Finite Group Databasebeta

← #28303357

Subgroups — #28303357

Order 13568.

526 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290442C212yesmaximal
2#20289591C214yesmaximal, complemented
3#20277120C211yesmaximal, characteristic
4C2×#13133274—22no—
5C2×#13133272C2214yes—
6#16811489—22no—
7C2×#13133175C412yes—
8C2×C42×D53C2211yescharacteristic
9#16810493C411yescharacteristic
10#16810488C2212yes—
11#16810480—22no—
12C4⋊C4×Dic53—22no—
13C53×#384C411yescharacteristic
14C22×#88324—24no—
15#13133274—28no—
16#13133272—28no—
17C22×C4×D53C2×C411yescharacteristic
18C22×C4×D53D412yes—
19#13133175—44no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53C2×C412yes—
22C2×#11719921—22no—
23C2×C4×Dic53—22no—
24C2×C4×Dic53D412yes—
25C2×C4×Dic53C2311yescharacteristic
26C42×D53—24no—
27#13133124—24no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53C2×C412yes—
30C2×C42×C53C2×C411yescharacteristic
31C23×D53C22⋊C411yescharacteristic
32C22×C4×C53—22no—
33C22×C4×C53C22⋊C412yes—
34C22×C4×C53C22×C411yescharacteristic
35C22×Dic53—22no—
36C22×Dic53C22⋊C411yescharacteristic
37C22×Dic53C2×D412yes—
38C2×C4×D53—22no—
39C2×C4×D53—22no—
40C2×C4×D53—22no—
41C2×C4×D53—24no—
42C2×C4×D53—24no—
43C2×C4×D53—42no—
44C2×#88324—28no—
45C2×#88324—44no—
46C2×#88324—44no—
47C4⋊C4×C53—44no—
48C4⋊C4×C53—42no—
49C42×C53—22no—
50#11719921—44no—
51C4⋊Dic53—44no—
52C4⋊Dic53—42no—
53C4×Dic53—24no—
54C4×Dic53—44no—
55C4×Dic53#4114yes—
56C23×C53C2×C22⋊C411yescharacteristic
57C22×D53—21no—
58C22×D53—21no—
59C22×D53—22no—
60C22×D53—22no—
61C22×D53#10112yes—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53—42no—
67C2×C4×C53—44no—
68C2×C4×C53#10414yes—
69C2×C4×C53#11712yes—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—42no—
75C2×Dic53—44no—
76C2×Dic53#10012yes—
77C2×Dic53#12314yes—
78C4×D53—28no—
79C4×D53—48no—
80#88324—48no—
81#8721—531nomaximal, Sylow
82C22×C53#38111yescharacteristic
83C22×C53C2×#10011yescharacteristic
84C22×C53C2×#10111yescharacteristic
85C22×C53C2×#10412yes—
86C22×C53#42112yes—
87C2×D53—22no—
88C2×D53—24no—
89C2×D53—41no—
90C2×D53—42no—
91C2×D53—41no—
92C4×C53—22no—
93C4×C53—24no—
94C4×C53—82no—
95Dic53—22no—
96Dic53—24no—
97Dic53—82no—
98#2285—531no—
99#1749—532no—
100#1295—534no—
101C2×C53#152511yescharacteristic
102C2×C53#153112yes—
103C2×C53#153611yescharacteristic
104C2×C53#182111yescharacteristic
105C2×C53#182612yes—
106D53—42no—
107C2×#117—532no—
108C22×C42—531no—
109C2×C22⋊C8—1062no—
110C2×#97—534no—
111#384—531no—
112#384#615711yescharacteristic
113#382—532no—
114#380—1062no—
115C4×C4⋊C4—1062no—
116#336—1062no—
117C53#872111yescharacteristic, complemented, Sylow
118C23×C4—531no—
119C23×C4—532no—
120C22×C8—1064no—
121#117—2124no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—1062no—
125C2×C4⋊C4C2×#615712yes—
126C2×C22⋊C4—1062no—
127C2×C42—531no—
128C2×C42—532no—
129C2×C42—1064no—
130C2×C42—1062no—
131C2×C42C2×#615711yescharacteristic
132C22⋊C8—1068no—
133#97—1068no—
134#95—1064no—
135C24—531no—
136C22×C4—22no—
137C22×C4—531no—
138C22×C4—532no—
139C22×C4—1062no—
140C22×C4—1064no—
141C22×C4—1062no—
142C22×C4—1062no—
143C22×C4—1064no—
144C22×C4—1062no—
145C22×C4—2122no—
146C22×C4C22⋊#615712yes—
147C22×C4C22×#615711yescharacteristic
148C2×C8—1068no—
149C2×C8—2124no—
150C2×C8—2124no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—2122no—
154C4⋊C4—2124no—
155C22⋊C4—2124no—
156C42—22no—
157C42—534no—
158C42—1064no—
159C42—2124no—
160C23—532no—
161C23—1061no—
162C23—1061no—
163C23—1062no—
164C23—1062no—
165C23C2×C22⋊#615711yescharacteristic
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—534no—
173C2×C4—532no—
174C2×C4—1068no—
175C2×C4—1061no—
176C2×C4—1061no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—2128no—
180C2×C4—2122no—
181C2×C4—2124no—
182C2×C4#1313328214yes—
183C2×C4#1313329612yes—
184C8—2128no—
185C22—1062no—
186C22—1064no—
187C22—2122no—
188C22—2121no—
189C22—2121no—
190C22#1681156611yescharacteristic
191C22D53⋊#10011yescharacteristic
192C22D53⋊#10111yescharacteristic
193C22C2×#1313328212yes—
194C22#1681162112yes—
195C4—22no—
196C4—24no—
197C4—82no—
198C4—1062no—
199C4—1064no—
200C4—4242no—
201C2—2122no—
202C2#2029017111yescharacteristic
203C2#2029018912yes—
204C2#2029020411yescharacteristic
205C2#2029061511yescharacteristic
206C2#2029062212yes—
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#8721
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. #20289591C2semidirect
117. C53#8721semidirect

Maximal subgroups

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

subgroupindex
#202904422
#202895912
#202771202
#872153