Finite Group Databasebeta

← #27293573

Subgroups — #27293573

Order 13568.

504 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 75 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#20277439C211yesmaximal, characteristic
2#20273052C212yesmaximal, complemented
3#20272658C212yesmaximal, complemented
4#20272647C211yesmaximal, characteristic, complemented
5C53×#1318C211yesmaximal, characteristic, complemented
6C2×#13133213C411yescharacteristic
7C2×D4×Dic53C411yescharacteristic
8C2×#13133179C2211yescharacteristic
9#16810585—22no—
10#16810525C411yescharacteristic
11#16810500—22no—
12#16810499—22no—
13#16810488C2211yescharacteristic
14#16810469C411yescharacteristic
15#16810349C2212yes—
16#16810346—22no—
17C53×#390C2211yescharacteristic
18C53×#324C2212yes—
19C22×D212C2×C411yescharacteristic
20C22×C22⋊D53—22no—
21C23×Dic53—21no—
22C22×C4×D53D412yes—
23C2×#11719912—42no—
24#13133213—22no—
25#13133213—22no—
26D4×Dic53—42no—
27#13133179—42no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C22⋊Dic53—21no—
30C2×#11719921—22no—
31C2×#11719921D412yes—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C22⋊#88322—22no—
35#13133129—42no—
36#13133124—22no—
37#13133124—21no—
38#13133124—21no—
39C22×D4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—22no—
44C24×C53—21no—
45C23×D53C22⋊C412yes—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53M1612yescomplemented
50C22×C4×C53—21no—
51C22×C4×C53C22×C411yescharacteristic
52C22×C4×C53C2×D412yes—
53C2×C22⋊D53—28no—
54C2×C22⋊D53—82no—
55C2×C22⋊D53—82no—
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53—42no—
59C22×Dic53—82no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C2×D212—21no—
63C2×D212—21no—
64C2×D212—41no—
65C2×D212—41no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—82no—
69C2×C8×C53—42no—
70C4⋊C4×C53—41no—
71C22⋊C4×C53—82no—
72C42×C53—41no—
73C22⋊Dic53—42no—
74#11719921—42no—
75#11719921—82no—
76C4⋊Dic53—41no—
77C4×Dic53—21no—
78C4×Dic53—22no—
79C4×Dic53M1612yescomplemented
80C2×#88322—42no—
81#11719912—84no—
82C23×C53—22no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—42no—
92C22×D53—42no—
93D4×C53—42no—
94D4×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53C2×M1611yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C22⋊D53—88no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53—41no—
113C2×Dic53—84no—
114C2×Dic53#9912yes—
115D212—42no—
116D212—42no—
117C4×D53—44no—
118C4×D53—82no—
119C8×C53—82no—
120#88322—82no—
121#11533—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—44no—
125C22×C53—42no—
126C22×C53—82no—
127C22×C53—81no—
128C22×C53—81no—
129C22×C53#40711yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53#41311yescharacteristic
132C2×D53—44no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—82no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53—81no—
141Dic53—22no—
142Dic53—42no—
143Dic53—41no—
144Dic53—81no—
145#2328—531no—
146#1465—532no—
147#1324—532no—
148#1318—531no—
149#1318D5311yescharacteristic, complemented
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—82no—
153C2×C53#146511yescharacteristic
154C2×C53#212711yescharacteristic
155C2×C53#213411yescharacteristic
156D53—82no—
157C2×#127—531no—
158C2×C4×D4—531no—
159C2×C4×D4—531no—
160#407—1062no—
161#390—531no—
162#390C2×D5311yescharacteristic
163#386—1062no—
164#386—1062no—
165#382—531no—
166#376—531no—
167#324—532no—
168#324C2×D5312yes—
169#323—1062no—
170C53#1153311yescharacteristic, complemented, Sylow
171C22×D4—531no—
172C22×D4—1062no—
173C22×D4C4×D5311yescharacteristic
174C23×C4—532no—
175C23×C4—1061no—
176C2×M16—2122no—
177#127—1062no—
178#127—1062no—
179C4×D4—2122no—
180C4×D4—2122no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4C22×D5311yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42C4×D5311yescharacteristic
190C22⋊C8—22no—
191C22⋊C8—1062no—
192C22⋊C8—2122no—
193#95—1061no—
194#95—1061no—
195#95—1062no—
196C24—21no—
197C24—532no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—1068no—
202C2×D4—1061no—
203C2×D4—1061no—
204C2×D4—4242no—
205C2×D4—4242no—
206C2×D4—2121no—
207C2×D4—2121no—
208C2×D4#1171991212yescomplemented
209C22×C4—21no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—4242no—
217C22×C4—4242no—
218C22×C4—2122no—
219C22×C4C2×C4×D5311yescharacteristic
220C22×C4D4×D5312yes—
221M16—4244no—
222C2×C8—42no—
223C2×C8—2122no—
224C4⋊C4—41no—
225C4⋊C4—2121no—
226C22⋊C4—82no—
227C22⋊C4—4242no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C42—41no—
231C42—532no—
232C42—1062no—
233C42—1061no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—1062no—
240C23—1062no—
241C23—1062no—
242C23—2122no—
243C23—2122no—
244C23C22⋊C4×D5311yescharacteristic
245D4—42no—
246D4—42no—
247D4—4248no—
248D4—2122no—
249D4—2122no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—532no—
258C2×C4—1062no—
259C2×C4—1061no—
260C2×C4—4242no—
261C2×C4—4244no—
262C2×C4—2124no—
263C2×C4—2121no—
264C2×C4—2122no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4M16×D5311yescharacteristic
270C2×C4C2×#1171991211yescharacteristic
271C8—82no—
272C8—4242no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—82no—
278C22—81no—
279C22—81no—
280C22—4242no—
281C22—2124no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22#1681058511yescharacteristic
286C22#99×D5311yescharacteristic
287C22#1681060411yescharacteristic
288C4—21no—
289C4—21no—
290C4—42no—
291C4—81no—
292C4—1062no—
293C4—4241no—
294C4—2122no—
295C4—2121no—
296C2—21no—
297C2—21no—
298C2—82no—
299C2—4242no—
300C2#2027305011yescharacteristic
301C2#2027638111yescharacteristic
302C2#2027640511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1318C2
socleC22×C53C2×#99
prime radicalC2×#13133213C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1318D53
hypocenterC53#11533
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #20273052C2semidirect
3. #20272658C2semidirect
4. #20272647C2semidirect
5. C53×#1318C2semidirect
49. C2×D4×C53M16semidirect
79. C4×Dic53M16semidirect
149. #1318D53semidirect
170. C53#11533semidirect
208. C2×D4#11719912semidirect

Maximal subgroups

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

subgroupindex
#202774392
#202730522
#202726582
#202726472
C53×#13182
#1153353