Finite Group Databasebeta

← #28303446

Subgroups — #28303446

Order 13568.

438 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289963C212yesmaximal
2#20289616C214yesmaximal
3#20277331C211yesmaximal, characteristic
4#16811593C2214yes—
5#16811468—22no—
6#16811468—22no—
7C2×#13133176C411yescharacteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810521—22no—
10#16810519—22no—
11#16810481C412yes—
12#16810481C2212yes—
13C2×#117×C53C411yescharacteristic
14C2×#11719940—24no—
15C22⋊#88324—24no—
16C22⋊#88324—24no—
17C23×Dic53C2×C411yescharacteristic
18C23×Dic53C2311yescharacteristic
19#13133176—42no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—24no—
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53D412yes—
24C2×#11719919—22no—
25C2×#11719919—22no—
26C2×C22⋊Dic53C2×C412yes—
27C2×C4×Dic53—21no—
28#13133124—22no—
29#13133124—22no—
30C23×C4×C53C2×C411yescharacteristic
31#117×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C22⋊C4×C53C2×C412yes—
34C2×C42×C53—21no—
35C24×C53C22×C411yescharacteristic
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C22⋊C411yescharacteristic
42C22×Dic53—22no—
43C22×Dic53—22no—
44C22×Dic53—81no—
45C22×Dic53—81no—
46C22×Dic53C22⋊C412yes—
47C22×Dic53C2×D412yes—
48C22×Dic53#4114yes—
49#11719940—84no—
50C2×#88324—44no—
51C4⋊C4×C53—42no—
52C22⋊C4×C53—42no—
53C42×C53—42no—
54C22⋊Dic53—42no—
55C4⋊Dic53—24no—
56C4⋊Dic53—24no—
57C4⋊Dic53—42no—
58#11719919—82no—
59#11719919—82no—
60C4×Dic53—42no—
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—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53#10012yes—
76C2×C4×C53#10112yes—
77C2×Dic53—22no—
78C2×Dic53—44no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—82no—
85C2×Dic53#12314yes—
86#88324—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—82no—
101Dic53—24no—
102Dic53—42no—
103Dic53—82no—
104#2316—531no—
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×#117—531no—
114C2×#117#615711yescharacteristic
115C22×C4⋊C4—531no—
116#407—534no—
117#389—1062no—
118#389—1062no—
119#380—532no—
120#380—532no—
121#324—1062no—
122#324—1062no—
123C53#875711yescharacteristic, complemented, Sylow
124C23×C4—531no—
125C23×C4—531no—
126C23×C4C2×#615711yescharacteristic
127C2×M16—1064no—
128#117—42no—
129#117—2122no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—1064no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—1062no—
136C2×C4⋊C4—1061no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4C2×#615712yes—
139C2×C42—21no—
140C2×C42—1061no—
141C22⋊C8—1064no—
142C22⋊C8—1064no—
143#95—1062no—
144#95—1062no—
145C24C22×#615711yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—532no—
151C22×C4—534no—
152C22×C4—532no—
153C22×C4—1062no—
154C22×C4—1062no—
155C22×C4—4241no—
156C22×C4—4241no—
157C22×C4C22⋊#615711yescharacteristic
158C22×C4C22⋊#615711yescharacteristic
159M16—4244no—
160C2×C8—2124no—
161C4⋊C4—42no—
162C4⋊C4—1064no—
163C4⋊C4—1064no—
164C4⋊C4—4242no—
165C4⋊C4—4242no—
166C4⋊C4—2122no—
167C22⋊C4—42no—
168C22⋊C4—2122no—
169C42—42no—
170C42—2122no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊#615711yescharacteristic
176C23#1313329612yes—
177C2×C4—28no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—534no—
186C2×C4—1062no—
187C2×C4—4242no—
188C2×C4—2122no—
189C2×C4—2124no—
190C2×C4—2122no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4#1313327712yes—
195C2×C4#1313327812yes—
196C8—4244no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22#1681156611yescharacteristic
205C22C2×#1313327711yescharacteristic
206C22C2×#1313327811yescharacteristic
207C4—22no—
208C4—22no—
209C4—82no—
210C4—1064no—
211C4—4242no—
212C4—2122no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2#2029061411yescharacteristic
217C2#2029062711yescharacteristic
218C2#2029062911yescharacteristic
subgroupG/H
centerC22#16811566
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC2×#117×C53C4
socleC22×C53#381
cosocleC23×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8757
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#117

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
#202899632
#202896162
#202773312
#875753