Finite Group Databasebeta

← #27281927

Subgroups — #27281927

Order 13568.

518 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277318C211yesmaximal, characteristic
2#20274814C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20272567C212yesmaximal
5C53×#2484C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810509—24no—
8#16810486—21no—
9#16810483—21no—
10C4×C22⋊Dic53C2212yes—
11#16810382—24no—
12#16810349—22no—
13C2×#117×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400C412yescomplemented
16C53×#398C412yescomplemented
17C53×#387—22no—
18C53×#385—21no—
19C53×#381—21no—
20C53×#381C2211yescharacteristic
21C2×#11719917—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53D414yes—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—24no—
29#13133124—22no—
30C23×C4×C53D411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C23×C4×C53C2311yescharacteristic
33#117×C53—22no—
34#117×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53D412yes—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C42×C53—21no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—21no—
49#95×C53C2×C412yes—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4112yes—
67C22×Dic53—24no—
68C4⋊C4×C53—24no—
69C4⋊C4×C53—24no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—44no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C42×C53—42no—
77C22⋊Dic53—44no—
78C4⋊Dic53—82no—
79C4×Dic53—44no—
80#11719917—44no—
81C2×#88322—42no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—41no—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53#12011yescharacteristic
87C23×C53#12311yescharacteristic
88C2×C4×C53—24no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
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—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—81no—
108C2×C4×C53C4×D412yes—
109C2×C4×C53#12112yes—
110C2×Dic53—44no—
111C2×Dic53—44no—
112C2×Dic53—44no—
113#88322—82no—
114#9482—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—44no—
123C22×C53#38611yescharacteristic
124C22×C53#42112yes—
125C22×C53#45812yes—
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—42no—
129C4×C53—42no—
130C4×C53—82no—
131C4×C53—81no—
132C4×C53—81no—
133Dic53—84no—
134#2484D5311yescharacteristic
135#2313—531no—
136#1826—532no—
137#1463—531no—
138#1284—532no—
139C2×C53—22no—
140C2×C53—22no—
141C2×C53—41no—
142C2×C53#189011yescharacteristic
143C2×C53#189711yescharacteristic
144C2×C53#191211yescharacteristic
145C2×#117C2×D5311yescharacteristic
146C22×C4⋊C4C2×D5311yescharacteristic
147#407—532no—
148#400Dic5312yescomplemented
149#398Dic5312yescomplemented
150#387—22no—
151#387—1064no—
152#385—21no—
153#381—21no—
154#381—1061no—
155#381C2×D5311yescharacteristic
156#380—1061no—
157C4×C22⋊C4—532no—
158#339—1064no—
159#324—1062no—
160C53#948211yescharacteristic, complemented, Sylow
161C23×C4C22⋊D5311yescharacteristic
162C23×C4C22⋊D5311yescharacteristic
163C23×C4C22×D5311yescharacteristic
164C2×M16—1062no—
165#117—22no—
166#117—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4C2×Dic5311yescharacteristic
172C2×C4⋊C4C2×Dic5311yescharacteristic
173C2×C4⋊C4C2×Dic5312yes—
174C2×C4⋊C4C22⋊D5312yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—1062no—
178C2×C42—21no—
179C2×C42—534no—
180C22⋊C8—1062no—
181C22⋊C8—1062no—
182#95—21no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—1062no—
187#95—1064no—
188#95—1062no—
189#95C2×Dic5312yes—
190C24C2×C22⋊D5311yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—1064no—
202C22×C4C22⋊Dic5312yes—
203C22×C4C22⋊Dic5312yes—
204C22×C4D4×D5312yes—
205C22×C4#1171994712yes—
206C22×C4C22×Dic5311yescharacteristic
207C22×C4C2×C22⋊D5311yescharacteristic
208M16—2124no—
209C2×C8—2122no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—4242no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—2124no—
220C42—42no—
221C42—2124no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×C22⋊Dic5311yescharacteristic
226C23#1313318511yescharacteristic
227C23#1313319811yescharacteristic
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—2124no—
249C2×C4—2124no—
250C2×C4—2124no—
251C2×C4D4×Dic5312yes—
252C2×C4#1313318912yes—
253C8—4242no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#1681050311yescharacteristic
263C22#1681064012yes—
264C22#1681081212yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—81no—
271C4—81no—
272C4—4244no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#2027512911yescharacteristic
277C2#2027516411yescharacteristic
278C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2484C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2484D53
hypocenterC53#9482
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
15. C53×#400C4semidirect
16. C53×#398C4semidirect
148. #400Dic53semidirect
149. #398Dic53semidirect
160. C53#9482semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202748142
#202730452
#202725672
C53×#24842
#948253