Finite Group Databasebeta

← #27281817

Subgroups — #27281817

Order 13568.

512 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277318C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20273248C212yesmaximal
4#20272568C212yesmaximal
5C53×#2503C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810509—24no—
8#16810486C2212yes—
9#16810483—21no—
10C4×C22⋊Dic53—21no—
11#16810392—22no—
12#16810360—24no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400—22no—
16C53×#391C412yescomplemented
17C53×#382C412yescomplemented
18C53×#381—21no—
19C53×#381—21no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C2×#11719917—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—22no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124D414yes—
31C22×Q8×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C22⋊Q8×C53—22no—
36C22⋊Q8×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C412yes—
42C2×C4⋊C4×C53D412yes—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C42×C53C2×C412yes—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53—21no—
51C24×C53C2×D411yescharacteristic
52C2×Q8×C53—22no—
53C2×Q8×C53—42no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53#4112yes—
70C22×Dic53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C42×C53—42no—
79C22⋊Dic53—44no—
80C4⋊Dic53—82no—
81C4×Dic53—82no—
82#11719917—44no—
83C2×#88322—42no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88C23×C53#12011yescharacteristic
89C23×C53#12311yescharacteristic
90Q8×C53—82no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
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—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—81no—
109C2×C4×C53—81no—
110C2×C4×C53C4×D412yes—
111C2×C4×C53#12112yes—
112C2×Dic53—44no—
113C2×Dic53—44no—
114C2×Dic53—44no—
115#88322—82no—
116#9460—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—44no—
125C22×C53#38611yescharacteristic
126C22×C53#41012yes—
127C22×C53#45512yes—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—42no—
132C4×C53—82no—
133C4×C53—81no—
134C4×C53—81no—
135Dic53—84no—
136#2503D5311yescharacteristic
137#2313—531no—
138#1821—531no—
139#1511—532no—
140#1284—532no—
141C2×C53—22no—
142C2×C53—22no—
143C2×C53—41no—
144C2×C53#188511yescharacteristic
145C2×C53#189911yescharacteristic
146C2×C53#191211yescharacteristic
147C2×C22⋊Q8C2×D5311yescharacteristic
148C22×C4⋊C4C2×D5311yescharacteristic
149#407—532no—
150#400—22no—
151#391Dic5312yescomplemented
152#387—1064no—
153#382Dic5312yescomplemented
154#381—21no—
155#381—21no—
156#381—532no—
157#380—1061no—
158C4×C22⋊C4—1061no—
159C4×C22⋊C4C2×D5311yescharacteristic
160#343—1062no—
161#328—1064no—
162C53#946011yescharacteristic, complemented, Sylow
163C22×Q8C2×Dic5311yescharacteristic
164C23×C4C22⋊D5311yescharacteristic
165C23×C4C22⋊D5311yescharacteristic
166C23×C4C22×D5311yescharacteristic
167C2×M16—1062no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4C2×Dic5311yescharacteristic
175C2×C4⋊C4C2×Dic5312yes—
176C2×C4⋊C4C22⋊D5312yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—1062no—
180C2×C42—1062no—
181C2×C42C2×Dic5312yes—
182C22⋊C8—1062no—
183C22⋊C8—1062no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—534no—
190#95—1062no—
191#95—1062no—
192#95—1062no—
193C24C2×C22⋊D5311yescharacteristic
194C2×Q8—22no—
195C2×Q8—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1064no—
207C22×C4C22⋊Dic5312yes—
208C22×C4C22⋊Dic5312yes—
209C22×C4D4×D5312yes—
210C22×C4#1171994712yes—
211C22×C4C22×Dic5311yescharacteristic
212C22×C4C2×C22⋊D5311yescharacteristic
213M16—2124no—
214C2×C8—2122no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—4242no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—2124no—
224C42—42no—
225C42—4242no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×C22⋊Dic5311yescharacteristic
230C23#1313318511yescharacteristic
231C23#1313319811yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—2124no—
253C2×C4—2124no—
254C2×C4—2124no—
255C2×C4D4×Dic5312yes—
256C2×C4#1313318912yes—
257C8—4242no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1681050311yescharacteristic
267C22#1681059712yes—
268C22#1681080012yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4—81no—
276C4—4244no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2027510411yescharacteristic
281C2#2027517411yescharacteristic
282C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2503C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2503D53
hypocenterC53#9460
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
16. C53×#391C4semidirect
17. C53×#382C4semidirect
151. #391Dic53semidirect
153. #382Dic53semidirect
162. C53#9460semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202747952
#202732482
#202725682
C53×#25032
#946053