Finite Group Databasebeta

← #27293563

Subgroups — #27293563

Order 13568.

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

normal subgroups only

This group has 444 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#20277254C211yesmaximal, characteristic
2#20273057C212yesmaximal, complemented
3#20272659C211yesmaximal, characteristic, complemented
4#20272646C212yesmaximal
5C53×#1324C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic
7C2×#13133182C2211yescharacteristic
8C2×#13133179C411yescharacteristic
9#16810677—22no—
10#16810528C411yescharacteristic
11#16810524C2211yescharacteristic
12#16810512—22no—
13#16810470—22no—
14C42×Dic53C411yescharacteristic
15#16810349C2212yes—
16#16810348—22no—
17C53×#382C2211yescharacteristic
18C53×#324C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22#13133205—22no—
23#13133205—22no—
24#13133182—41no—
25#13133182—41no—
26#13133179—41no—
27#13133179—41no—
28C2×#11719919D412yes—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37#13133146—42no—
38C22⋊#88322—22no—
39#13133127—42no—
40#13133124—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C22⋊C8×C53—22no—
45#95×C53—22no—
46C23×D53—21no—
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—41no—
61C2×C4×D53—41no—
62C2×Dic106—21no—
63C2×Dic106—21no—
64C2×Dic106—41no—
65C2×Dic106—41no—
66C2×C8×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C42×C53—21no—
70C42×C53M1612yes—
71#11719921—28no—
72#11719921—44no—
73#11719921—42no—
74#11719919—42no—
75C4×Dic53—22no—
76C4×Dic53—22no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—42no—
80C4×Dic53—42no—
81C4×Dic53—41no—
82C2×#88322—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84C22×D53—22no—
85C22×D53—42no—
86C22×D53—42no—
87C22×D53—41no—
88C22×D53—41no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53C2×M1611yescharacteristic
99C2×C4×C53C2×M1611yescharacteristic
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53#9911yescharacteristic
110C2×Dic53#9911yescharacteristic
111D212—42no—
112D212—42no—
113C4×D53—82no—
114C4×D53—82no—
115Dic106—42no—
116Dic106—42no—
117C8×C53—82no—
118#88322—82no—
119#11530—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#40711yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41311yescharacteristic
125C2×D53—44no—
126C2×D53—42no—
127C2×D53—82no—
128C2×D53—81no—
129C2×D53—81no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53—42no—
134C4×C53—81no—
135Dic53—21no—
136Dic53—42no—
137Dic53—41no—
138Dic53—42no—
139Dic53#35512yes—
140C4×#124—531no—
141#1467—532no—
142#1324—531no—
143#1324D5311yescharacteristic, complemented
144#1318—532no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53#146511yescharacteristic
148C2×C53#212611yescharacteristic
149C2×C53C2×#35511yescharacteristic
150D53—82no—
151C2×#124—531no—
152C2×C4×Q8—531no—
153C2×C4×D4—531no—
154#432—1062no—
155#391—531no—
156#390—531no—
157#388—1062no—
158#382C2×D5311yescharacteristic
159C4×C22⋊C4—1062no—
160C43—531no—
161#324—532no—
162#324—1062no—
163#324C2×D5312yes—
164C53#1153011yescharacteristic, complemented, Sylow
165C22×Q8—531no—
166C22×D4—531no—
167C23×C4—1061no—
168#124—1062no—
169#124—1062no—
170C4×Q8—2121no—
171C4×Q8—2121no—
172C4×D4—2121no—
173C4×D4—2121no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4C4×D5311yescharacteristic
176C2×C4⋊C4C22×D5311yescharacteristic
177C2×C22⋊C4—1062no—
178C2×C22⋊C4—1061no—
179C2×C22⋊C4—1062no—
180C2×C42—531no—
181C2×C42—532no—
182C2×C42—531no—
183C2×C42—1061no—
184C2×C42—1061no—
185C2×C42C4×D5311yescharacteristic
186C4⋊C8—2122no—
187C22⋊C8—22no—
188C22⋊C8—1062no—
189#97—2122no—
190#95—22no—
191#95—1062no—
192C24—1061no—
193C2×Q8—1061no—
194C2×Q8—1061no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C2×D4—532no—
198C2×D4—1061no—
199C2×D4—2121no—
200C2×D4—2121no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—531no—
204C22×C4—532no—
205C22×C4—531no—
206C22×C4—1061no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4—2124no—
210C22×C4C2×C4×D5311yescharacteristic
211C22×C4D4×D5312yes—
212C2×C8—42no—
213C2×C8—2122no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—2122no—
217C22⋊C4—1068no—
218C22⋊C4—2122no—
219C22⋊C4—2124no—
220C42—21no—
221C42—1062no—
222C42—1062no—
223C42—1061no—
224C42—1061no—
225C42—2121no—
226C42—2122no—
227C42—2122no—
228C42#1171991212yes—
229C23—1062no—
230C23—2122no—
231C23—2122no—
232C23—2121no—
233C23—2121no—
234C23C22⋊C4×D5311yescharacteristic
235Q8—2122no—
236Q8—2122no—
237D4—2122no—
238D4—2122no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—531no—
249C2×C4—531no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1061no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—4242no—
257C2×C4—4242no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4M16×D5311yescharacteristic
262C2×C4C2×#1171991211yescharacteristic
263C8—82no—
264C8—4242no—
265C22—21no—
266C22—21no—
267C22—4242no—
268C22—4241no—
269C22—4241no—
270C22—2124no—
271C22—2122no—
272C22#1681058511yescharacteristic
273C22#99×D5311yescharacteristic
274C22#1681060611yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—81no—
280C4—532no—
281C4—1061no—
282C4—2122no—
283C4—2121no—
284C4—2122no—
285C2—21no—
286C2—21no—
287C2—4242no—
288C2#2027305211yescharacteristic
289C2#2027637911yescharacteristic
290C2#355×D5311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1324C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1324D53
hypocenterC53#11530
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
2. #20273057C2semidirect
3. #20272659C2semidirect
5. C53×#1324C2semidirect
143. #1324D53semidirect
164. C53#11530semidirect

Maximal subgroups

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

subgroupindex
#202772542
#202730572
#202726592
#202726462
C53×#13242
#1153053