Finite Group Databasebeta

← #27293565

Subgroups — #27293565

Order 13568.

460 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 77 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#20277258C211yesmaximal, characteristic
2#20273052C212yesmaximal
3#20272658C212yesmaximal, complemented
4#20272649C211yesmaximal, characteristic, complemented
5C53×#1319C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133179C2211yescharacteristic
9#16810585—22no—
10#16810525C411yescharacteristic
11#16810515—22no—
12#16810488C2211yescharacteristic
13#16810470—22no—
14C42×Dic53C411yescharacteristic
15#16810349C2212yes—
16#16810346—22no—
17C53×#391C2211yescharacteristic
18C53×#324C2212yes—
19C22×D212C2×C411yescharacteristic
20C22×C4×D53D412yes—
21C2×#11719912—42no—
22#13133206—22no—
23#13133206—22no—
24Q8×Dic53—41no—
25Q8×Dic53—41no—
26#13133179—41no—
27#13133179—41no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921D412yes—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2×C411yescharacteristic
38C22⋊#88322—22no—
39#13133129—42no—
40#13133124—22no—
41C22×Q8×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C22⋊C8×C53—22no—
45#95×C53—22no—
46C23×D53C22⋊C412yes—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53M1612yescomplemented
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D412yes—
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C2×D212—21no—
60C2×D212—21no—
61C2×D212—41no—
62C2×D212—41no—
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×C4×D53—44no—
66C2×C8×C53—42no—
67C4⋊C4×C53—41no—
68C42×C53—41no—
69#11719921—28no—
70#11719921—42no—
71#11719921—44no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53—42no—
78C4×Dic53—42no—
79C4×Dic53—41no—
80C4×Dic53—41no—
81C4×Dic53M1612yescomplemented
82C2×#88322—42no—
83#11719912—84no—
84C23×C53C2×C22⋊C411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—42no—
89C22×D53—42no—
90Q8×C53—42no—
91Q8×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53C2×M1611yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—41no—
112C2×Dic53#9911yescharacteristic
113C2×Dic53#9911yescharacteristic
114D212—42no—
115D212—42no—
116C4×D53—44no—
117C4×D53—82no—
118C8×C53—82no—
119#88322—82no—
120#11531—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#40711yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C22×C53#41311yescharacteristic
126C2×D53—44no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—82no—
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53—42no—
135C4×C53—42no—
136Dic53—21no—
137Dic53—42no—
138Dic53—41no—
139Dic53—81no—
140Dic53#35412yes—
141C4×#124—531no—
142#1465—532no—
143#1324—532no—
144#1319—531no—
145#1319D5311yescharacteristic, complemented
146C2×C53—21no—
147C2×C53—21no—
148C2×C53#146511yescharacteristic
149C2×C53C2×#35411yescharacteristic
150C2×C53#213211yescharacteristic
151D53—82no—
152C2×#124—531no—
153C2×C4×Q8—531no—
154C2×C4×D4—531no—
155#407—1062no—
156#391C2×D5311yescharacteristic
157#390—531no—
158#388—1062no—
159#382—531no—
160C4×C22⋊C4—1062no—
161C43—531no—
162#324—532no—
163#324C2×D5312yes—
164#323—1062no—
165C53#1153111yescharacteristic, complemented, Sylow
166C22×Q8C4×D5311yescharacteristic
167C22×D4—531no—
168C23×C4—532no—
169C2×M16—2122no—
170#124—1062no—
171#124—1062no—
172C4×Q8—2121no—
173C4×Q8—2121no—
174C4×D4—2121no—
175C4×D4—2121no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4C22×D5311yescharacteristic
179C2×C22⋊C4—532no—
180C2×C22⋊C4—1062no—
181C2×C22⋊C4—1062no—
182C2×C42—531no—
183C2×C42—531no—
184C2×C42—1061no—
185C2×C42—1061no—
186C2×C42—1061no—
187C2×C42C4×D5311yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—1062no—
190C22⋊C8—2122no—
191#95—22no—
192#95—1062no—
193C24—532no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8#1171991212yescomplemented
198C2×D4—1061no—
199C2×D4—1061no—
200C2×D4—2121no—
201C2×D4—2121no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—1062no—
207C22×C4—1062no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—2124no—
211C22×C4C2×C4×D5311yescharacteristic
212C22×C4D4×D5312yes—
213M16—4244no—
214C2×C8—42no—
215C2×C8—2122no—
216C4⋊C4—41no—
217C4⋊C4—2121no—
218C4⋊C4—2122no—
219C22⋊C4—1068no—
220C22⋊C4—2122no—
221C22⋊C4—2124no—
222C42—41no—
223C42—532no—
224C42—1061no—
225C42—1061no—
226C42—1061no—
227C42—2122no—
228C42—2121no—
229C42—2121no—
230C42—2122no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23—2122no—
235C23—2122no—
236C23C22⋊C4×D5311yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—2122no—
240D4—2122no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—531no—
252C2×C4—531no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1061no—
256C2×C4—4242no—
257C2×C4—2124no—
258C2×C4—2122no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4M16×D5311yescharacteristic
264C2×C4C2×#1171991211yescharacteristic
265C8—82no—
266C8—4242no—
267C22—21no—
268C22—21no—
269C22—4242no—
270C22—2124no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22#1681058511yescharacteristic
275C22#99×D5311yescharacteristic
276C22#1681060411yescharacteristic
277C4—21no—
278C4—21no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—532no—
283C4—1061no—
284C4—4241no—
285C4—2122no—
286C4—2121no—
287C2—21no—
288C2—21no—
289C2—4242no—
290C2#2027305011yescharacteristic
291C2#354×D5311yescharacteristic
292C2#2027639911yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1319C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1319D53
hypocenterC53#11531
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #20272658C2semidirect
4. #20272649C2semidirect
5. C53×#1319C2semidirect
50. C2×Q8×C53M16semidirect
81. C4×Dic53M16semidirect
145. #1319D53semidirect
165. C53#11531semidirect
197. C2×Q8#11719912semidirect

Maximal subgroups

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

subgroupindex
#202772582
#202730522
#202726582
#202726492
C53×#13192
#1153153