Finite Group Databasebeta

← #28311345

Subgroups — #28311345

Order 13568.

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

normal subgroups only

This group has 614 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#20277303C211yesmaximal, characteristic
4C2×#13133278C2214yes—
5C2×#13133277—22no—
6#16811595—22no—
7C22×#11719945C2211yescharacteristic
8C22×C4⋊Dic53C411yescharacteristic
9#16810529C2212yes—
10#16810525C412yes—
11#16810500—22no—
12#16810480—22no—
13C2×#117×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—
24C23×Dic53—21no—
25C22×C4×D53D412yes—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—42no—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719921—24no—
30#13133124—24no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53—21no—
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×C53C22⋊C411yescharacteristic
43C22×C4×C53C22×C411yescharacteristic
44C2×C22⋊D53—24no—
45C2×C22⋊D53—24no—
46C2×C22⋊D53—42no—
47C22×Dic53—22no—
48C22×Dic53—42no—
49C22×Dic53—82no—
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—
64C4⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C42×C53—42no—
67#11719921—84no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53#9912yes—
75C23×C53C2×C22⋊C411yescharacteristic
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—42no—
80C22×D53—42no—
81C2×C4×C53—28no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53#9912yes—
90C2×C4×C53#11712yes—
91C22⋊D53—48no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—44no—
95C2×Dic53—42no—
96C2×Dic53—84no—
97C2×Dic53#12314yes—
98D212—44no—
99C4×D53—48no—
100Dic106—44no—
101#88324—84no—
102#12863—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—22no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53—41no—
109C22×C53—41no—
110C22×C53#38111yescharacteristic
111C22×C53C2×#9911yescharacteristic
112C22×C53C2×#9911yescharacteristic
113C2×D53—44no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—42no—
117C2×D53—82no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—82no—
121Dic53—24no—
122Dic53—82no—
123#2311—531no—
124#1785—532no—
125#1394—534no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53—42no—
129C2×C53#181911yescharacteristic
130C2×C53#213411yescharacteristic
131C2×C53#213511yescharacteristic
132D53—82no—
133C22×#41—531no—
134C2×#117#615711yescharacteristic
135C22×C4⋊C4—531no—
136C2×#101—534no—
137C2×#100—1062no—
138#407—1062no—
139#391—532no—
140#390—532no—
141#386—1062no—
142#380—1062no—
143C53#1286311yescharacteristic, complemented, Sylow
144C2×#41—1064no—
145C2×#41—2122no—
146C2×#41—2122no—
147C22×Q8—531no—
148C22×D4—532no—
149C22×D4—531no—
150C23×C4—532no—
151C23×C4—1061no—
152C23×C4C2×#615711yescharacteristic
153C2×M16—1064no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—2122no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—1064no—
161C2×C42C2×#615712yes—
162#101—1068no—
163#100—2124no—
164C22⋊C8—2124no—
165#95—1064no—
166C24—532no—
167C24C22⋊#615711yescharacteristic
168#41—21216no—
169C2×Q8—534no—
170C2×Q8—2121no—
171C2×Q8—2121no—
172C2×D4—1064no—
173C2×D4—1064no—
174C2×D4—1062no—
175C2×D4—2121no—
176C2×D4—2121no—
177C2×D4—2122no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—532no—
183C22×C4—1064no—
184C22×C4—1064no—
185C22×C4—1062no—
186C22×C4—4242no—
187C22×C4—2122no—
188C22×C4—2122no—
189C22×C4C22⋊#615711yescharacteristic
190C22×C4C22×#615711yescharacteristic
191M16—2124no—
192M16—2124no—
193C2×C8—2124no—
194C4⋊C4—42no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C22⋊C4—42no—
198C22⋊C4—4244no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23—1062no—
205C23—1062no—
206C23—1062no—
207C23—2122no—
208C23—2122no—
209C23#1313327512yes—
210C23C2×C22⋊#615711yescharacteristic
211Q8—2124no—
212D4—2124no—
213D4—2128no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—534no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—4244no—
226C2×C4—2128no—
227C2×C4—2124no—
228C2×C4—2122no—
229C2×C4#1313327612yes—
230C2×C4#1313329612yes—
231C8—4244no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—42no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22—4242no—
240C22—2124no—
241C22—2122no—
242C22—2122no—
243C22—2122no—
244C22#1681156611yescharacteristic
245C22C2×#1313327511yescharacteristic
246C22C2×#1313327611yescharacteristic
247C4—22no—
248C4—22no—
249C4—82no—
250C4—1064no—
251C4—4242no—
252C2—21no—
253C2—21no—
254C2—42no—
255C2—4242no—
256C2#2029061011yescharacteristic
257C2#2029101611yescharacteristic
258C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC22×#11719945C22
cosocleC22×Dic106C23
hypercenterC2×#117#6157
hypocenterC53#12863
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289812C2semidirect
143. C53#12863semidirect

Maximal subgroups

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

subgroupindex
#202905512
#202898122
#202773032
#1286353