Finite Group Databasebeta

← #27281924

Subgroups — #27281924

Order 13568.

518 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 99 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#20279002C211yesmaximal, characteristic
2#20277318C211yesmaximal, characteristic
3#20274812C212yesmaximal
4#20272569C212yesmaximal
5C53×#1463C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810562C412yescomplemented
8#16810553C412yescomplemented
9#16810509—24no—
10#16810504—22no—
11#16810496—21no—
12#16810486—21no—
13#16810486—21no—
14#16810486C2211yescharacteristic
15#16810483—21no—
16C4×C22⋊Dic53C2212yes—
17#16810384—24no—
18C22×C4⋊C4×C53C2211yescharacteristic
19C53×#407C2212yes—
20C53×#324—22no—
21#13133176—22no—
22#13133176—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53D414yes—
32#13133124—22no—
33#13133124—24no—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—21no—
38#13133124—22no—
39#13133124C2×C412yes—
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43C2×M16×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C4⋊C4×C53D412yes—
48C22⋊C8×C53—22no—
49C22⋊C8×C53—22no—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53#4112yes—
64C22×Dic53—22no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—24no—
68M16×C53—44no—
69C2×C8×C53—42no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—24no—
72C22⋊Dic53—44no—
73C22⋊Dic53—42no—
74C22⋊Dic53—41no—
75C22⋊Dic53—41no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4⋊Dic53—82no—
79#11719919—44no—
80C4×Dic53—42no—
81C4×Dic53—44no—
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—81no—
101C2×C4×C53C4×D412yes—
102C2×C4×C53#12112yes—
103C2×Dic53—42no—
104C2×Dic53—44no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—44no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—44no—
113C8×C53—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—
130Dic53—84no—
131Dic53—82no—
132Dic53—81no—
133Dic53—81no—
134#2484—531no—
135#2313—531no—
136#1826—532no—
137#1463D5311yescharacteristic
138#1284—532no—
139C2×C53—22no—
140C2×C53—22no—
141C2×C53—41no—
142C2×C53#189011yescharacteristic
143C2×C53#189711yescharacteristic
144C2×C53#191211yescharacteristic
145C2×#117—531no—
146C22×C4⋊C4C2×D5311yescharacteristic
147#407C2×D5312yes—
148#400—532no—
149#398—532no—
150#387—1064no—
151#387—1062no—
152#385—1061no—
153#381—531no—
154#381—1061no—
155#381—1061no—
156#380—1061no—
157C4×C22⋊C4—532no—
158#339—1064no—
159#324—22no—
160C53#948211yescharacteristic, complemented, Sylow
161C23×C4D21211yescharacteristic
162C23×C4D21211yescharacteristic
163C23×C4C22×D5311yescharacteristic
164C2×M16—22no—
165#117—1062no—
166#117—1062no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4C4×D5312yes—
174C2×C4⋊C4C22⋊D5312yes—
175C2×C22⋊C4—1061no—
176C2×C22⋊C4—1062no—
177C2×C22⋊C4—1061no—
178C2×C42—534no—
179C2×C42—1061no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182#95—532no—
183#95—1062no—
184#95—1064no—
185#95—1062no—
186#95—1061no—
187#95—1062no—
188#95—1062no—
189#95—1062no—
190C24C2×D21211yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—1062no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—1064no—
202C22×C4#1171992112yes—
203C22×C4#1171992112yes—
204C22×C4C2×C4×D5311yescharacteristic
205C22×C4D4×D5312yes—
206C22×C4#1171994712yes—
207C22×C4C2×C22⋊D5311yescharacteristic
208M16—44no—
209C2×C8—42no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—4242no—
215C4⋊C4—2124no—
216C22⋊C4—2122no—
217C22⋊C4—2124no—
218C22⋊C4—2121no—
219C22⋊C4—2121no—
220C42—2124no—
221C42—2122no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×#1171992111yescharacteristic
226C23#1313318711yescharacteristic
227C23#1313320111yescharacteristic
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—81no—
241C2×C4—2122no—
242C2×C4—2124no—
243C2×C4—2122no—
244C2×C4—2124no—
245C2×C4—2122no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2124no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4#1313317712yes—
252C2×C4#1313318912yes—
253C8—82no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#1681050011yescharacteristic
263C22#1681063912yes—
264C22#1681081412yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—4244no—
270C4—4242no—
271C4—4241no—
272C4—4241no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#2027512611yescharacteristic
277C2#2027516111yescharacteristic
278C2#2027525011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1463C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1463D53
hypocenterC53#9482
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
7. #16810562C4semidirect
8. #16810553C4semidirect
160. C53#9482semidirect

Maximal subgroups

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

subgroupindex
#202790022
#202773182
#202748122
#202725692
C53×#14632
#948253