no—
96C2×C53—44no—
97C2×C53—42no—
98C2×C53#182011yescharacteristic
99C2×C53#182811yescharacteristic
100C2×C53#182811yescharacteristic
101C22×C22⋊C4C2×D5311yescharacteristic
102#388—24no—
103#380Dic5312yescomplemented
104#379—534no—
105#379C2×D5312yes—
106C4×C22⋊C4Dic5312yescomplemented
107#342—1064no—
108C53#875011yescharacteristic, complemented, Sylow
109C25C22×D5311yescharacteristic
110C23×C4C2×Dic5311yescharacteristic
111C23×C4C2×Dic5311yescharacteristic
112C2×C4⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—1064no—
117C2×C22⋊C4—1064no—
118C2×C22⋊C4—1064no—
119C2×C22⋊C4C2×Dic5314yes—
120C2×C22⋊C4C22⋊D5314yes—
121C2×C42—22no—
122#95—22no—
123#95—22no—
124C24—42no—
125C24—81no—
126C24—81no—
127C24#1171994514yes—
128C24C22×Dic5311yescharacteristic
129C24C2×C22⋊D5312yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4—2124no—
136C22×C4C22⋊Dic5312yes—
137C22×C4C22⋊Dic5312yes—
138C4⋊C4—82no—
139C22⋊C4—28no—
140C22⋊C4—44no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—4244no—
144C22⋊C4—4244no—
145C22⋊C4—2128no—
146C42—82no—
147C23—22no—
148C23—24no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23—88no—
157C23C2×C22⋊Dic5311yescharacteristic
158C23#1313317612yes—
159C23#1313320014yes—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—4244no—
170C2×C4—4244no—
171C2×C4—4244no—
172C22—21no—
173C22—21no—
174C22—24no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—24no—
180C22—44no—
181C22—84no—
182C22—88no—
183C22#1681048611yescharacteristic
184C22#1681059712yes—
185C4—42no—
186C4—42no—
187C4—84no—
188C4—8484no—
189C2—22no—
190C2—44no—
191C2—42no—
192C2#2027479211yescharacteristic
193C2#2027482311yescharacteristic
194C2#2027482311yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#2312C2
socleC22×C53#381
prime radical, cosocleC25×C53C23
hypercenter#2312D53
hypocenterC53#8750
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #2312

Complements

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

NH ≅ G/Ntype
8. C53×#380C4semidirect
10. C4×C22⋊C4×C53C4semidirect
103. #380Dic53semidirect
106. C4×C22⋊C4Dic53semidirect
108. C53#8750semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202726192
C53×#23122
#875053