Finite Group Databasebeta

← #27281907

Subgroups — #27281907

Order 13568.

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

normal subgroups only

This group has 610 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#20274814C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20272567C212yesmaximal
5C53×#2484C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810587C2212yes—
8C4⋊C4×Dic53—24no—
9C4×C22⋊Dic53—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—
22#13133176—28no—
23C2×C4⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53D414yes—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124—22no—
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35#117×C53—22no—
36#117×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C42×C53—21no—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53—21no—
51#95×C53C2×C412yes—
52C24×C53C2×D411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—24no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53#4112yes—
69C22×Dic53—24no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—44no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C42×C53—42no—
79C22⋊Dic53—28no—
80C4⋊Dic53—28no—
81C4×Dic53—28no—
82C4×Dic53—44no—
83C4×Dic53—82no—
84#11719917—44no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#12011yescharacteristic
91C23×C53#12311yescharacteristic
92C2×C4×C53—24no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—24no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—81no—
112C2×C4×C53#10414yes—
113C2×C4×C53C4×D412yes—
114C2×C4×C53#12112yes—
115C2×Dic53—28no—
116C2×Dic53—44no—
117C2×Dic53—44no—
118#88322—82no—
119#9478—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—44no—
128C22×C53#38611yescharacteristic
129C22×C53C2×#10412yes—
130C22×C53#45812yes—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—24no—
134C4×C53—42no—
135C4×C53—82no—
136C4×C53—81no—
137C4×C53—81no—
138Dic53—48no—
139#2484D5311yescharacteristic
140#2274—531no—
141#1826—532no—
142#1463—531no—
143#1284—532no—
144C2×C53—22no—
145C2×C53—22no—
146C2×C53—41no—
147C2×C53#189011yescharacteristic
148C2×C53#189611yescharacteristic
149C2×C53#191211yescharacteristic
150C2×#117—1061no—
151C2×#117C2×D5311yescharacteristic
152C22×C4⋊C4C2×D5311yescharacteristic
153#407—532no—
154#400Dic5312yescomplemented
155#398Dic5312yescomplemented
156#387—22no—
157#385—21no—
158#381—21no—
159#381C2×D5311yescharacteristic
160C4×C4⋊C4—1064no—
161C4×C22⋊C4—532no—
162C4×C22⋊C4—1061no—
163#339—1064no—
164#324—1062no—
165C53#947811yescharacteristic, complemented, Sylow
166C23×C4C22⋊D5311yescharacteristic
167C23×C4C22⋊D5311yescharacteristic
168C23×C4C22×D5311yescharacteristic
169C2×M16—1062no—
170#117—22no—
171#117—22no—
172#117—1068no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—1062no—
177C2×C4⋊C4C2×Dic5311yescharacteristic
178C2×C4⋊C4C2×Dic5311yescharacteristic
179C2×C4⋊C4C2×Dic5312yes—
180C2×C4⋊C4C22⋊D5312yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—1062no—
184C2×C42—21no—
185C2×C42—534no—
186C2×C42—1062no—
187C2×C42—1062no—
188C22⋊C8—1062no—
189C22⋊C8—1062no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—1062no—
195#95—1062no—
196#95C2×Dic5312yes—
197C24C2×C22⋊D5311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—1064no—
209C22×C4C22⋊Dic5312yes—
210C22×C4C22⋊Dic5312yes—
211C22×C4D4×D5312yes—
212C22×C4#1171994712yes—
213C22×C4C22×Dic5311yescharacteristic
214C22×C4C2×C22⋊D5311yescharacteristic
215M16—2124no—
216C2×C8—2122no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—1068no—
223C22⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—1068no—
227C42—42no—
228C42—1068no—
229C42—4242no—
230C42—2124no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×C22⋊Dic5311yescharacteristic
235C23#1313318511yescharacteristic
236C23#1313319811yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—1068no—
258C2×C4—2124no—
259C2×C4—2124no—
260C2×C4#1313314414yes—
261C2×C4D4×Dic5312yes—
262C2×C4#1313318912yes—
263C8—4242no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1681050311yescharacteristic
273C22C2×#1313314412yes—
274C22#1681081212yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4—81no—
282C4—2128no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2027512911yescharacteristic
287C2#2027515911yescharacteristic
288C2#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#9478
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
154. #400Dic53semidirect
155. #398Dic53semidirect
165. C53#9478semidirect

Maximal subgroups

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

subgroupindex
#202770722
#202748142
#202730452
#202725672
C53×#24842
#947853