Finite Group Databasebeta

← #27281562

Subgroups — #27281562

Order 13568.

552 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 103 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#20280195C211yesmaximal, characteristic
2#20273230C212yesmaximal
3#20273045C211yesmaximal, characteristic, complemented
4#20272601C212yesmaximal, complemented
5C53×#2313C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7#16810587C2212yes—
8#16810562C412yescomplemented
9#16810540C412yescomplemented
10#16810513—22no—
11#16810502—21no—
12#16810502—21no—
13C4×C22⋊Dic53C2211yescharacteristic
14#16810356—24no—
15#16810349—22no—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#387—24no—
18C53×#381—21no—
19C53×#380—21no—
20C4×C22⋊C4×C53C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23C2×#11719917—22no—
24#13133191—22no—
25#13133191—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53—21no—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46C2×C42×C53—22no—
47#95×C53—22no—
48#95×C53—24no—
49#95×C53D414yes—
50C24×C53C2×D411yescharacteristic
51C23×D53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—24no—
57C22×C4×C53—42no—
58C22×C4×C53—81no—
59C22×C4×C53—81no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D412yes—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53#4112yes—
66C2×C22⋊D53—44no—
67C2×C22⋊D53—41no—
68C2×C22⋊D53—41no—
69C22×Dic53—21no—
70C22×Dic53—22no—
71C2×D212—22no—
72C2×D212—81no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—82no—
76C22⋊C4×C53—44no—
77C42×C53—82no—
78C22⋊Dic53—42no—
79#11719921—44no—
80#11719921—42no—
81#11719921—82no—
82C4⋊Dic53—22no—
83C4×Dic53—81no—
84#11719917—82no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#12011yescharacteristic
91C23×C53#12311yescharacteristic
92C22×D53—42no—
93C22×D53—42no—
94C22×D53—41no—
95C22×D53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—44no—
109C2×C4×C53—44no—
110C2×C4×C53—44no—
111C2×C4×C53—81no—
112C2×C4×C53C4×D412yes—
113C2×C4×C53#12112yes—
114C22⋊D53—82no—
115C22⋊D53—82no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121D212—82no—
122#88322—82no—
123#9414—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×C53#42112yes—
134C22×C53#45412yes—
135C2×D53—84no—
136C2×D53—81no—
137C2×D53—81no—
138C2×D53—81no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—42no—
142C4×C53—42no—
143C4×C53—84no—
144Dic53—82no—
145Dic53—81no—
146#2591—531no—
147#2313D5311yescharacteristic, complemented
148#1507—532no—
149#1463—531no—
150#1297—532no—
151C2×C53—22no—
152C2×C53—22no—
153C2×C53—41no—
154C2×C53#188811yescharacteristic
155C2×C53#189711yescharacteristic
156C2×C53#191311yescharacteristic
157D53—161no—
158C2×#121—531no—
159C22×C4⋊C4C2×D5311yescharacteristic
160#407—532no—
161#400—532no—
162#394—532no—
163#388—1062no—
164#387—24no—
165#386—1061no—
166#386—1061no—
167#381—21no—
168#380—21no—
169C4×C22⋊C4—531no—
170C4×C22⋊C4C2×D5312yes—
171#327—1064no—
172#324—1062no—
173C53#941411yescharacteristic, complemented, Sylow
174C22×D4—531no—
175C22×D4—1061no—
176C23×C4C22⋊D5311yescharacteristic
177C23×C4C22⋊D5311yescharacteristic
178C23×C4C22×D5311yescharacteristic
179C2×M16—1062no—
180#121—1062no—
181#121—1062no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4C4×D5312yes—
187C2×C4⋊C4D21212yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1061no—
191C2×C22⋊C4—1061no—
192C2×C22⋊C4—1062no—
193C2×C42—22no—
194C2×C42—1061no—
195C22⋊C8—1062no—
196C22⋊C8—1062no—
197#95—22no—
198#95—24no—
199#95—532no—
200#95—1062no—
201#95—1062no—
202#95C22⋊D5314yes—
203C24—1061no—
204C24C2×C22⋊D5311yescharacteristic
205C2×D4—1062no—
206C2×D4—4241no—
207C2×D4—2124no—
208C2×D4—2121no—
209C2×D4—2121no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—1062no—
219C22×C4—1061no—
220C22×C4#1171992112yes—
221C22×C4#1171992112yes—
222C22×C4C2×C4×D5311yescharacteristic
223C22×C4C2×D21211yescharacteristic
224C22×C4#1171994512yes—
225C22×C4C2×C22⋊D5312yes—
226M16—4242no—
227C2×C8—2122no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—82no—
231C4⋊C4—1062no—
232C22⋊C4—44no—
233C22⋊C4—4242no—
234C22⋊C4—2124no—
235C22⋊C4—2122no—
236C22⋊C4—2122no—
237C42—82no—
238C42—4241no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—2122no—
243C23—2122no—
244C23—2121no—
245C23—2122no—
246C23C2×#1171992111yescharacteristic
247C23#1313318611yescharacteristic
248C23#1313320011yescharacteristic
249D4—4242no—
250D4—4242no—
251D4—4242no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—81no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2121no—
272C2×C4—2122no—
273C2×C4C4×C22⋊D5312yes—
274C2×C4#1313319112yes—
275C8—4242no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—4244no—
285C22—4241no—
286C22—4241no—
287C22—4241no—
288C22#1681050211yescharacteristic
289C22#1681064112yes—
290C22#1681079612yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—4242no—
297C4—4241no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—8481no—
302C2#2027511811yescharacteristic
303C2#2027516311yescharacteristic
304C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2313C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2313D53
hypocenterC53#9414
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
3. #20273045C2semidirect
4. #20272601C2semidirect
5. C53×#2313C2semidirect
8. #16810562C4semidirect
9. #16810540C4semidirect
147. #2313D53semidirect
173. C53#9414semidirect

Maximal subgroups

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

subgroupindex
#202801952
#202732302
#202730452
#202726012
C53×#23132
#941453