Finite Group Databasebeta

← #27277628

Subgroups — #27277628

Order 13568.

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

normal subgroups only

This group has 438 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#20273045C212yesmaximal
2#20272621C214yesmaximal
3C53×#2316C211yesmaximal, characteristic
4#16810587C2214yes—
5#16810349—22no—
6#16810349—22no—
7C2×#117×C53C411yescharacteristic
8C2×#117×C53C411yescharacteristic
9C22×C4⋊C4×C53C2211yescharacteristic
10C53×#389—22no—
11C53×#389—22no—
12C53×#380C412yes—
13C53×#380C2212yes—
14C2×#11719917—24no—
15C22⋊#88322—24no—
16C22⋊#88322—24no—
17C23×C4×C53C2×C411yescharacteristic
18C23×C4×C53C2×C411yescharacteristic
19C23×C4×C53C2311yescharacteristic
20#117×C53—42no—
21#117×C53—42no—
22C2×C4⋊C4×C53—21no—
23C2×C4⋊C4×C53—21no—
24C2×C4⋊C4×C53—22no—
25C2×C4⋊C4×C53—22no—
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53D412yes—
28C2×C4⋊C4×C53D412yes—
29C2×C22⋊C4×C53C2×C412yes—
30C2×C22⋊C4×C53C2×C412yes—
31C2×C42×C53—21no—
32C2×C42×C53—21no—
33#95×C53—22no—
34#95×C53—22no—
35C24×C53C22×C411yescharacteristic
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—81no—
43C22×C4×C53—81no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C412yes—
47C22×C4×C53C2×D412yes—
48C22×C4×C53#4114yes—
49C4⋊C4×C53—24no—
50C4⋊C4×C53—24no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—82no—
54C4⋊C4×C53—82no—
55C22⋊C4×C53—42no—
56C22⋊C4×C53—42no—
57C42×C53—42no—
58C42×C53—42no—
59#11719917—84no—
60C2×#88322—44no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—41no—
64C23×C53—41no—
65C23×C53C2×C22⋊C411yescharacteristic
66C23×C53#11712yes—
67C2×C4×C53—28no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—42no—
74C2×C4×C53—44no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—82no—
83C2×C4×C53#10012yes—
84C2×C4×C53#10112yes—
85C2×C4×C53#12314yes—
86#88322—84no—
87#8757—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—22no—
91C22×C53—22no—
92C22×C53—22no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53#38111yescharacteristic
96C22×C53C2×#10011yescharacteristic
97C22×C53C2×#10111yescharacteristic
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—24no—
101C4×C53—42no—
102C4×C53—82no—
103C4×C53—82no—
104#2316D5311yescharacteristic
105#1463—532no—
106#1305—534no—
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—42no—
110C2×C53#182111yescharacteristic
111C2×C53#182811yescharacteristic
112C2×C53#183011yescharacteristic
113C2×#117Dic5311yescharacteristic
114C2×#117Dic5311yescharacteristic
115C22×C4⋊C4C2×D5311yescharacteristic
116#407—534no—
117#389—22no—
118#389—22no—
119#380Dic5312yes—
120#380C2×D5312yes—
121#324—1062no—
122#324—1062no—
123C53#875711yescharacteristic, complemented, Sylow
124C23×C4C2×Dic5311yescharacteristic
125C23×C4C2×Dic5311yescharacteristic
126C23×C4C22×D5311yescharacteristic
127C2×M16—1064no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C22⋊D5312yes—
136C2×C4⋊C4C22⋊D5312yes—
137C2×C22⋊C4C2×Dic5312yes—
138C2×C22⋊C4C2×Dic5312yes—
139C2×C42—21no—
140C2×C42—21no—
141C22⋊C8—1064no—
142C22⋊C8—1064no—
143#95—22no—
144#95—22no—
145C24C22×Dic5311yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C22⋊Dic5311yescharacteristic
155C22×C4C22⋊Dic5311yescharacteristic
156C22×C4C22⋊Dic5312yes—
157C22×C4#1171994514yes—
158C22×C4C2×C22⋊D5312yes—
159M16—4244no—
160C2×C8—2124no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—42no—
170C42—42no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊Dic5311yescharacteristic
176C23#1313317612yes—
177C2×C4—28no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4#1313313412yes—
194C2×C4#1313313612yes—
195C2×C4#1313320014yes—
196C8—4244no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22#1681048611yescharacteristic
205C22C2×#1313313411yescharacteristic
206C22C2×#1313313611yescharacteristic
207C4—22no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—82no—
212C4—82no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2#2027479511yescharacteristic
217C2#2027482311yescharacteristic
218C2#2027482911yescharacteristic
subgroupG/H
centerC22#16810486
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2316C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#2316D53
hypocenterC53#8757
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
123. C53#8757semidirect

Maximal subgroups

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

subgroupindex
#202730452
#202726212
C53×#23162
#875753