Finite Group Databasebeta

← C53×#7644 #27215156

Subgroups — C53×#7644 #27215156

Order 13568.

402 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2295C211yesmaximal, characteristic
2C53×#1839C211yesmaximal, characteristic
3C53×#1831C211yesmaximal, characteristic
4C53×#1272C212yesmaximal
5C53×#1271C212yesmaximal
6C2×C4×Q8×C53C411yescharacteristic
7C2×C4⋊C8×C53—21no—
8C53×#391C411yescharacteristic
9C53×#391C2211yescharacteristic
10C53×#389C411yescharacteristic
11C53×#389C2211yescharacteristic
12C53×#387—21no—
13C53×#387—21no—
14C4×C4⋊C4×C53—21no—
15C4×C4⋊C4×C53C411yescharacteristic
16C53×#376—21no—
17C43×C53C2211yescharacteristic
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—
26C4×Q8×C53—22no—
27C4×Q8×C53—22no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C42×C53—21no—
35C2×C42×C53—21no—
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53D411yescharacteristic
39C2×C42×C53D411yescharacteristic
40C2×C42×C53D411yescharacteristic
41C2×C42×C53D411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53—42no—
44#95×C53—21no—
45#95×C53—21no—
46#95×C53—21no—
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53M1612yes—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C2×C8×C53—42no—
67C2×C8×C53—42no—
68C2×C8×C53—42no—
69C2×C8×C53—42no—
70C2×C8×C53—42no—
71C2×C8×C53—42no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53M1612yes—
79C42×C53—24no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—22no—
83C42×C53—41no—
84C42×C53—41no—
85C42×C53—41no—
86C42×C53—41no—
87C42×C53—42no—
88C42×C53SD1612yes—
89C42×C53Q1612yes—
90C23×C53C2×C22⋊C411yescharacteristic
91Q8×C53—42no—
92Q8×C53—42no—
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—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#10312yes—
122C2×C4×C53#10312yes—
123C2×C4×C53C2×M1611yescharacteristic
124C2×C4×C53C2×M1611yescharacteristic
125C2×C4×C53C2×SD1611yescharacteristic
126C2×C4×C53C2×Q1611yescharacteristic
127C2×C4×C53#13512yes—
128C8×C53—82no—
129C8×C53—82no—
130#7644C5311yesmaximal, characteristic, direct product factor, Sylow
131C22×C53#40711yescharacteristic
132C22×C53#41011yescharacteristic
133C22×C53#41311yescharacteristic
134C22×C53C2×#10311yescharacteristic
135C22×C53#41611yescharacteristic
136C22×C53#42111yescharacteristic
137C22×C53#42111yescharacteristic
138C4×C53—22no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—22no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—42no—
146C4×C53—41no—
147C4×C53—42no—
148C4×C53—41no—
149C4×C53—81no—
150C4×C53—81no—
151#2295C2×C5311yescharacteristic
152#1839C2×C5311yescharacteristic
153#1831C2×C5311yescharacteristic
154#1272C2×C5312yes—
155#1271C2×C5312yes—
156C2×C53#146411yescharacteristic
157C2×C53#148711yescharacteristic
158C2×C53#148811yescharacteristic
159C2×C53#151811yescharacteristic
160C2×C53#152111yescharacteristic
161C2×C53#153511yescharacteristic
162C2×C53#154211yescharacteristic
163C2×C4×Q8C4×C5311yescharacteristic
164C2×C4⋊C8—21no—
165#391C4×C5311yescharacteristic
166#391C22×C5311yescharacteristic
167#389C4×C5311yescharacteristic
168#389C22×C5311yescharacteristic
169#387—21no—
170#387—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C4×C5311yescharacteristic
173#376—21no—
174C43C22×C5311yescharacteristic
175#336—21no—
176#336—21no—
177#336—21no—
178#336C22×C5312yes—
179#336C22×C5312yes—
180C53#764411yescharacteristic, direct product factor, Sylow
181C22×Q8C2×C4×C5311yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2×C4×C5311yescharacteristic
190C2×C4⋊C4C2×C4×C5311yescharacteristic
191C2×C4⋊C4C2×C4×C5311yescharacteristic
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C2×C4×C5311yescharacteristic
195C2×C42C2×C4×C5311yescharacteristic
196C2×C42D4×C5311yescharacteristic
197C2×C42D4×C5311yescharacteristic
198C2×C42D4×C5311yescharacteristic
199C2×C42D4×C5311yescharacteristic
200C2×C42C23×C5311yescharacteristic
201C4⋊C8—42no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8M16×C5312yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C22⋊C4×C5311yescharacteristic
218C22×C4C22⋊C4×C5311yescharacteristic
219C22×C4C22⋊C4×C5311yescharacteristic
220C22×C4C22⋊C4×C5311yescharacteristic
221C22×C4C22×C4×C5311yescharacteristic
222C22×C4C2×D4×C5311yescharacteristic
223C22×C4C2×D4×C5311yescharacteristic
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—24no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4M16×C5312yes—
237C42—24no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—42no—
246C42SD16×C5312yes—
247C42Q16×C5312yes—
248C23C2×C22⋊C4×C5311yescharacteristic
249Q8—42no—
250Q8—42no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4#103×C5312yes—
280C2×C4#103×C5312yes—
281C2×C4C2×M16×C5311yescharacteristic
282C2×C4C2×M16×C5311yescharacteristic
283C2×C4C2×SD16×C5311yescharacteristic
284C2×C4C2×Q16×C5311yescharacteristic
285C2×C4#135×C5312yes—
286C8—82no—
287C8—82no—
288C22C53×#40711yescharacteristic
289C22C53×#41011yescharacteristic
290C22C53×#41311yescharacteristic
291C22C2×#103×C5311yescharacteristic
292C22C53×#41611yescharacteristic
293C22C53×#42111yescharacteristic
294C22C53×#42111yescharacteristic
295C4—21no—
296C4—21no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—41no—
306C4—81no—
307C4—81no—
308C2C53×#146411yescharacteristic
309C2C53×#148711yescharacteristic
310C2C53×#148811yescharacteristic
311C2C53×#151811yescharacteristic
312C2C53×#152111yescharacteristic
313C2C53×#153511yescharacteristic
314C2C53×#154211yescharacteristic
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
130. #7644C53direct (N × H)
180. C53#7644direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#22952
C53×#18392
C53×#18312
C53×#12722
C53×#12712
#764453