Finite Group Databasebeta

← #27293499

Subgroups — #27293499

Order 13568.

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

normal subgroups only

This group has 434 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#20277336C211yesmaximal, characteristic
2#20273043C212yesmaximal
3#20272661C211yesmaximal, characteristic, complemented
4#20272658C212yesmaximal, complemented
5C53×#1325C211yesmaximal, characteristic, complemented
6C2×#13133175C411yescharacteristic
7C2×#13133174C411yescharacteristic
8C2×C4⋊C4×D53C2211yescharacteristic
9#16810585—22no—
10#16810518—22no—
11#16810494—22no—
12#16810492C411yescharacteristic
13#16810489C411yescharacteristic
14#16810488C2211yescharacteristic
15#16810349C2212yes—
16#16810348—22no—
17C53×#384C2211yescharacteristic
18C53×#324C2212yes—
19C22×C4×D53C2×C411yescharacteristic
20C22×C4×D53D412yes—
21C2×#11719912—42no—
22#13133175—42no—
23#13133174—22no—
24#13133174—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—22no—
29C2×#11719919D412yes—
30C4⋊C4×D53—42no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C22⋊#88322—22no—
36#13133129—42no—
37#13133124—22no—
38#13133124—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—22no—
44C23×D53C22⋊C411yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22×C411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—21no—
50C22×Dic53—21no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C412yes—
53C2×C4×D53—22no—
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—82no—
60C2×C8×C53—42no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—82no—
64C42×C53—21no—
65C42×C53M1612yescomplemented
66#11719921—24no—
67#11719921—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70#11719919—24no—
71#11719919—42no—
72#11719919—82no—
73C4×Dic53—21no—
74C4×Dic53—42no—
75C4×Dic53M1612yescomplemented
76C2×#88322—42no—
77#11719912—84no—
78C23×C53C2×C22⋊C411yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—41no—
82C22×D53—41no—
83C22×D53#9912yes—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104C2×Dic53#10112yes—
105C4×D53—24no—
106C4×D53—24no—
107C4×D53—44no—
108C4×D53—82no—
109C8×C53—82no—
110#88322—82no—
111#11514—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#40711yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C2×D53—22no—
118C2×D53—22no—
119C2×D53—22no—
120C2×D53—42no—
121C2×D53—42no—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—42no—
125C4×C53—42no—
126C4×C53—81no—
127Dic53—22no—
128Dic53—42no—
129Dic53—42no—
130Dic53—81no—
131#2317—531no—
132#1463—532no—
133#1325—531no—
134#1325D5311yescharacteristic, complemented
135#1324—532no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53#146311yescharacteristic
139C2×C53#212711yescharacteristic
140C2×C53#213311yescharacteristic
141D53—42no—
142C2×#117—531no—
143C2×#117—531no—
144C22×C4⋊C4—531no—
145#407—1062no—
146#389—1062no—
147#385—1062no—
148#384—531no—
149#384C2×D5311yescharacteristic
150#382—531no—
151#382—531no—
152#324—532no—
153#324—1062no—
154#324C2×D5312yes—
155C53#1151411yescharacteristic, complemented, Sylow
156C23×C4—531no—
157C23×C4—532no—
158C2×M16—2122no—
159#117—1062no—
160#117—1062no—
161#117—2122no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4—1062no—
168C2×C4⋊C4—2122no—
169C2×C4⋊C4C4×D5311yescharacteristic
170C2×C4⋊C4C22×D5311yescharacteristic
171C2×C22⋊C4—1061no—
172C2×C22⋊C4—1061no—
173C2×C42—531no—
174C2×C42—1061no—
175C2×C42C4×D5311yescharacteristic
176C22⋊C8—22no—
177C22⋊C8—1062no—
178C22⋊C8—2122no—
179#95—1062no—
180#95—1062no—
181C24—531no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—531no—
185C22×C4—532no—
186C22×C4—1062no—
187C22×C4—1062no—
188C22×C4—1062no—
189C22×C4—1062no—
190C22×C4—1061no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4—1061no—
194C22×C4—4242no—
195C22×C4C2×C4×D5311yescharacteristic
196C22×C4D4×D5312yes—
197M16—4244no—
198C2×C8—42no—
199C2×C8—2122no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—82no—
203C4⋊C4—1064no—
204C4⋊C4—4242no—
205C4⋊C4—2122no—
206C4⋊C4—2121no—
207C4⋊C4—2121no—
208C22⋊C4—1064no—
209C22⋊C4—2122no—
210C42—21no—
211C42—532no—
212C42—1061no—
213C42—2122no—
214C42#1171991212yescomplemented
215C23—532no—
216C23—1061no—
217C23—1061no—
218C23—2121no—
219C23—2121no—
220C23C22⋊C4×D5311yescharacteristic
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—532no—
231C2×C4—1064no—
232C2×C4—1064no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—4242no—
238C2×C4—2124no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4—2122no—
244C2×C4M16×D5311yescharacteristic
245C2×C4C2×#1171991211yescharacteristic
246C8—82no—
247C8—4242no—
248C22—21no—
249C22—21no—
250C22—1062no—
251C22—1062no—
252C22—1062no—
253C22—2122no—
254C22—2122no—
255C22#1681058511yescharacteristic
256C22#99×D5311yescharacteristic
257C22#101×D5311yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—1062no—
264C4—4241no—
265C4—2122no—
266C4—2122no—
267C2—21no—
268C2—21no—
269C2—2122no—
270C2#2027304311yescharacteristic
271C2#2027638111yescharacteristic
272C2#2027640311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1325C2
socleC22×C53C2×#99
prime radicalC23×D53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1325D53
hypocenterC53#11514
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #20272661C2semidirect
4. #20272658C2semidirect
5. C53×#1325C2semidirect
65. C42×C53M16semidirect
75. C4×Dic53M16semidirect
134. #1325D53semidirect
155. C53#11514semidirect
214. C42#11719912semidirect

Maximal subgroups

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

subgroupindex
#202773362
#202730432
#202726612
#202726582
C53×#13252
#1151453