Finite Group Databasebeta

← C53×#7610 #27215122

Subgroups — C53×#7610 #27215122

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#389C211yesmaximal, characteristic
2C53×#1817C211yesmaximal, characteristic
3C53×#1728C211yesmaximal, characteristic
4C53×#1271C214yesmaximal, complemented
5C22×M16×C53—21no—
6C22×C4⋊C4×C53—21no—
7C22×C4⋊C4×C53C411yescharacteristic
8C22×C4⋊C4×C53C411yescharacteristic
9C22×C42×C53C2211yescharacteristic
10C2×C22⋊C8×C53—21no—
11C53×#389—21no—
12C53×#389—21no—
13C53×#389C412yes—
14C53×#389C2212yes—
15C2×#95×C53—21no—
16C53×#336—21no—
17C53×#336—21no—
18C53×#336C2212yes—
19C53×#336C2212yes—
20C23×C4×C53—21no—
21C23×C4×C53—21no—
22C23×C4×C53C2×C411yescharacteristic
23C23×C4×C53D411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C2×M16×C53—24no—
26C2×M16×C53—42no—
27C2×M16×C53—42no—
28C22×C8×C53—22no—
29C22×C8×C53—22no—
30C2×C4⋊C4×C53—24no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—42no—
36C2×C4⋊C4×C53—42no—
37C2×C4⋊C4×C53—42no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C42×C53—42no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53C2311yescharacteristic
45C22⋊C8×C53—42no—
46#95×C53—22no—
47#95×C53—22no—
48C24×C53C22⋊C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—24no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—41no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53D812yes—
69C22×C4×C53SD1612yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73M16×C53—48no—
74C2×C8×C53—24no—
75C2×C8×C53—42no—
76C2×C8×C53—42no—
77C2×C8×C53—42no—
78C2×C8×C53—42no—
79C2×C8×C53—42no—
80C4⋊C4×C53—24no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—44no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53M1614yescomplemented
88C42×C53—41no—
89C42×C53—41no—
90C42×C53—41no—
91C42×C53—41no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53#9912yes—
97C23×C53#10212yes—
98C23×C53#10412yes—
99C23×C53C2×C22⋊C411yescharacteristic
100C2×C4×C53—28no—
101C2×C4×C53—24no—
102C2×C4×C53—24no—
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—21no—
112C2×C4×C53—21no—
113C2×C4×C53—24no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—44no—
118C2×C4×C53—48no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—84no—
125C2×C4×C53#10112yes—
126C2×C4×C53#10212yes—
127C2×C4×C53C2×M1612yes—
128C2×C4×C53C2×D811yescharacteristic
129C2×C4×C53C2×SD1611yescharacteristic
130C2×C4×C53#13712yes—
131C8×C53—44no—
132C8×C53—82no—
133#7610C5311yesmaximal, characteristic, direct product factor, Sylow
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53#32714yes—
141C22×C53#40711yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53C2×#10111yescharacteristic
144C22×C53C2×#10211yescharacteristic
145C22×C53#41911yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C22×C53#42111yescharacteristic
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—22no—
152C4×C53—21no—
153C4×C53—22no—
154C4×C53—44no—
155C4×C53—41no—
156C4×C53—82no—
157C2×#389C2×C5311yescharacteristic
158#1817C2×C5311yescharacteristic
159#1728C2×C5311yescharacteristic
160#1271C2×C5314yescomplemented
161C2×C53—24no—
162C2×C53#146311yescharacteristic
163C2×C53#148611yescharacteristic
164C2×C53#149111yescharacteristic
165C2×C53C2×#32711yescharacteristic
166C2×C53#151111yescharacteristic
167C2×C53#152811yescharacteristic
168C2×C53#154611yescharacteristic
169C22×M16—21no—
170C22×C4⋊C4—21no—
171C22×C4⋊C4C4×C5311yescharacteristic
172C22×C4⋊C4C4×C5311yescharacteristic
173C22×C42C22×C5311yescharacteristic
174C2×C22⋊C8—21no—
175#389—21no—
176#389—21no—
177#389C4×C5312yes—
178#389C22×C5312yes—
179C2×#95—21no—
180#336—21no—
181#336—21no—
182#336C22×C5312yes—
183#336C22×C5312yes—
184C53#761011yescharacteristic, direct product factor, Sylow
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×C4×C5311yescharacteristic
188C23×C4D4×C5311yescharacteristic
189C23×C4D4×C5311yescharacteristic
190C2×M16—24no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—22no—
194C22×C8—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×C4×C5312yes—
204C2×C4⋊C4C2×C4×C5312yes—
205C2×C42—42no—
206C2×C42C2×C4×C5311yescharacteristic
207C2×C42D4×C5311yescharacteristic
208C2×C42D4×C5311yescharacteristic
209C2×C42C23×C5311yescharacteristic
210C22⋊C8—42no—
211#95—22no—
212#95—22no—
213C24C22⋊C4×C5311yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊C4×C5311yescharacteristic
231C22×C4C22⋊C4×C5311yescharacteristic
232C22×C4C22⋊C4×C5311yescharacteristic
233C22×C4D8×C5312yes—
234C22×C4SD16×C5312yes—
235C22×C4C22×C4×C5311yescharacteristic
236C22×C4C2×D4×C5311yescharacteristic
237C22×C4C2×D4×C5311yescharacteristic
238M16—48no—
239C2×C8—24no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4M16×C5314yescomplemented
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#99×C5312yes—
262C23#102×C5312yes—
263C23#104×C5312yes—
264C23C2×C22⋊C4×C5311yescharacteristic
265C2×C4—28no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4#101×C5312yes—
291C2×C4#102×C5312yes—
292C2×C4C2×M16×C5312yes—
293C2×C4C2×D8×C5311yescharacteristic
294C2×C4C2×SD16×C5311yescharacteristic
295C2×C4#137×C5312yes—
296C8—44no—
297C8—82no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22C53×#32714yes—
305C22C53×#40711yescharacteristic
306C22C2×#99×C5311yescharacteristic
307C22C2×#101×C5311yescharacteristic
308C22C2×#102×C5311yescharacteristic
309C22C53×#41911yescharacteristic
310C22C2×#104×C5311yescharacteristic
311C22C53×#42111yescharacteristic
312C4—22no—
313C4—22no—
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—44no—
319C4—41no—
320C4—82no—
321C2—24no—
322C2C53×#146311yescharacteristic
323C2C53×#148611yescharacteristic
324C2C53×#149111yescharacteristic
325C2C2×C53×#32711yescharacteristic
326C2C53×#151111yescharacteristic
327C2C53×#152811yescharacteristic
328C2C53×#154611yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC24×C53C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1271C2semidirect
87. C4⋊C4×C53M16semidirect
133. #7610C53direct (N × H)
160. #1271C2×C53semidirect
184. C53#7610direct (N × H)
252. C4⋊C4M16×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#3892
C53×#18172
C53×#17282
C53×#12712
#761053