Finite Group Databasebeta

← #28303365

Subgroups — #28303365

Order 13568.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290572C212yesmaximal
2#20289593C214yesmaximal, complemented
3#384×D53C211yesmaximal, characteristic
4C2×#13133284C2214yes—
5C2×#13133274—22no—
6#16811489—22no—
7C2×C4⋊C4×D53—22no—
8C2×C4⋊C4×D53C412yes—
9C2×C42×D53C2211yescharacteristic
10#16810493C411yescharacteristic
11#16810492—22no—
12#16810492C2212yes—
13C53×#384C411yescharacteristic
14C22×#88324—24no—
15#13133284—28no—
16#13133274—44no—
17C22×C4×D53—22no—
18C22×C4×D53C2×C411yescharacteristic
19C22×C4×D53D412yes—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53C2×C412yes—
22C2×#11719919—22no—
23C2×#11719919—24no—
24C4⋊C4×D53—48no—
25C4⋊C4×D53—44no—
26C2×C4×Dic53D412yes—
27C2×C4×Dic53C2311yescharacteristic
28C42×D53—24no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53C2×C412yes—
31C2×C42×C53C2×C411yescharacteristic
32C23×D53C22⋊C411yescharacteristic
33C22×C4×C53—22no—
34C22×C4×C53C22⋊C412yes—
35C22×C4×C53C22×C411yescharacteristic
36C22×Dic53—22no—
37C22×Dic53C22⋊C411yescharacteristic
38C22×Dic53C2×D412yes—
39C2×C4×D53—22no—
40C2×C4×D53—22no—
41C2×C4×D53—22no—
42C2×C4×D53—24no—
43C2×C4×D53—24no—
44C2×C4×D53—48no—
45C2×C4×D53—44no—
46C2×C4×D53SD1614yes—
47C2×C4×D53Q1614yes—
48C2×#88324—44no—
49C2×#88324—44no—
50C2×#88324—44no—
51C4⋊C4×C53—44no—
52C4⋊C4×C53—42no—
53C42×C53—22no—
54C4⋊Dic53—44no—
55C4⋊Dic53—42no—
56#11719919—44no—
57#11719919—48no—
58C4×Dic53—24no—
59C4×Dic53#4114yes—
60C23×C53C2×C22⋊C411yescharacteristic
61C22×D53—21no—
62C22×D53—21no—
63C22×D53—22no—
64C22×D53#10012yes—
65C22×D53#10314yes—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—42no—
71C2×C4×C53—44no—
72C2×C4×C53#10314yes—
73C2×C4×C53#11712yes—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—42no—
77C2×Dic53—44no—
78C2×Dic53#10112yes—
79C2×Dic53#12314yes—
80C2×Dic53C2×SD1612yes—
81C2×Dic53C2×Q1612yes—
82C2×Dic53AGL(1,ℤ/8)14yes—
83C4×D53—28no—
84C4×D53—28no—
85C4×D53—28no—
86C4×D53—88no—
87#88324—84no—
88#8725—531nomaximal, Sylow
89C22×C53#38111yescharacteristic
90C22×C53C2×#10011yescharacteristic
91C22×C53C2×#10111yescharacteristic
92C22×C53C2×#10312yes—
93C22×C53#41812yes—
94C2×D53—24no—
95C2×D53—22no—
96C2×D53—22no—
97C2×D53—24no—
98C2×D53#33314yes—
99C4×C53—22no—
100C4×C53—24no—
101C4×C53—82no—
102Dic53—22no—
103Dic53—82no—
104Dic53#48114yes—
105Dic53#48414yes—
106C2×#384—531no—
107#1797—532no—
108#1296—534no—
109C2×C53C2×#33311yescharacteristic
110C2×C53#154912yes—
111C2×C53#155511yescharacteristic
112C2×C53#182111yescharacteristic
113C2×C53#182412yes—
114D53—24no—
115C22×C4⋊C4—532no—
116C22×C4⋊C4—1062no—
117C22×C42—531no—
118C2×C4⋊C8—534no—
119C2×C22⋊C8—1062no—
120#384—532no—
121#384—531no—
122#384—1062no—
123#384#615711yescharacteristic
124#336—1062no—
125C53#872511yescharacteristic, complemented, Sylow
126C23×C4—532no—
127C23×C4—531no—
128C23×C4—1062no—
129C22×C8—1064no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—1062no—
133C2×C4⋊C4—1064no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—2124no—
136C2×C4⋊C4—2128no—
137C2×C4⋊C4C2×#615712yes—
138C2×C42—531no—
139C2×C42—532no—
140C2×C42—1064no—
141C2×C42C2×#615711yescharacteristic
142C4⋊C8—1068no—
143C22⋊C8—2124no—
144C24—531no—
145C22×C4—22no—
146C22×C4—532no—
147C22×C4—531no—
148C22×C4—534no—
149C22×C4—534no—
150C22×C4—1064no—
151C22×C4—1064no—
152C22×C4—1062no—
153C22×C4—1062no—
154C22×C4—1062no—
155C22×C4—1062no—
156C22×C4—2124no—
157C22×C4—2128no—
158C22×C4C22⋊#615712yes—
159C22×C4C22×#615711yescharacteristic
160C2×C8—2124no—
161C2×C8—2124no—
162C2×C8—2124no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—2122no—
166C4⋊C4—2124no—
167C4⋊C4—2128no—
168C4⋊C4—2124no—
169C42—22no—
170C42—534no—
171C42—1064no—
172C23—532no—
173C23—534no—
174C23—1061no—
175C23—1061no—
176C23—1062no—
177C23C2×C22⋊#615711yescharacteristic
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—534no—
185C2×C4—532no—
186C2×C4—532no—
187C2×C4—534no—
188C2×C4—532no—
189C2×C4—1068no—
190C2×C4—1068no—
191C2×C4—1068no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—4248no—
195C2×C4—2122no—
196C2×C4—2124no—
197C2×C4#1313328014yes—
198C2×C4#1313329612yes—
199C8—4244no—
200C22—534no—
201C22—1062no—
202C22—1064no—
203C22—1064no—
204C22—1062no—
205C22#1681156611yescharacteristic
206C22D53⋊#10011yescharacteristic
207C22D53⋊#10111yescharacteristic
208C22C2×#1313328012yes—
209C22#1681161512yes—
210C4—22no—
211C4—24no—
212C4—82no—
213C4—534no—
214C4—534no—
215C4—1062no—
216C4—4242no—
217C2—1064no—
218C2D53⋊#33311yescharacteristic
219C2#2029023112yes—
220C2#2029024311yescharacteristic
221C2#2029061511yescharacteristic
222C2#2029061912yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#8725
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #20289593C2semidirect
125. C53#8725semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202895932
#384×D532
#872553