Finite Group Databasebeta

← C53×#7611 #27215123

Subgroups — C53×#7611 #27215123

Order 13568.

408 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2295C211yesmaximal, characteristic
2C53×#1839C211yesmaximal, characteristic
3C53×#1745C211yesmaximal, characteristic
4C53×#1272C212yesmaximal
5C53×#1271C212yesmaximal
6C2×C4×Q8×C53C411yescharacteristic
7C2×#97×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
43#97×C53—24no—
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—24no—
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—22no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—22no—
84C42×C53—22no—
85C42×C53—41no—
86C42×C53—41no—
87C42×C53—41no—
88C42×C53—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90Q8×C53—42no—
91Q8×C53—42no—
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—22no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
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#10412yes—
122C2×C4×C53C2×M1611yescharacteristic
123C2×C4×C53C2×M1611yescharacteristic
124C2×C4×C53#13512yes—
125C2×C4×C53#13712yes—
126C8×C53—44no—
127C8×C53—82no—
128#7611C5311yesmaximal, characteristic, direct product factor, Sylow
129C22×C53#40711yescharacteristic
130C22×C53#41011yescharacteristic
131C22×C53#41311yescharacteristic
132C22×C53#41611yescharacteristic
133C22×C53#41911yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C22×C53#42111yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—42no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53—81no—
147C4×C53—81no—
148#2295C2×C5311yescharacteristic
149#1839C2×C5311yescharacteristic
150#1745C2×C5311yescharacteristic
151#1272C2×C5312yes—
152#1271C2×C5312yes—
153C2×C53#146411yescharacteristic
154C2×C53#148811yescharacteristic
155C2×C53#149111yescharacteristic
156C2×C53#150011yescharacteristic
157C2×C53#151511yescharacteristic
158C2×C53#152911yescharacteristic
159C2×C53#155411yescharacteristic
160C2×C4×Q8C4×C5311yescharacteristic
161C2×#97—21no—
162#391C4×C5311yescharacteristic
163#391C22×C5311yescharacteristic
164#389C4×C5311yescharacteristic
165#389C22×C5311yescharacteristic
166#387—21no—
167#387—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4C4×C5311yescharacteristic
170#376—21no—
171C43C22×C5311yescharacteristic
172#336—21no—
173#336—21no—
174#336—21no—
175#336C22×C5312yes—
176#336C22×C5312yes—
177C53#761111yescharacteristic, direct product factor, Sylow
178C22×Q8C2×C4×C5311yescharacteristic
179C22×C8—22no—
180C22×C8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C2×C4×C5311yescharacteristic
187C2×C4⋊C4C2×C4×C5311yescharacteristic
188C2×C4⋊C4C2×C4×C5311yescharacteristic
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C2×C4×C5311yescharacteristic
192C2×C42C2×C4×C5311yescharacteristic
193C2×C42D4×C5311yescharacteristic
194C2×C42D4×C5311yescharacteristic
195C2×C42D4×C5311yescharacteristic
196C2×C42D4×C5311yescharacteristic
197C2×C42C23×C5311yescharacteristic
198#97—24no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8M16×C5312yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4C22⋊C4×C5311yescharacteristic
215C22×C4C22⋊C4×C5311yescharacteristic
216C22×C4C22⋊C4×C5311yescharacteristic
217C22×C4C22⋊C4×C5311yescharacteristic
218C22×C4C22×C4×C5311yescharacteristic
219C22×C4C2×D4×C5311yescharacteristic
220C22×C4C2×D4×C5311yescharacteristic
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C4⋊C4—24no—
228C4⋊C4—21no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4M16×C5312yes—
234C42—24no—
235C42—22no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—42no—
244C23C2×C22⋊C4×C5311yescharacteristic
245Q8—42no—
246Q8—42no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#104×C5312yes—
277C2×C4C2×M16×C5311yescharacteristic
278C2×C4C2×M16×C5311yescharacteristic
279C2×C4#135×C5312yes—
280C2×C4#137×C5312yes—
281C8—44no—
282C8—82no—
283C22C53×#40711yescharacteristic
284C22C53×#41011yescharacteristic
285C22C53×#41311yescharacteristic
286C22C53×#41611yescharacteristic
287C22C53×#41911yescharacteristic
288C22C2×#104×C5311yescharacteristic
289C22C53×#42111yescharacteristic
290C4—21no—
291C4—21no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C2C53×#146411yescharacteristic
303C2C53×#148811yescharacteristic
304C2C53×#149111yescharacteristic
305C2C53×#150011yescharacteristic
306C2C53×#151511yescharacteristic
307C2C53×#152911yescharacteristic
308C2C53×#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
128. #7611C53direct (N × H)
177. C53#7611direct (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×#17452
C53×#12722
C53×#12712
#761153