Finite Group Databasebeta

← C53×#9026 #27216538

Subgroups — C53×#9026 #27216538

Order 13568.

616 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C53×#398C211yesmaximal, characteristic
2C53×#2313C211yesmaximal, characteristic
3C53×#1823C212yesmaximal
4C2×C53×#324C211yesmaximal, characteristic
5C53×#1294C212yesmaximal
6C22×C4⋊C4×C53—21no—
7C22×C4⋊C4×C53C2211yescharacteristic
8C22×C4⋊C4×C53C2211yescharacteristic
9C2×C22⋊C8×C53C2212yes—
10C53×#398—22no—
11C53×#398C414yescomplemented
12C53×#387—24no—
13C53×#381—21no—
14C53×#380C2212yes—
15C4×C22⋊C4×C53—21no—
16C2×#95×C53—21no—
17C2×#95×C53C2211yescharacteristic
18C53×#340—24no—
19C53×#324—22no—
20C23×C4×C53—21no—
21C23×C4×C53—21no—
22C23×C4×C53D411yescharacteristic
23C23×C4×C53D411yescharacteristic
24C23×C4×C53C2311yescharacteristic
25C22×C8×C53—22no—
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—44no—
33C2×C4⋊C4×C53C2×C412yes—
34C2×C4⋊C4×C53C2×C412yes—
35C2×C4⋊C4×C53D414yes—
36C2×C4⋊C4×C53D412yes—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53—22no—
39C22⋊C8×C53—22no—
40C22⋊C8×C53—22no—
41#95×C53—22no—
42#95×C53—24no—
43#95×C53—24no—
44#95×C53—21no—
45#95×C53—22no—
46#95×C53C2×C412yes—
47C24×C53C2×D411yescharacteristic
48C22×C4×C53—24no—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—24no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D412yes—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4112yes—
72C2×C8×C53—44no—
73C2×C8×C53—42no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—48no—
78C4⋊C4×C53—44no—
79C4⋊C4×C53—42no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C42×C53—82no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53#12011yescharacteristic
87C23×C53#12311yescharacteristic
88C23×C53#13512yes—
89C23×C53AGL(1,ℤ/8)12yes—
90C2×C4×C53—24no—
91C2×C4×C53—24no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—48no—
100C2×C4×C53—42no—
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—42no—
108C2×C4×C53—42no—
109C2×C4×C53—44no—
110C2×C4×C53—44no—
111C2×C4×C53—44no—
112C2×C4×C53—81no—
113C2×C4×C53—82no—
114C2×C4×C53—82no—
115C2×C4×C53C4×D412yes—
116C2×C4×C53#12112yes—
117C8×C53—82no—
118#9026C5311yesmaximal, characteristic, direct product factor, Sylow
119C22×C53—24no—
120C22×C53—24no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53#38611yescharacteristic
126C22×C53#41611yescharacteristic
127C22×C53#41811yescharacteristic
128C22×C53#44912yes—
129C22×C53#45612yes—
130C22×C53#45812yes—
131C22×C53#48212yes—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—44no—
135C4×C53—42no—
136C4×C53—42no—
137C4×C53—84no—
138C4×C53—82no—
139C2×#398C2×C5311yescharacteristic
140#2313C2×C5311yescharacteristic
141#1823C2×C5312yes—
142C2×#324C2×C5311yescharacteristic
143#1294C2×C5312yes—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—22no—
147C2×C53#160914yes—
148C2×C53#187511yescharacteristic
149C2×C53#188611yescharacteristic
150C2×C53#189011yescharacteristic
151C22×C4⋊C4—21no—
152C22×C4⋊C4C22×C5311yescharacteristic
153C22×C4⋊C4C22×C5311yescharacteristic
154C2×C22⋊C8C22×C5312yes—
155#398—22no—
156#398C4×C5314yescomplemented
157#387—24no—
158#381—21no—
159#380C22×C5312yes—
160C4×C22⋊C4—21no—
161C2×#95—21no—
162C2×#95C22×C5311yescharacteristic
163#340—24no—
164#324—22no—
165C53#902611yescharacteristic, direct product factor, Sylow
166C23×C4—21no—
167C23×C4—21no—
168C23×C4D4×C5311yescharacteristic
169C23×C4D4×C5311yescharacteristic
170C23×C4C23×C5311yescharacteristic
171C22×C8—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C2×C4×C5312yes—
180C2×C4⋊C4C2×C4×C5312yes—
181C2×C4⋊C4D4×C5312yes—
182C2×C4⋊C4D4×C5314yes—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187#95—22no—
188#95—24no—
189#95—21no—
190#95—22no—
191#95—24no—
192#95C2×C4×C5312yes—
193C24C2×D4×C5311yescharacteristic
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C22⋊C4×C5312yes—
213C22×C4C22⋊C4×C5312yes—
214C22×C4C22×C4×C5311yescharacteristic
215C22×C4C2×D4×C5312yes—
216C22×C4C2×D4×C5311yescharacteristic
217C22×C4#41×C5312yes—
218C2×C8—44no—
219C2×C8—42no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—82no—
229C23—22no—
230C23—22no—
231C23C2×C22⋊C4×C5311yescharacteristic
232C23#120×C5311yescharacteristic
233C23#123×C5311yescharacteristic
234C23#135×C5312yes—
235C23AGL(1,ℤ/8)×C5312yes—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4C4×D4×C5312yes—
262C2×C4#121×C5312yes—
263C8—82no—
264C22—24no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22C53×#38611yescharacteristic
271C22C53×#41611yescharacteristic
272C22C53×#41811yescharacteristic
273C22C53×#44912yes—
274C22C53×#45612yes—
275C22C53×#45812yes—
276C22C53×#48212yes—
277C4—22no—
278C4—22no—
279C4—44no—
280C4—42no—
281C4—42no—
282C4—84no—
283C4—82no—
284C2—21no—
285C2—21no—
286C2—22no—
287C2C53×#160914yes—
288C2C53×#187511yescharacteristic
289C2C53×#188611yescharacteristic
290C2C53×#189011yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
11. C53×#398C4semidirect
118. #9026C53direct (N × H)
156. #398C4×C53semidirect
165. C53#9026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#3982
C53×#23132
C53×#18232
C2×C53×#3242
C53×#12942
#902653