Finite Group Databasebeta

← #27281552

Subgroups — #27281552

Order 13568.

526 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 103 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#20280095C211yesmaximal, characteristic
2#20273234C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20272601C212yesmaximal
5C53×#2313C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810587C2212yes—
8#16810553C412yescomplemented
9#16810543C412yescomplemented
10#16810518—22no—
11#16810496—21no—
12#16810496—21no—
13C4×C22⋊Dic53C2211yescharacteristic
14#16810359—24no—
15#16810349—22no—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#387—24no—
18C53×#381—21no—
19C53×#380—21no—
20C4×C22⋊C4×C53C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C2×#11719917—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46C2×C42×C53—22no—
47#95×C53—22no—
48#95×C53—24no—
49#95×C53D414yes—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—24no—
56C22×C4×C53—42no—
57C22×C4×C53—81no—
58C22×C4×C53—81no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4112yes—
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C2×Dic106—22no—
69C2×Dic106—81no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—82no—
73C22⋊C4×C53—44no—
74C42×C53—82no—
75C22⋊Dic53—24no—
76C22⋊Dic53—41no—
77C22⋊Dic53—41no—
78C4⋊Dic53—22no—
79#11719919—44no—
80#11719919—42no—
81#11719919—82no—
82C4×Dic53—81no—
83#11719917—82no—
84C2×#88322—42no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53#12011yescharacteristic
90C23×C53#12311yescharacteristic
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—44no—
104C2×C4×C53—44no—
105C2×C4×C53—44no—
106C2×C4×C53—81no—
107C2×C4×C53C4×D412yes—
108C2×C4×C53#12112yes—
109C2×Dic53—24no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116Dic106—82no—
117#88322—82no—
118#9412—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53#38611yescharacteristic
128C22×C53#42112yes—
129C22×C53#45612yes—
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—42no—
133C4×C53—42no—
134C4×C53—84no—
135Dic53—44no—
136Dic53—81no—
137Dic53—81no—
138#2583—531no—
139#2313D5311yescharacteristic
140#1508—532no—
141#1463—531no—
142#1297—532no—
143C2×C53—22no—
144C2×C53—22no—
145C2×C53—41no—
146C2×C53#188611yescharacteristic
147C2×C53#189711yescharacteristic
148C2×C53#191311yescharacteristic
149C2×C22⋊Q8—531no—
150C22×C4⋊C4C2×D5311yescharacteristic
151#407—532no—
152#398—532no—
153#395—532no—
154#389—1062no—
155#387—24no—
156#385—1061no—
157#385—1061no—
158#381—21no—
159#380—21no—
160C4×C22⋊C4—531no—
161C4×C22⋊C4C2×D5312yes—
162#328—1064no—
163#324—1062no—
164C53#941211yescharacteristic, complemented, Sylow
165C22×Q8—531no—
166C23×C4C22⋊D5311yescharacteristic
167C23×C4C22⋊D5311yescharacteristic
168C23×C4C22×D5311yescharacteristic
169C2×M16—1062no—
170C22⋊Q8—1062no—
171C22⋊Q8—1062no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—1062no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1062no—
179C2×C4⋊C4C4×D5312yes—
180C2×C4⋊C4D21212yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1061no—
183C2×C22⋊C4—1061no—
184C2×C42—22no—
185C2×C42—1061no—
186C22⋊C8—1062no—
187C22⋊C8—1062no—
188#95—22no—
189#95—24no—
190#95—532no—
191#95—1062no—
192#95—1062no—
193#95C22⋊D5314yes—
194C24C2×C22⋊D5311yescharacteristic
195C2×Q8—1062no—
196C2×Q8—4241no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—1062no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4#1171992112yes—
209C22×C4#1171992112yes—
210C22×C4C2×C4×D5311yescharacteristic
211C22×C4C2×D21211yescharacteristic
212C22×C4#1171994512yes—
213C22×C4C2×C22⋊D5312yes—
214M16—4242no—
215C2×C8—2122no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—82no—
219C4⋊C4—1062no—
220C4⋊C4—4242no—
221C4⋊C4—2124no—
222C4⋊C4—2122no—
223C22⋊C4—44no—
224C22⋊C4—1064no—
225C22⋊C4—2121no—
226C22⋊C4—2121no—
227C42—82no—
228C42—4241no—
229C23—22no—
230C23—22no—
231C23—41no—
232C23C2×#1171992111yescharacteristic
233C23#1313318611yescharacteristic
234C23#1313320011yescharacteristic
235Q8—4242no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—81no—
252C2×C4—1064no—
253C2×C4—2122no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4C4×C22⋊D5312yes—
260C2×C4#1313319112yes—
261C8—4242no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1681050211yescharacteristic
271C22#1681064112yes—
272C22#1681080612yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—84no—
278C4—4241no—
279C4—4241no—
280C4—2124no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2027510811yescharacteristic
285C2#2027516311yescharacteristic
286C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2313C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2313D53
hypocenterC53#9412
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
8. #16810553C4semidirect
9. #16810543C4semidirect
164. C53#9412semidirect

Maximal subgroups

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

subgroupindex
#202800952
#202732342
#202730452
#202726012
C53×#23132
#941253