Finite Group Databasebeta

← #28303358

Subgroups — #28303358

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#20290441C212yesmaximal
2#20289591C214yesmaximal, complemented
3#20277118C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272C2214yes—
6#16811489—22no—
7C2×#13133176C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810492C412yes—
10#16810488C2212yes—
11#16810481—22no—
12C4×#11719919—22no—
13C2×#117×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16#13133272—28no—
17C23×Dic53D412yes—
18#13133176—42no—
19C2×C4⋊Dic53C2×C412yes—
20C2×#11719919—22no—
21C2×#11719919—22no—
22C2×C22⋊Dic53—21no—
23C2×C4×Dic53—24no—
24C2×C4×Dic53—21no—
25C2×C4×Dic53C2×C411yescharacteristic
26C2×C4×Dic53D412yes—
27C2×C4×Dic53C2311yescharacteristic
28#13133124—24no—
29C23×C4×C53C2×C411yescharacteristic
30#117×C53—42no—
31C2×C4⋊C4×C53C2×C412yes—
32C2×C22⋊C4×C53—21no—
33C2×C42×C53—21no—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53C22⋊C411yescharacteristic
40C22×C4×C53C22×C411yescharacteristic
41C22×Dic53—24no—
42C22×Dic53—24no—
43C22×Dic53—22no—
44C22×Dic53—42no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C2×D412yes—
47C2×#88324—28no—
48C2×#88324—44no—
49C2×#88324—44no—
50C4⋊C4×C53—42no—
51C22⋊C4×C53—42no—
52C42×C53—42no—
53C22⋊Dic53—42no—
54C4⋊Dic53—42no—
55#11719919—44no—
56#11719919—44no—
57C4×Dic53—22no—
58C4×Dic53—24no—
59C4×Dic53—42no—
60C4×Dic53#4114yes—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53#10012yes—
66C23×C53C2×C22⋊C411yescharacteristic
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#10112yes—
76C2×C4×C53#11712yes—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—48no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53#10414yes—
86C2×Dic53#12314yes—
87#88324—48no—
88#8722—531nomaximal, Sylow
89C22×C53—22no—
90C22×C53—24no—
91C22×C53—41no—
92C22×C53—42no—
93C22×C53—41no—
94C22×C53#38111yescharacteristic
95C22×C53C2×#10011yescharacteristic
96C22×C53C2×#10111yescharacteristic
97C22×C53C2×#10412yes—
98C22×C53#42112yes—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—82no—
102Dic53—24no—
103Dic53—24no—
104Dic53—82no—
105#2285—531no—
106#1749—532no—
107#1295—534no—
108C2×C53—42no—
109C2×C53#152611yescharacteristic
110C2×C53#153012yes—
111C2×C53#153711yescharacteristic
112C2×C53#182111yescharacteristic
113C2×C53#182612yes—
114C2×#117—531no—
115C2×#117#615711yescharacteristic
116C22×C42—531no—
117C2×C22⋊C8—1062no—
118C2×#97—534no—
119#384—532no—
120#382—532no—
121#380—1062no—
122C4×C4⋊C4—1062no—
123#336—1062no—
124C53#872211yescharacteristic, complemented, Sylow
125C23×C4—532no—
126C23×C4C2×#615711yescharacteristic
127C22×C8—1064no—
128#117—42no—
129#117—2122no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—1062no—
132C2×C4⋊C4—1062no—
133C2×C4⋊C4C2×#615712yes—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—1061no—
136C2×C42—21no—
137C2×C42—531no—
138C2×C42—532no—
139C2×C42—531no—
140C2×C42—1064no—
141C2×C42—1061no—
142C22⋊C8—1068no—
143#97—1068no—
144#95—1064no—
145C24C22⋊#615711yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—1064no—
153C22×C4—1064no—
154C22×C4—1062no—
155C22×C4—2122no—
156C22×C4C22⋊#615711yescharacteristic
157C22×C4C22×#615711yescharacteristic
158C2×C8—1068no—
159C2×C8—2124no—
160C2×C8—2124no—
161C4⋊C4—42no—
162C4⋊C4—2122no—
163C4⋊C4—2124no—
164C4⋊C4—2124no—
165C22⋊C4—42no—
166C22⋊C4—2122no—
167C42—42no—
168C42—534no—
169C42—1064no—
170C42—1062no—
171C42—2122no—
172C23—21no—
173C23—21no—
174C23—22no—
175C23—22no—
176C23#1313327712yes—
177C23C2×C22⋊#615711yescharacteristic
178C2×C4—28no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—534no—
187C2×C4—534no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—2128no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4#1313327812yes—
197C2×C4#1313329612yes—
198C8—2128no—
199C22—22no—
200C22—24no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22#1681156611yescharacteristic
205C22C2×#1313327711yescharacteristic
206C22C2×#1313327811yescharacteristic
207C22D53⋊#10412yes—
208C22#1681162012yes—
209C4—22no—
210C4—22no—
211C4—82no—
212C4—1064no—
213C4—1064no—
214C4—4242no—
215C2—42no—
216C2#2029017211yescharacteristic
217C2#2029018412yes—
218C2#2029020611yescharacteristic
219C2#2029061411yescharacteristic
220C2#2029062312yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8722
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289591C2semidirect
124. C53#8722semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895912
#202771182
#872253