Finite Group Databasebeta

← C53×#9478 #27216990

Subgroups — C53×#9478 #27216990

Order 13568.

610 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C53×#2484C211yesmaximal, characteristic
2C53×#2274C211yesmaximal, characteristic
3C53×#1826C212yesmaximal
4C53×#1463C211yesmaximal, characteristic
5C53×#1284C212yesmaximal
6C2×#117×C53—21no—
7C2×#117×C53C2211yescharacteristic
8C22×C4⋊C4×C53C2211yescharacteristic
9C53×#407C2212yes—
10C53×#400C412yescomplemented
11C53×#398C412yescomplemented
12C53×#387—22no—
13C53×#385—21no—
14C53×#381—21no—
15C53×#381C2211yescharacteristic
16C4×C4⋊C4×C53—24no—
17C4×C22⋊C4×C53—21no—
18C4×C22⋊C4×C53C2212yes—
19C53×#339—24no—
20C53×#324—22no—
21C23×C4×C53D411yescharacteristic
22C23×C4×C53D411yescharacteristic
23C23×C4×C53C2311yescharacteristic
24C2×M16×C53—22no—
25#117×C53—28no—
26#117×C53—22no—
27#117×C53—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C412yes—
35C2×C4⋊C4×C53D412yes—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—21no—
39C2×C42×C53—22no—
40C2×C42×C53—22no—
41C2×C42×C53—21no—
42C2×C42×C53D414yes—
43C22⋊C8×C53—22no—
44C22⋊C8×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
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—24no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4112yes—
70M16×C53—44no—
71C2×C8×C53—42no—
72C4⋊C4×C53—28no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—44no—
78C22⋊C4×C53—28no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—28no—
83C42×C53—44no—
84C42×C53—42no—
85C42×C53—82no—
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—28no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—44no—
104C2×C4×C53—42no—
105C2×C4×C53—44no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—81no—
115C2×C4×C53#10414yes—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12112yes—
118C8×C53—82no—
119#9478C5311yesmaximal, characteristic, direct product factor, 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—48no—
135C4×C53—42no—
136C4×C53—82no—
137C4×C53—81no—
138C4×C53—81no—
139#2484C2×C5311yescharacteristic
140#2274C2×C5311yescharacteristic
141#1826C2×C5312yes—
142#1463C2×C5311yescharacteristic
143#1284C2×C5312yes—
144C2×C53—22no—
145C2×C53—22no—
146C2×C53—41no—
147C2×C53#189011yescharacteristic
148C2×C53#189611yescharacteristic
149C2×C53#191211yescharacteristic
150C2×#117—21no—
151C2×#117C22×C5311yescharacteristic
152C22×C4⋊C4C22×C5311yescharacteristic
153#407C22×C5312yes—
154#400C4×C5312yescomplemented
155#398C4×C5312yescomplemented
156#387—22no—
157#385—21no—
158#381—21no—
159#381C22×C5311yescharacteristic
160C4×C4⋊C4—24no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4C22×C5312yes—
163#339—24no—
164#324—22no—
165C53#947811yescharacteristic, direct product factor, Sylow
166C23×C4D4×C5311yescharacteristic
167C23×C4D4×C5311yescharacteristic
168C23×C4C23×C5311yescharacteristic
169C2×M16—22no—
170#117—28no—
171#117—22no—
172#117—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×C4×C5311yescharacteristic
178C2×C4⋊C4C2×C4×C5311yescharacteristic
179C2×C4⋊C4C2×C4×C5312yes—
180C2×C4⋊C4D4×C5312yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42D4×C5314yes—
188C22⋊C8—22no—
189C22⋊C8—22no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95C2×C4×C5312yes—
197C24C2×D4×C5311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4C22⋊C4×C5312yes—
210C22×C4C22⋊C4×C5312yes—
211C22×C4C22×C4×C5311yescharacteristic
212C22×C4C2×D4×C5312yes—
213C22×C4C2×D4×C5311yescharacteristic
214C22×C4#41×C5312yes—
215M16—44no—
216C2×C8—42no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C22⋊C4—28no—
224C22⋊C4—42no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C42—28no—
228C42—44no—
229C42—42no—
230C42—82no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×C22⋊C4×C5311yescharacteristic
235C23#120×C5311yescharacteristic
236C23#123×C5311yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4#104×C5314yes—
261C2×C4C4×D4×C5312yes—
262C2×C4#121×C5312yes—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22C53×#38611yescharacteristic
273C22C2×#104×C5312yes—
274C22C53×#45812yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4—81no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2C53×#189011yescharacteristic
287C2C53×#189611yescharacteristic
288C2C53×#191211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
10. C53×#400C4semidirect
11. C53×#398C4semidirect
119. #9478C53direct (N × H)
154. #400C4×C53semidirect
155. #398C4×C53semidirect
165. C53#9478direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24842
C53×#22742
C53×#18262
C53×#14632
C53×#12842
#947853