Finite Group Databasebeta

← C53×#7729 #27215241

Subgroups — C53×#7729 #27215241

Order 13568.

346 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2863C211yesmaximal, characteristic
2C53×#1839C211yesmaximal, characteristic
3C53×#1839C211yesmaximal, characteristic
4C53×#1272C212yesmaximal
5C53×#1271C212yesmaximal
6C53×#391C411yescharacteristic
7C53×#391C411yescharacteristic
8C53×#391C2211yescharacteristic
9C53×#389C411yescharacteristic
10C53×#389C411yescharacteristic
11C53×#389C2211yescharacteristic
12C53×#387—21no—
13C53×#387—21no—
14C53×#387—21no—
15C53×#383—21no—
16C43×C53C2211yescharacteristic
17C53×#336—21no—
18C53×#336—21no—
19C53×#336—21no—
20C53×#336—21no—
21C53×#336C2212yes—
22C53×#336C2212yes—
23C22×Q8×C53C2×C411yescharacteristic
24C22×C8×C53—22no—
25C22×C8×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53C2×C411yescharacteristic
30C2×C4⋊C4×C53C2×C411yescharacteristic
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C42×C53C2×C411yescharacteristic
33C2×C42×C53C2×C411yescharacteristic
34C2×C42×C53D411yescharacteristic
35C2×C42×C53D411yescharacteristic
36C2×C42×C53D411yescharacteristic
37C2×C42×C53D411yescharacteristic
38C2×C42×C53C2311yescharacteristic
39#95×C53—21no—
40#95×C53—21no—
41#95×C53—21no—
42#95×C53—21no—
43#95×C53—21no—
44#95×C53—21no—
45#95×C53—21no—
46#95×C53—21no—
47#95×C53—21no—
48C2×Q8×C53—22no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C2×C8×C53—42no—
65C2×C8×C53—42no—
66C2×C8×C53—42no—
67C2×C8×C53—42no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C42×C53—22no—
75C42×C53—22no—
76C42×C53—22no—
77C42×C53—41no—
78C42×C53—41no—
79C42×C53—41no—
80C42×C53—41no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—82no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#13112yes—
114C2×C4×C53#13512yes—
115C2×C4×C53#13712yes—
116C8×C53—82no—
117C8×C53—82no—
118#7729C5311yesmaximal, characteristic, direct product factor, Sylow
119C22×C53#40811yescharacteristic
120C22×C53#41011yescharacteristic
121C22×C53#41311yescharacteristic
122C22×C53#41611yescharacteristic
123C22×C53#41911yescharacteristic
124C22×C53#42111yescharacteristic
125C22×C53#42111yescharacteristic
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—81no—
134C4×C53—81no—
135C4×C53—81no—
136C4×C53—81no—
137#2863C2×C5311yescharacteristic
138#1839C2×C5311yescharacteristic
139#1839C2×C5311yescharacteristic
140#1272C2×C5312yes—
141#1271C2×C5312yes—
142C2×C53#147311yescharacteristic
143C2×C53#148211yescharacteristic
144C2×C53#149711yescharacteristic
145C2×C53#151511yescharacteristic
146C2×C53#152111yescharacteristic
147C2×C53#153511yescharacteristic
148C2×C53#155411yescharacteristic
149#391C4×C5311yescharacteristic
150#391C4×C5311yescharacteristic
151#391C22×C5311yescharacteristic
152#389C4×C5311yescharacteristic
153#389C4×C5311yescharacteristic
154#389C22×C5311yescharacteristic
155#387—21no—
156#387—21no—
157#387—21no—
158#383—21no—
159C43C22×C5311yescharacteristic
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336C22×C5312yes—
165#336C22×C5312yes—
166C53#772911yescharacteristic, direct product factor, Sylow
167C22×Q8C2×C4×C5311yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C2×C4×C5311yescharacteristic
174C2×C4⋊C4C2×C4×C5311yescharacteristic
175C2×C4⋊C4C2×C4×C5311yescharacteristic
176C2×C42C2×C4×C5311yescharacteristic
177C2×C42C2×C4×C5311yescharacteristic
178C2×C42D4×C5311yescharacteristic
179C2×C42D4×C5311yescharacteristic
180C2×C42D4×C5311yescharacteristic
181C2×C42D4×C5311yescharacteristic
182C2×C42C23×C5311yescharacteristic
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4C22⋊C4×C5311yescharacteristic
200C22×C4C22⋊C4×C5311yescharacteristic
201C22×C4C22⋊C4×C5311yescharacteristic
202C22×C4C22⋊C4×C5311yescharacteristic
203C22×C4C22×C4×C5311yescharacteristic
204C22×C4C2×D4×C5311yescharacteristic
205C22×C4C2×D4×C5311yescharacteristic
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C4⋊C4—22no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C42—22no—
219C42—22no—
220C42—22no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—41no—
225C23C2×C22⋊C4×C5311yescharacteristic
226Q8—82no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4#131×C5312yes—
258C2×C4#135×C5312yes—
259C2×C4#137×C5312yes—
260C8—82no—
261C8—82no—
262C22C53×#40811yescharacteristic
263C22C53×#41011yescharacteristic
264C22C53×#41311yescharacteristic
265C22C53×#41611yescharacteristic
266C22C53×#41911yescharacteristic
267C22C53×#42111yescharacteristic
268C22C53×#42111yescharacteristic
269C4—22no—
270C4—22no—
271C4—22no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C2C53×#147311yescharacteristic
281C2C53×#148211yescharacteristic
282C2C53×#149711yescharacteristic
283C2C53×#151511yescharacteristic
284C2C53×#152111yescharacteristic
285C2C53×#153511yescharacteristic
286C2C53×#155411yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C43×C53 ◁ G

Complements

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

NH ≅ G/Ntype
118. #7729C53direct (N × H)
166. C53#7729direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28632
C53×#18392
C53×#12722
C53×#12712
#772953