Finite Group Databasebeta

← #27393545

Subgroups — #27393545

Order 13568.

1130 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 315 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#20277313C212yesmaximal
2#20275701C218yesmaximal, complemented
3#20274789C214yesmaximal
4C2×C53×#543C211yesmaximal, characteristic
5#16810516—28no—
6#16810483C2214yes—
7#16810479C2214yes—
8C4×C22⋊Dic53C2214yes—
9#16810390C22116yes—
10C53×#543—28no—
11C2×C22⋊Q8×C53C418yescomplemented
12C2×C22⋊Q8×C53C2214yes—
13C22×C22⋊C4×C53C2212yes—
14C22×C22⋊C4×C53C2211yescharacteristic
15C2×C4⋊Dic53—24no—
16C2×C22⋊Dic53—24no—
17C2×C22⋊Dic53—24no—
18C2×C22⋊Dic53C2318yes—
19C2×C4×Dic53—24no—
20#13133124—24no—
21#13133124—24no—
22C25×C53C2311yescharacteristic
23C22×Q8×C53C2×C414yes—
24C23×C4×C53C2×C414yes—
25C23×C4×C53C2312yes—
26C22⋊Q8×C53—232no—
27C22⋊Q8×C53—28no—
28C22⋊Q8×C53—28no—
29C2×C4⋊C4×C53C2×C4112yes—
30C2×C22⋊C4×C53—28no—
31C2×C22⋊C4×C53C2×C414yes—
32C2×C22⋊C4×C53C2×C414yes—
33C2×C22⋊C4×C53D418yes—
34C2×C22⋊C4×C53C2314yes—
35C24×C53—24no—
36C24×C53—82no—
37C24×C53C22×C412yes—
38C24×C53C2×D414yes—
39C24×C53C2411yescharacteristic
40C2×Q8×C53—28no—
41C2×Q8×C53—84no—
42C22×C4×C53—28no—
43C22×C4×C53—82no—
44C22×C4×C53—84no—
45C22×C4×C53C22×C418yes—
46C22×C4×C53C22×C414yes—
47C22×C4×C53C2×D414yes—
48C22×C4×C53C2×D414yes—
49C22×Dic53—28no—
50C4⋊C4×C53—224no—
51C22⋊C4×C53—28no—
52C22⋊C4×C53—28no—
53C22⋊C4×C53C22⋊C4116yes—
54C22⋊Dic53—216no—
55C22⋊Dic53—84no—
56C22⋊Dic53—84no—
57C4⋊Dic53—84no—
58C4×Dic53—84no—
59C23×C53—24no—
60C23×C53—28no—
61C23×C53—44no—
62C23×C53—42no—
63C23×C53—44no—
64C23×C53—44no—
65C23×C53—88no—
66C23×C53C2×C22⋊C414yes—
67C23×C53C23×C411yescharacteristic
68C23×C53C22×D412yes—
69C23×C532−1+414yes—
70Q8×C53—88no—
71C2×C4×C53—28no—
72C2×C4×C53—44no—
73C2×C4×C53—44no—
74C2×C4×C53—44no—
75C2×C4×C53—48no—
76C2×C4×C53—88no—
77C2×C4×C53C2×C22⋊C418yes—
78C2×Dic53—48no—
79C2×Dic53—48no—
80C2×Dic53—48no—
81#21827—531nomaximal, Sylow
82C22×C53—22no—
83C22×C53—22no—
84C22×C53—22no—
85C22×C53—24no—
86C22×C53—28no—
87C22×C53—44no—
88C22×C53—48no—
89C22×C53—88no—
90C22×C53C22×C22⋊C411yescharacteristic
91C22×C53#51912yes—
92C22×C53#53614yes—
93C4×C53—28no—
94C4×C53—48no—
95Dic53—88no—
96C2×#543D5311yescharacteristic
97#2312—532no—
98#1996—538no—
99#1819—534no—
100C2×C53—24no—
101C2×C53—22no—
102C2×C53—44no—
103C2×C53#231011yescharacteristic
104C2×C53#288312yes—
105#543—28no—
106C2×C22⋊Q8Dic5318yescomplemented
107C2×C22⋊Q8C2×D5314yes—
108C22×C22⋊C4C2×D5312yes—
109C22×C22⋊C4C2×D5311yescharacteristic
110#388—1068no—
111#380—534no—
112#379—534no—
113C4×C22⋊C4—534no—
114#342—5316no—
115C53#2182711yescharacteristic, complemented, Sylow
116C25C22×D5311yescharacteristic
117C22×Q8C2×Dic5314yes—
118C23×C4C2×Dic5314yes—
119C23×C4C22×D5312yes—
120C22⋊Q8—28no—
121C22⋊Q8—28no—
122C22⋊Q8—232no—
123C2×C4⋊C4—1064no—
124C2×C4⋊C4C2×Dic53112yes—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—538no—
127C2×C22⋊C4—1064no—
128C2×C22⋊C4—1064no—
129C2×C22⋊C4C2×Dic5314yes—
130C2×C22⋊C4C2×Dic5314yes—
131C2×C22⋊C4C22⋊D5318yes—
132C2×C22⋊C4C22×D5314yes—
133C2×C42—1064no—
134#95—1064no—
135#95—1064no—
136C24—24no—
137C24—82no—
138C24C22×Dic5312yes—
139C24C2×C22⋊D5314yes—
140C24C23×D5311yescharacteristic
141C2×Q8—28no—
142C2×Q8—84no—
143C22×C4—28no—
144C22×C4—82no—
145C22×C4—84no—
146C22×C4—1068no—
147C22×C4C22×Dic5318yes—
148C22×C4C22×Dic5314yes—
149C22×C4C2×C22⋊D5314yes—
150C22×C4C2×C22⋊D5314yes—
151C4⋊C4—224no—
152C4⋊C4—4244no—
153C22⋊C4—28no—
154C22⋊C4—28no—
155C22⋊C4—10616no—
156C22⋊C4—4244no—
157C22⋊C4—4244no—
158C22⋊C4C22⋊Dic53116yes—
159C42—4244no—
160C23—24no—
161C23—28no—
162C23—44no—
163C23—42no—
164C23—44no—
165C23—44no—
166C23—88no—
167C23C2×C22⋊Dic5314yes—
168C23C23×Dic5311yescharacteristic
169C23C22×C22⋊D5312yes—
170C23#1313326514yes—
171Q8—88no—
172C2×C4—28no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—48no—
177C2×C4—88no—
178C2×C4—2128no—
179C2×C4—2128no—
180C2×C4—2128no—
181C2×C4C2×C22⋊Dic5318yes—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—28no—
186C22—24no—
187C22—44no—
188C22—48no—
189C22—88no—
190C22C22×C22⋊Dic5311yescharacteristic
191C22#1681110112yes—
192C22#1681119714yes—
193C4—28no—
194C4—48no—
195C4—4248no—
196C2—24no—
197C2—22no—
198C2—44no—
199C2#2027730111yescharacteristic
200C2#2028317612yes—
subgroupG/H
centerC22C22×C22⋊Dic53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC2×C53×#543C2
socleC22×C53C22×C22⋊C4
prime radicalC25×C53C23
cosocleC24×C53C24
hypercenterC2×#543D53
hypocenterC53#21827
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #20275701C2semidirect
11. C2×C22⋊Q8×C53C4semidirect
106. C2×C22⋊Q8Dic53semidirect
115. C53#21827semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202757012
#202747892
C2×C53×#5432
#2182753