Finite Group Databasebeta

← #27281533

Subgroups — #27281533

Order 13568.

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

normal subgroups only

This group has 618 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
2#20273235C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20272601C212yesmaximal
5C53×#2583C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810587C2212yes—
8C4⋊C4×Dic53—24no—
9C4×C22⋊Dic53—21no—
10C4×C22⋊Dic53C2212yes—
11#16810360—24no—
12#16810349—22no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#398C412yescomplemented
16C53×#395C412yescomplemented
17C53×#389—22no—
18C53×#385—21no—
19C53×#385—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×Q8×C53C2×C411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53C2311yescharacteristic
36C22⋊Q8×C53—22no—
37C22⋊Q8×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C412yes—
45C2×C4⋊C4×C53D412yes—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C42×C53—21no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53C2×C412yes—
52C24×C53C2×D411yescharacteristic
53C2×Q8×C53—22no—
54C2×Q8×C53—81no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—24no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—81no—
64C22×C4×C53—81no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4112yes—
71C22×Dic53—24no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—44no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—82no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C42×C53—81no—
82C22⋊Dic53—28no—
83C4⋊Dic53—28no—
84C4×Dic53—28no—
85C4×Dic53—82no—
86C4×Dic53—82no—
87#11719917—82no—
88C2×#88322—42no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—41no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53#12011yescharacteristic
94C23×C53#12311yescharacteristic
95Q8×C53—82no—
96C2×C4×C53—24no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—24no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—81no—
116C2×C4×C53#10414yes—
117C2×C4×C53C4×D412yes—
118C2×C4×C53#12112yes—
119C2×Dic53—28no—
120C2×Dic53—44no—
121C2×Dic53—44no—
122#88322—82no—
123#9408—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—44no—
132C22×C53#38611yescharacteristic
133C22×C53C2×#10412yes—
134C22×C53#45612yes—
135C4×C53—24no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—44no—
139C4×C53—42no—
140C4×C53—81no—
141C4×C53—81no—
142Dic53—48no—
143#2583D5311yescharacteristic
144#2274—531no—
145#1508—532no—
146#1463—531no—
147#1297—532no—
148C2×C53—22no—
149C2×C53—22no—
150C2×C53—41no—
151C2×C53#188611yescharacteristic
152C2×C53#189611yescharacteristic
153C2×C53#191311yescharacteristic
154C2×C22⋊Q8C2×D5311yescharacteristic
155C2×#117—1061no—
156C22×C4⋊C4C2×D5311yescharacteristic
157#407—532no—
158#398Dic5312yescomplemented
159#395Dic5312yescomplemented
160#389—22no—
161#385—21no—
162#385—21no—
163C4×C4⋊C4—1064no—
164C4×C22⋊C4—532no—
165C4×C22⋊C4—1061no—
166C4×C22⋊C4C2×D5311yescharacteristic
167#328—1064no—
168#324—1062no—
169C53#940811yescharacteristic, complemented, Sylow
170C22×Q8C2×Dic5311yescharacteristic
171C23×C4C22⋊D5311yescharacteristic
172C23×C4C22⋊D5311yescharacteristic
173C23×C4C22×D5311yescharacteristic
174C2×M16—1062no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#117—1068no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—1062no—
184C2×C4⋊C4C2×Dic5311yescharacteristic
185C2×C4⋊C4C2×Dic5312yes—
186C2×C4⋊C4C22⋊D5312yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—1062no—
190C2×C42—21no—
191C2×C42—1062no—
192C2×C42—1062no—
193C2×C42—1062no—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196#95—22no—
197#95—22no—
198#95—534no—
199#95—1062no—
200#95C2×Dic5312yes—
201C24C2×C22⋊D5311yescharacteristic
202C2×Q8—22no—
203C2×Q8—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—1064no—
215C22×C4C22⋊Dic5312yes—
216C22×C4C22⋊Dic5312yes—
217C22×C4D4×D5312yes—
218C22×C4#1171994712yes—
219C22×C4C22×Dic5311yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221M16—4242no—
222C2×C8—2122no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—1068no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—1068no—
234C42—81no—
235C42—1068no—
236C42—4242no—
237C42—4242no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×C22⋊Dic5311yescharacteristic
242C23#1313318511yescharacteristic
243C23#1313319811yescharacteristic
244Q8—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—1068no—
266C2×C4—2124no—
267C2×C4—2124no—
268C2×C4#1313314414yes—
269C2×C4D4×Dic5312yes—
270C2×C4#1313318912yes—
271C8—4242no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1681050311yescharacteristic
281C22C2×#1313314412yes—
282C22#1681080512yes—
283C4—22no—
284C4—22no—
285C4—24no—
286C4—44no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—2128no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2027510911yescharacteristic
295C2#2027515911yescharacteristic
296C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2583C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2583D53
hypocenterC53#9408
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
15. C53×#398C4semidirect
16. C53×#395C4semidirect
158. #398Dic53semidirect
159. #395Dic53semidirect
169. C53#9408semidirect

Maximal subgroups

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

subgroupindex
#202770722
#202732352
#202730452
#202726012
C53×#25832
#940853