Finite Group Databasebeta

← #28311339

Subgroups — #28311339

Order 13568.

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

normal subgroups only

This group has 642 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#20290551C212yesmaximal
2#20289812C214yesmaximal, complemented
3#20277205C211yesmaximal, characteristic
4C2×#13133278C2214yes—
5C2×#13133277—22no—
6#16811595—22no—
7C22×#11719945C2211yescharacteristic
8C2×#13133176C411yescharacteristic
9#16810529C2212yes—
10#16810525C412yes—
11#16810502—22no—
12C4×#11719921—22no—
13C22×C42×C53C411yescharacteristic
14C2×#11719940—24no—
15#13133278—28no—
16#13133277—44no—
17#13133274—44no—
18C2×#11719945—24no—
19C2×#11719945—42no—
20C2×#11719945—42no—
21C22×Dic106C2311yescharacteristic
22C22×D212C2×C411yescharacteristic
23C22×C22⋊D53D412yes—
24C22×C4×D53D412yes—
25#13133176—42no—
26C2×C4⋊Dic53C2×C412yes—
27C2×C22⋊Dic53—21no—
28C2×#11719921—24no—
29C2×C4×Dic53—21no—
30#13133124—24no—
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C2×C42×C53—21no—
34C2×C42×C53—42no—
35C2×C42×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C23×D53C22⋊C412yes—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—44no—
43C22×C4×C53—42no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22×C411yescharacteristic
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—42no—
49C22×Dic53—22no—
50C22×Dic53C2×D412yes—
51#11719945—416no—
52C2×D212—22no—
53C2×D212—41no—
54C2×D212—41no—
55C2×C4×D53—24no—
56C2×C4×D53—24no—
57C2×C4×D53—42no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×Dic106#4114yes—
61#11719940—44no—
62#11719940—44no—
63C2×#88324—44no—
64C42×C53—42no—
65C42×C53—42no—
66C22⋊Dic53—42no—
67#11719921—48no—
68C4⋊Dic53—42no—
69C4×Dic53—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—42no—
73C23×C53#9912yes—
74C23×C53C2×C22⋊C411yescharacteristic
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—44no—
79C2×C4×C53—28no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—24no—
85C2×C4×C53—42no—
86C2×C4×C53—48no—
87C2×C4×C53—42no—
88C2×C4×C53#9912yes—
89C2×C4×C53#11712yes—
90C22⋊D53—48no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—44no—
94C2×Dic53—42no—
95C2×Dic53#12314yes—
96D212—44no—
97C4×D53—48no—
98Dic106—44no—
99#88324—84no—
100#12861—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—22no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53#38111yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#9911yescharacteristic
109C2×D53—44no—
110C2×D53—42no—
111C2×D53—44no—
112C2×D53—82no—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—44no—
116Dic53—24no—
117Dic53—82no—
118#2297—531no—
119#1785—532no—
120#1394—534no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—42no—
124C2×C53#181911yescharacteristic
125C2×C53#213212yes—
126D53—82no—
127C22×#41—531no—
128C2×#117—531no—
129C22×C42#615711yescharacteristic
130C2×#101—534no—
131C2×#100—1062no—
132#407—1062no—
133#391—532no—
134#390—532no—
135#386—1062no—
136C4×C22⋊C4—1062no—
137C53#1286111yescharacteristic, complemented, Sylow
138C2×#41—1064no—
139C2×#41—2122no—
140C2×#41—2122no—
141C22×Q8—531no—
142C22×D4—532no—
143C22×D4—531no—
144C23×C4—21no—
145C23×C4—532no—
146C23×C4C2×#615711yescharacteristic
147C2×M16—1064no—
148#117—2122no—
149C2×C4⋊C4—532no—
150C2×C22⋊C4—1064no—
151C2×C22⋊C4—1061no—
152C2×C42—21no—
153C2×C42—42no—
154C2×C42—1061no—
155C2×C42C2×#615712yes—
156#101—1068no—
157#100—2124no—
158C22⋊C8—2124no—
159#95—1064no—
160C24—532no—
161C24C22⋊#615711yescharacteristic
162#41—21216no—
163C2×Q8—534no—
164C2×Q8—2121no—
165C2×Q8—2121no—
166C2×D4—1064no—
167C2×D4—1064no—
168C2×D4—1062no—
169C2×D4—2121no—
170C2×D4—2121no—
171C2×D4—2122no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—532no—
179C22×C4—1064no—
180C22×C4—1064no—
181C22×C4—1062no—
182C22×C4—2122no—
183C22×C4C22⋊#615711yescharacteristic
184C22×C4C22×#615711yescharacteristic
185M16—2124no—
186M16—2124no—
187C2×C8—2124no—
188C4⋊C4—2122no—
189C22⋊C4—2128no—
190C22⋊C4—2122no—
191C42—42no—
192C42—42no—
193C42—2122no—
194C23—21no—
195C23—21no—
196C23—42no—
197C23—1062no—
198C23—1062no—
199C23—1062no—
200C23—2124no—
201C23#1313327512yes—
202C23C2×C22⋊#615711yescharacteristic
203Q8—2124no—
204D4—2124no—
205D4—2128no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—42no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—534no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—2128no—
219C2×C4—2124no—
220C2×C4—2122no—
221C2×C4#1313327612yes—
222C2×C4#1313329612yes—
223C8—4244no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—4242no—
230C22—2124no—
231C22—2122no—
232C22—2124no—
233C22#1681156611yescharacteristic
234C22C2×#1313327511yescharacteristic
235C22C2×#1313327611yescharacteristic
236C4—22no—
237C4—22no—
238C4—44no—
239C4—1064no—
240C4—4242no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—4242no—
245C2#2029061011yescharacteristic
246C2#2029100812yes—
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC22×#11719945C22
cosocleC22×Dic106C23
hypercenterC22×C42#6157
hypocenterC53#12861
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289812C2semidirect
137. C53#12861semidirect

Maximal subgroups

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

subgroupindex
#202905512
#202898122
#202772052
#1286153