Finite Group Databasebeta

← #27272352

Subgroups — #27272352

Order 13568.

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

normal subgroups only

This group has 520 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#20277258C211yesmaximal, characteristic
2#20274567C212yesmaximal, complemented
3#20272541C211yesmaximal, characteristic, complemented
4#20272535C212yesmaximal
5C53×#1272C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133179C411yescharacteristic
9C2×#13133146—22no—
10C2×#13133127—22no—
11#16810525C2211yescharacteristic
12#16810515—22no—
13#16810488C411yescharacteristic
14#16810470—22no—
15C42×Dic53C2211yescharacteristic
16#16810378C2212yes—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×D212C2×C411yescharacteristic
20C22×C4×D53—21no—
21C22×#88322—22no—
22#13133206—22no—
23#13133206—22no—
24Q8×Dic53—42no—
25#13133179—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35C2×C4×Dic53D412yes—
36#13133146—28no—
37#13133127—44no—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C22×C8×C53—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44C23×D53—21no—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53M1612yes—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22×C411yescharacteristic
52C22×C4×C53C2×D412yes—
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C2×D212—21no—
58C2×D212—41no—
59C2×D212—41no—
60C2×D212M1612yes—
61C2×C4×D53—42no—
62C2×C4×D53—44no—
63C2×C8×C53—24no—
64C2×C8×C53—42no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C42×C53—41no—
68#11719921—28no—
69#11719921—42no—
70#11719921—42no—
71#11719921—42no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53—22no—
79C4×Dic53—24no—
80C4×Dic53—42no—
81C2×#88322—24no—
82C2×#88322—42no—
83C2×#88322—42no—
84C23×C53C2×C22⋊C411yescharacteristic
85C22×D53—22no—
86C22×D53—41no—
87C22×D53—41no—
88C22×D53—42no—
89C22×D53—42no—
90Q8×C53—42no—
91Q8×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53C2×M1611yescharacteristic
103C2×C4×C53AGL(1,ℤ/8)12yes—
104C2×C4×C53#13712yes—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53#10012yes—
113C2×Dic53#10414yes—
114D212—42no—
115D212—42no—
116C4×D53—84no—
117C8×C53—44no—
118#88322—44no—
119#7635—531nomaximal, Sylow
120C22×C53#40711yescharacteristic
121C22×C53#41011yescharacteristic
122C22×C53C2×#10011yescharacteristic
123C22×C53#41712yes—
124C22×C53C2×#10412yes—
125C2×D53—42no—
126C2×D53—44no—
127C2×D53—81no—
128C2×D53—82no—
129C2×D53—81no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—42no—
135C4×C53—81no—
136Dic53—41no—
137Dic53—42no—
138Dic53—81no—
139Dic53#32914yes—
140C4×#124—531no—
141#1767—532no—
142#1272—531no—
143#1272D5311yescharacteristic, complemented
144#1270—532no—
145C2×C53#146611yescharacteristic
146C2×C53#148912yes—
147C2×C53#150912yes—
148C2×C53C2×#32911yescharacteristic
149C2×C53#156011yescharacteristic
150D53—82no—
151C2×#124—531no—
152C2×C4×Q8—531no—
153C2×C4×D4—531no—
154C2×C4⋊C8—1062no—
155C2×#97—1062no—
156#391C2×D5311yescharacteristic
157#390—531no—
158#388—1062no—
159#382—531no—
160C4×C22⋊C4—1062no—
161C43—531no—
162#336—532no—
163#336C2×D5312yes—
164C53#763511yescharacteristic, complemented, Sylow
165C22×Q8C4×D5311yescharacteristic
166C22×D4—531no—
167C23×C4—1061no—
168C22×C8—22no—
169C22×C8—1062no—
170#124—1062no—
171#124—1062no—
172C4×Q8—2122no—
173C4×D4—2122no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4C4×D5311yescharacteristic
177C2×C22⋊C4—1062no—
178C2×C22⋊C4—1061no—
179C2×C22⋊C4—1062no—
180C2×C42—531no—
181C2×C42—531no—
182C2×C42—532no—
183C2×C42—532no—
184C2×C42—1061no—
185C2×C42C22×D5311yescharacteristic
186C4⋊C8—1068no—
187#97—2124no—
188#95—22no—
189#95—1062no—
190C24—1061no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8#1171991212yes—
195C2×D4—532no—
196C2×D4—1061no—
197C2×D4—2121no—
198C2×D4—2121no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—531no—
202C22×C4—532no—
203C22×C4—531no—
204C22×C4—1061no—
205C22×C4—2122no—
206C22×C4—2124no—
207C22×C4C2×C4×D5311yescharacteristic
208C22×C4D4×D5312yes—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—1064no—
213C2×C8—2122no—
214C2×C8—2122no—
215C4⋊C4—41no—
216C4⋊C4—2121no—
217C4⋊C4—2122no—
218C22⋊C4—1068no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C22⋊C4—2122no—
222C42—41no—
223C42—1064no—
224C42—1062no—
225C42—1061no—
226C42—1061no—
227C42—1062no—
228C42—1062no—
229C42—2122no—
230C23—1062no—
231C23—2121no—
232C23—2121no—
233C23—2122no—
234C23—2122no—
235C23C22⋊C4×D5311yescharacteristic
236Q8—42no—
237Q8—42no—
238D4—2122no—
239D4—2122no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—532no—
250C2×C4—534no—
251C2×C4—1062no—
252C2×C4—1061no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—4244no—
257C2×C4—2121no—
258C2×C4—2122no—
259C2×C4M16×D5311yescharacteristic
260C2×C4C2×#1171991211yescharacteristic
261C2×C4#1313324412yes—
262C2×C4#1313324912yes—
263C8—44no—
264C8—2124no—
265C22—4241no—
266C22—4242no—
267C22—4241no—
268C22—2124no—
269C22—2122no—
270C22#1681058511yescharacteristic
271C22#1681059511yescharacteristic
272C22#100×D5311yescharacteristic
273C22#1681062312yes—
274C22#104×D5312yes—
275C4—22no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—42no—
280C4—81no—
281C4—534no—
282C4—4241no—
283C4—2122no—
284C4—2121no—
285C2—4242no—
286C2#2027305411yescharacteristic
287C2#2027315312yes—
288C2#2027323912yes—
289C2#329×D5311yescharacteristic
290C2#2027344911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7635
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #20274567C2semidirect
3. #20272541C2semidirect
5. C53×#1272C2semidirect
143. #1272D53semidirect
164. C53#7635semidirect

Maximal subgroups

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

subgroupindex
#202772582
#202745672
#202725412
#202725352
C53×#12722
#763553