Finite Group Databasebeta

← #27277600

Subgroups — #27277600

Order 13568.

786 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274630C212yesmaximal
2#20272619C214yesmaximal, complemented
3C53×#2309C211yesmaximal, characteristic
4C2×#13133132—24no—
5#16810479C2214yes—
6C23×D4×C53C2211yescharacteristic
7C2×#117×C53C412yescomplemented
8C53×#390—24no—
9C53×#379C412yescomplemented
10C53×#379C2212yes—
11C2×C22⋊Dic53—24no—
12C2×C22⋊Dic53—28no—
13#13133132—48no—
14C25×C53C2×C411yescharacteristic
15C25×C53C2311yescharacteristic
16C22×D4×C53—24no—
17C22×D4×C53—44no—
18C22×D4×C53D414yes—
19C23×C4×C53C2×C411yescharacteristic
20#117×C53—44no—
21C2×C4⋊C4×C53—22no—
22C2×C22⋊C4×C53—24no—
23C2×C22⋊C4×C53—24no—
24C2×C22⋊C4×C53C2×C414yes—
25C2×C42×C53—22no—
26C24×C53—22no—
27C24×C53—42no—
28C24×C53—42no—
29C24×C53—82no—
30C24×C53—82no—
31C24×C53C22⋊C412yes—
32C24×C53C22×C411yescharacteristic
33C24×C53C2×D412yes—
34C24×C53#4114yes—
35C2×D4×C53—28no—
36C2×D4×C53—28no—
37C2×D4×C53—44no—
38C2×D4×C53—416no—
39C22×C4×C53—22no—
40C22×C4×C53—24no—
41C22×C4×C53—24no—
42C22×C4×C53C22⋊C412yes—
43C22×Dic53—44no—
44C4⋊C4×C53—44no—
45C22⋊C4×C53—44no—
46C22⋊C4×C53—84no—
47C22⋊C4×C53—84no—
48C42×C53—44no—
49C22⋊Dic53—48no—
50C22⋊Dic53—88no—
51C23×C53—22no—
52C23×C53—24no—
53C23×C53—22no—
54C23×C53—24no—
55C23×C53—42no—
56C23×C53—44no—
57C23×C53—44no—
58C23×C53—48no—
59C23×C53—44no—
60C23×C53—44no—
61C23×C53—84no—
62C23×C53—88no—
63C23×C53—88no—
64C23×C53C2×C22⋊C411yescharacteristic
65C23×C53#11712yes—
66C23×C53#12314yes—
67D4×C53—416no—
68C2×C4×C53—28no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—48no—
72C2×C4×C53—44no—
73C2×C4×C53#9914yes—
74C2×Dic53—84no—
75C2×Dic53—88no—
76#8749—531nomaximal, Sylow
77C22×C53—22no—
78C22×C53—24no—
79C22×C53—22no—
80C22×C53—28no—
81C22×C53—24no—
82C22×C53—44no—
83C22×C53—42no—
84C22×C53—44no—
85C22×C53—84no—
86C22×C53—84no—
87C22×C53—88no—
88C22×C53—88no—
89C22×C53#38111yescharacteristic
90C22×C53C2×#9912yes—
91C4×C53—24no—
92C4×C53—84no—
93Dic53—164no—
94#2309D5311yescharacteristic
95#1781—532no—
96#1304—534no—
97C2×C53—22no—
98C2×C53—44no—
99C2×C53—42no—
100C2×C53—82no—
101C2×C53#181911yescharacteristic
102C2×C53#182812yes—
103C23×D4C2×D5311yescharacteristic
104C2×#117Dic5312yescomplemented
105C2×#99—1064no—
106#390—24no—
107#379—534no—
108#379Dic5312yescomplemented
109#379C2×D5312yes—
110C53#874911yescharacteristic, complemented, Sylow
111C25C2×Dic5311yescharacteristic
112C25C22×D5311yescharacteristic
113C22×D4—24no—
114C22×D4—44no—
115C22×D4C22⋊D5314yes—
116C23×C4C2×Dic5311yescharacteristic
117#117—44no—
118C2×C4⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—1068no—
122C2×C22⋊C4—1064no—
123C2×C22⋊C4C2×Dic5314yes—
124C2×C42—22no—
125#99—2128no—
126C24—22no—
127C24—42no—
128C24—42no—
129C24—82no—
130C24—82no—
131C24C22⋊Dic5312yes—
132C24#1171994514yes—
133C24C22×Dic5311yescharacteristic
134C24C2×C22⋊D5312yes—
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—44no—
138C2×D4—416no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—2124no—
143C22×C4C22⋊Dic5312yes—
144C4⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—84no—
147C22⋊C4—84no—
148C22⋊C4—4248no—
149C22⋊C4—2128no—
150C42—44no—
151C23—22no—
152C23—24no—
153C23—22no—
154C23—24no—
155C23—42no—
156C23—44no—
157C23—48no—
158C23—44no—
159C23—44no—
160C23—44no—
161C23—84no—
162C23—88no—
163C23—88no—
164C23C2×C22⋊Dic5311yescharacteristic
165C23#1313317612yes—
166C23#1313320014yes—
167D4—416no—
168C2×C4—28no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—4248no—
174C2×C4—4244no—
175C2×C4#1313313214yes—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—28no—
180C22—24no—
181C22—44no—
182C22—42no—
183C22—44no—
184C22—84no—
185C22—84no—
186C22—88no—
187C22—88no—
188C22#1681048611yescharacteristic
189C22C2×#1313313212yes—
190C4—24no—
191C4—84no—
192C4—8484no—
193C2—22no—
194C2—44no—
195C2—42no—
196C2—82no—
197C2#2027478911yescharacteristic
198C2#2027482312yes—
subgroupG/H
centerC22#16810486
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#2309C2
socleC22×C53#381
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenter#2309D53
hypocenterC53#8749
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #20272619C2semidirect
7. C2×#117×C53C4semidirect
9. C53×#379C4semidirect
104. C2×#117Dic53semidirect
108. #379Dic53semidirect
110. C53#8749semidirect

Maximal subgroups

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

subgroupindex
#202746302
#202726192
C53×#23092
#874953