Finite Group Databasebeta

← #27293581

Subgroups — #27293581

Order 13568.

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

normal subgroups only

This group has 440 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#20277429C211yesmaximal, characteristic
2#20273052C212yesmaximal
3#20272660C212yesmaximal, complemented
4#20272649C211yesmaximal, characteristic, complemented
5C53×#1319C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133179C2211yescharacteristic
9#16810585—22no—
10#16810525C411yescharacteristic
11#16810512—22no—
12#16810497—22no—
13#16810492C2211yescharacteristic
14#16810469C411yescharacteristic
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—42no—
25#13133179—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921D412yes—
32C2×C4×Dic53—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C22⋊#88322—22no—
36#13133129—42no—
37#13133124—21no—
38#13133124—21no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C22⋊C8×C53—22no—
43#95×C53—22no—
44C23×D53C22⋊C412yes—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53M1612yescomplemented
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22×C411yescharacteristic
52C22×C4×C53C2×D412yes—
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C2×D212—21no—
58C2×D212—21no—
59C2×D212—41no—
60C2×D212—41no—
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—82no—
64C2×C8×C53—42no—
65C4⋊C4×C53—41no—
66C42×C53—41no—
67#11719921—28no—
68#11719921—42no—
69#11719921—82no—
70C4⋊Dic53—42no—
71C4⋊Dic53—41no—
72#11719919—82no—
73C4×Dic53—21no—
74C4×Dic53—22no—
75C4×Dic53—42no—
76C4×Dic53M1612yescomplemented
77C2×#88322—42no—
78#11719912—84no—
79C23×C53C2×C22⋊C411yescharacteristic
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—42no—
84C22×D53—42no—
85Q8×C53—42no—
86Q8×C53—42no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C2×M1611yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106C2×Dic53—41no—
107C2×Dic53#9912yes—
108D212—42no—
109D212—42no—
110C4×D53—44no—
111C4×D53—82no—
112C8×C53—82no—
113#88322—82no—
114#11535—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53#41311yescharacteristic
120C2×D53—44no—
121C2×D53—42no—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—82no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53—42no—
129C4×C53—81no—
130Dic53—22no—
131Dic53—42no—
132Dic53—41no—
133Dic53—81no—
134#2326—531no—
135#1465—532no—
136#1325—532no—
137#1319—531no—
138#1319D5311yescharacteristic, complemented
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#146511yescharacteristic
142C2×C53#212711yescharacteristic
143C2×C53#213511yescharacteristic
144D53—82no—
145C2×#124—531no—
146C2×C4×Q8—531no—
147C2×C4×D4—531no—
148#407—1062no—
149#391C2×D5311yescharacteristic
150#390—531no—
151#388—1062no—
152#385—1062no—
153#384—531no—
154#376—531no—
155#324—532no—
156#324C2×D5312yes—
157#323—1062no—
158C53#1153511yescharacteristic, complemented, Sylow
159C22×Q8C4×D5311yescharacteristic
160C22×D4—531no—
161C23×C4—532no—
162C2×M16—2122no—
163#124—1062no—
164#124—1062no—
165C4×Q8—2122no—
166C4×D4—2122no—
167C2×C4⋊C4—531no—
168C2×C4⋊C4—1062no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4C22×D5311yescharacteristic
171C2×C22⋊C4—532no—
172C2×C22⋊C4—1062no—
173C2×C22⋊C4—1062no—
174C2×C42—531no—
175C2×C42—531no—
176C2×C42—1061no—
177C2×C42C4×D5311yescharacteristic
178C22⋊C8—22no—
179C22⋊C8—1062no—
180C22⋊C8—2122no—
181#95—22no—
182#95—1061no—
183#95—1061no—
184C24—532no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8#1171991212yescomplemented
189C2×D4—1061no—
190C2×D4—1061no—
191C2×D4—2121no—
192C2×D4—2121no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—1062no—
198C22×C4—1062no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—4242no—
202C22×C4C2×C4×D5311yescharacteristic
203C22×C4D4×D5312yes—
204M16—4244no—
205C2×C8—42no—
206C2×C8—2122no—
207C4⋊C4—41no—
208C4⋊C4—4242no—
209C4⋊C4—2121no—
210C4⋊C4—2122no—
211C22⋊C4—1068no—
212C22⋊C4—4242no—
213C22⋊C4—2122no—
214C42—41no—
215C42—532no—
216C42—1062no—
217C42—1061no—
218C42—2122no—
219C23—1062no—
220C23—1062no—
221C23—1062no—
222C23—2122no—
223C23—2122no—
224C23C22⋊C4×D5311yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—2122no—
228D4—2122no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—532no—
240C2×C4—1062no—
241C2×C4—1061no—
242C2×C4—4242no—
243C2×C4—2124no—
244C2×C4—2121no—
245C2×C4—2122no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4M16×D5311yescharacteristic
251C2×C4C2×#1171991211yescharacteristic
252C8—82no—
253C8—4242no—
254C22—21no—
255C22—21no—
256C22—4242no—
257C22—2124no—
258C22—2122no—
259C22—2122no—
260C22—2122no—
261C22#1681058511yescharacteristic
262C22#99×D5311yescharacteristic
263C22#1681060411yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—1062no—
270C4—4241no—
271C4—2122no—
272C4—2121no—
273C2—21no—
274C2—21no—
275C2—4242no—
276C2#2027305011yescharacteristic
277C2#2027638311yescharacteristic
278C2#2027640811yescharacteristic
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#11535
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. #20272660C2semidirect
4. #20272649C2semidirect
5. C53×#1319C2semidirect
48. C2×Q8×C53M16semidirect
76. C4×Dic53M16semidirect
138. #1319D53semidirect
158. C53#11535semidirect
188. C2×Q8#11719912semidirect

Maximal subgroups

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

subgroupindex
#202774292
#202730522
#202726602
#202726492
C53×#13192
#1153553