Finite Group Databasebeta

← #27272406

Subgroups — #27272406

Order 13568.

468 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 99 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#20274857C212yesmaximal, complemented
3#20272541C211yesmaximal, characteristic, complemented
4#20272535C212yesmaximal
5C53×#1272C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133179C411yescharacteristic
9#16810525C2211yescharacteristic
10#16810515—22no—
11#16810488C411yescharacteristic
12#16810470—22no—
13C42×Dic53C2211yescharacteristic
14#16810378C2212yes—
15#16810377—22no—
16#16810377—22no—
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#13133124—22no—
37C22×Q8×C53C2×C411yescharacteristic
38C22×C8×C53—22no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C42×C53C2311yescharacteristic
41#95×C53—22no—
42C23×D53—21no—
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53M1612yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22×C411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—21no—
52C22×Dic53C22⋊C412yes—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C2×D212—21no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212M1612yes—
59C2×C4×D53—44no—
60C2×C4×D53—42no—
61C2×C8×C53—42no—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C42×C53—41no—
66#11719921—28no—
67#11719921—42no—
68#11719921—42no—
69#11719921—42no—
70C4⋊Dic53—42no—
71C4⋊Dic53—41no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—22no—
77C4×Dic53—42no—
78C4×Dic53—42no—
79C2×#88322—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83C22×D53—22no—
84C22×D53—41no—
85C22×D53—41no—
86C22×D53—42no—
87C22×D53—42no—
88Q8×C53—42no—
89Q8×C53—42no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53C2×M1611yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53#13512yes—
102C2×C4×C53#13512yes—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53#10012yes—
112D212—42no—
113D212—42no—
114C4×D53—84no—
115C8×C53—82no—
116#88322—82no—
117#7649—531nomaximal, Sylow
118C22×C53#40711yescharacteristic
119C22×C53#41011yescharacteristic
120C22×C53C2×#10011yescharacteristic
121C22×C53#41612yes—
122C22×C53#42112yes—
123C2×D53—42no—
124C2×D53—44no—
125C2×D53—81no—
126C2×D53—82no—
127C2×D53—81no—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—42no—
133C4×C53—81no—
134Dic53—22no—
135Dic53—41no—
136Dic53—42no—
137Dic53—81no—
138C4×#124—531no—
139#1839—532no—
140#1272—531no—
141#1272D5311yescharacteristic, complemented
142#1270—532no—
143C2×C53#146611yescharacteristic
144C2×C53#148812yes—
145C2×C53#152112yes—
146C2×C53#153311yescharacteristic
147C2×C53#154311yescharacteristic
148D53—82no—
149C2×#124—531no—
150C2×C4×Q8—531no—
151C2×C4×D4—531no—
152#391C2×D5311yescharacteristic
153#390—531no—
154#388—1062no—
155#382—531no—
156C4×C22⋊C4—1062no—
157C43—531no—
158#336—532no—
159#336—1062no—
160#336—1062no—
161#336C2×D5312yes—
162C53#764911yescharacteristic, complemented, Sylow
163C22×Q8C4×D5311yescharacteristic
164C22×D4—531no—
165C23×C4—1061no—
166C22×C8—22no—
167C22×C8—1062no—
168#124—1062no—
169#124—1062no—
170C4×Q8—2122no—
171C4×D4—2122no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4C4×D5311yescharacteristic
175C2×C22⋊C4—1062no—
176C2×C22⋊C4—1061no—
177C2×C22⋊C4—1062no—
178C2×C42—531no—
179C2×C42—531no—
180C2×C42—532no—
181C2×C42—532no—
182C2×C42—1061no—
183C2×C42C22×D5311yescharacteristic
184#95—22no—
185#95—1062no—
186C24—1061no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#1171991212yes—
191C2×D4—532no—
192C2×D4—1061no—
193C2×D4—2121no—
194C2×D4—2121no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—531no—
198C22×C4—532no—
199C22×C4—531no—
200C22×C4—1061no—
201C22×C4—2122no—
202C22×C4—2124no—
203C22×C4C2×C4×D5311yescharacteristic
204C22×C4D4×D5312yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—2122no—
209C2×C8—2122no—
210C2×C8—2122no—
211C4⋊C4—41no—
212C4⋊C4—2121no—
213C4⋊C4—2122no—
214C22⋊C4—1068no—
215C22⋊C4—2122no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C42—41no—
219C42—1062no—
220C42—1061no—
221C42—1061no—
222C42—1062no—
223C42—1062no—
224C42—2122no—
225C42—2122no—
226C23—1062no—
227C23—2121no—
228C23—2121no—
229C23—2122no—
230C23—2122no—
231C23C22⋊C4×D5311yescharacteristic
232Q8—42no—
233Q8—42no—
234D4—2122no—
235D4—2122no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—532no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—4244no—
253C2×C4—2121no—
254C2×C4—2122no—
255C2×C4M16×D5311yescharacteristic
256C2×C4C2×#1171991211yescharacteristic
257C2×C4#1313323812yes—
258C2×C4#1313324012yes—
259C8—82no—
260C8—4242no—
261C22—4242no—
262C22—4241no—
263C22—4241no—
264C22—2122no—
265C22—2124no—
266C22#1681058511yescharacteristic
267C22#1681059511yescharacteristic
268C22#100×D5311yescharacteristic
269C22#1681061812yes—
270C22#1681063812yes—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—1062no—
278C4—4241no—
279C4—2121no—
280C4—2122no—
281C2—4242no—
282C2#2027305411yescharacteristic
283C2#2027314812yes—
284C2#2027329112yes—
285C2#2027333711yescharacteristic
286C2#2027337611yescharacteristic
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#7649
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. #20274857C2semidirect
3. #20272541C2semidirect
5. C53×#1272C2semidirect
141. #1272D53semidirect
162. C53#7649semidirect

Maximal subgroups

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

subgroupindex
#202772582
#202748572
#202725412
#202725352
C53×#12722
#764953