Finite Group Databasebeta

← #27281543

Subgroups — #27281543

Order 13568.

644 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277072C211yesmaximal, characteristic, complemented
2#20273231C212yesmaximal
3#20273045C211yesmaximal, characteristic, complemented
4#20272601C212yesmaximal, complemented
5C53×#2591C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810587C2212yes—
8C4⋊C4×Dic53—24no—
9C4×C22⋊Dic53—21no—
10C4×C22⋊Dic53C2212yes—
11#16810357—24no—
12#16810349—22no—
13C2×#121×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400C412yescomplemented
16C53×#394C412yescomplemented
17C53×#388—22no—
18C53×#386—21no—
19C53×#386—21no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C2×#11719917—22no—
22#13133176—28no—
23C2×C4⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53—22no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124D414yes—
32C22×D4×C53—21no—
33C22×D4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37#121×C53—22no—
38#121×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—21no—
48C2×C42×C53—21no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53C2×C412yes—
52C24×C53—21no—
53C24×C53C2×D411yescharacteristic
54C2×D4×C53—22no—
55C2×D4×C53—44no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—81no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—24no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—81no—
67C22×C4×C53—81no—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22⋊C412yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4112yes—
74C22×Dic53—24no—
75C4⋊C4×C53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—22no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—82no—
82C42×C53—81no—
83C22⋊Dic53—28no—
84C4⋊Dic53—28no—
85C4×Dic53—28no—
86C4×Dic53—82no—
87C4×Dic53—82no—
88#11719917—82no—
89C2×#88322—42no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—41no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—41no—
96C23×C53—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53#12011yescharacteristic
99C23×C53#12311yescharacteristic
100D4×C53—82no—
101D4×C53—82no—
102D4×C53—82no—
103C2×C4×C53—24no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—81no—
121C2×C4×C53#10414yes—
122C2×C4×C53C4×D412yes—
123C2×C4×C53#12112yes—
124C2×Dic53—28no—
125C2×Dic53—44no—
126C2×Dic53—44no—
127#88322—82no—
128#9410—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—44no—
137C22×C53—84no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53#38611yescharacteristic
142C22×C53C2×#10412yes—
143C22×C53#45412yes—
144C4×C53—24no—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—42no—
148C4×C53—82no—
149C4×C53—81no—
150Dic53—48no—
151#2591D5311yescharacteristic
152#2274—531no—
153#1507—532no—
154#1463—531no—
155#1297—532no—
156C2×C53—22no—
157C2×C53—22no—
158C2×C53—41no—
159C2×C53—161no—
160C2×C53#188811yescharacteristic
161C2×C53#189611yescharacteristic
162C2×C53#191311yescharacteristic
163C2×#121C2×D5311yescharacteristic
164C2×#117—1061no—
165C22×C4⋊C4C2×D5311yescharacteristic
166#407—532no—
167#400Dic5312yescomplemented
168#394Dic5312yescomplemented
169#388—22no—
170#386—21no—
171#386—21no—
172C4×C4⋊C4—1064no—
173C4×C22⋊C4—532no—
174C4×C22⋊C4—1061no—
175C4×C22⋊C4C2×D5311yescharacteristic
176#327—1064no—
177#324—1062no—
178C53#941011yescharacteristic, complemented, Sylow
179C22×D4—21no—
180C22×D4C2×Dic5311yescharacteristic
181C23×C4C22⋊D5311yescharacteristic
182C23×C4C22⋊D5311yescharacteristic
183C23×C4C22×D5311yescharacteristic
184C2×M16—1062no—
185#121—22no—
186#121—22no—
187#117—1068no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—1062no—
191C2×C4⋊C4C2×Dic5311yescharacteristic
192C2×C4⋊C4C2×Dic5312yes—
193C2×C4⋊C4C22⋊D5312yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1062no—
199C2×C42—21no—
200C2×C42—1062no—
201C2×C42—1062no—
202C2×C42—1062no—
203C22⋊C8—1062no—
204C22⋊C8—1062no—
205#95—22no—
206#95—22no—
207#95—534no—
208#95—1062no—
209#95C2×Dic5312yes—
210C24—21no—
211C24C2×C22⋊D5311yescharacteristic
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—1064no—
227C22×C4C22⋊Dic5312yes—
228C22×C4C22⋊Dic5312yes—
229C22×C4D4×D5312yes—
230C22×C4#1171994712yes—
231C22×C4C22×Dic5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—4242no—
234C2×C8—2122no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—1068no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—1068no—
244C42—81no—
245C42—1068no—
246C42—4242no—
247C42—4242no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—42no—
255C23C2×C22⋊Dic5311yescharacteristic
256C23#1313318511yescharacteristic
257C23#1313319811yescharacteristic
258D4—82no—
259D4—82no—
260D4—82no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—1068no—
280C2×C4—2124no—
281C2×C4—2124no—
282C2×C4#1313314414yes—
283C2×C4D4×Dic5312yes—
284C2×C4#1313318912yes—
285C8—4242no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1681050311yescharacteristic
299C22C2×#1313314412yes—
300C22#1681079512yes—
301C4—22no—
302C4—22no—
303C4—24no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—2128no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2#2027511911yescharacteristic
313C2#2027515911yescharacteristic
314C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2591C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2591D53
hypocenterC53#9410
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. #20277072C2semidirect
3. #20273045C2semidirect
4. #20272601C2semidirect
15. C53×#400C4semidirect
16. C53×#394C4semidirect
167. #400Dic53semidirect
168. #394Dic53semidirect
178. C53#9410semidirect

Maximal subgroups

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

subgroupindex
#202770722
#202732312
#202730452
#202726012
C53×#25912
#941053