Finite Group Databasebeta

← C53×#7580 #27215092

Subgroups — C53×#7580 #27215092

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2293C211yesmaximal, characteristic
2C53×#1839C211yesmaximal, characteristic, complemented
3C53×#1744C211yesmaximal, characteristic, complemented
4C53×#1271C212yesmaximal, complemented
5C53×#1270C212yesmaximal
6C2×C4×D4×C53C411yescharacteristic
7C2×#117×C53—21no—
8C2×C4×C8×C53—21no—
9C53×#390C411yescharacteristic
10C53×#390C2211yescharacteristic
11C53×#389C411yescharacteristic
12C53×#389C2211yescharacteristic
13C53×#388—21no—
14C53×#388—21no—
15C4×C4⋊C4×C53C411yescharacteristic
16C4×C22⋊C4×C53—21no—
17C43×C53C2211yescharacteristic
18C53×#336—21no—
19C53×#336—21no—
20C53×#336—21no—
21C53×#336C2212yes—
22C53×#336C2212yes—
23C22×D4×C53C2×C411yescharacteristic
24C23×C4×C53—21no—
25C22×C8×C53—22no—
26C22×C8×C53—22no—
27C4×D4×C53—22no—
28C4×D4×C53C814yescomplemented
29#117×C53—44no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53C2311yescharacteristic
48C4×C8×C53—24no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52C24×C53—21no—
53C2×D4×C53—21no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53C2×C812yes—
57C22×C4×C53—21no—
58C22×C4×C53—24no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C2×C8×C53—24no—
71C2×C8×C53—42no—
72C2×C8×C53—42no—
73C2×C8×C53—42no—
74C2×C8×C53—42no—
75C2×C8×C53—42no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53C2×C812yes—
82C22⋊C4×C53—24no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—22no—
88C42×C53—22no—
89C42×C53—22no—
90C42×C53—41no—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53—42no—
94C42×C53C2×C812yes—
95C42×C53M1612yes—
96C23×C53—22no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104D4×C53—42no—
105D4×C53—42no—
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—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—48no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53C22⋊C812yes—
132C2×C4×C53C22⋊C812yes—
133C2×C4×C53#10412yes—
134C2×C4×C53C22×C811yescharacteristic
135C2×C4×C53C2×M1611yescharacteristic
136C2×C4×C53#13512yes—
137C2×C4×C53AGL(1,ℤ/8)12yes—
138C8×C53—44no—
139C8×C53—82no—
140#7580C5311yesmaximal, characteristic, direct product factor, Sylow
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53C2×C22⋊C811yescharacteristic
149C22×C53#41011yescharacteristic
150C22×C53#41311yescharacteristic
151C22×C53#41611yescharacteristic
152C22×C53#41811yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C22×C53#42111yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—42no—
164C4×C53—81no—
165C4×C53—81no—
166#2293C2×C5311yescharacteristic
167#1839C2×C5311yescharacteristic, complemented
168#1744C2×C5311yescharacteristic, complemented
169#1271C2×C5312yescomplemented
170#1270C2×C5312yes—
171C2×C53—82no—
172C2×C53#145811yescharacteristic
173C2×C53#147611yescharacteristic
174C2×C53#147811yescharacteristic
175C2×C53#150511yescharacteristic
176C2×C53#152111yescharacteristic
177C2×C53#153211yescharacteristic
178C2×C53#155311yescharacteristic
179C2×C4×D4C4×C5311yescharacteristic
180C2×#117—21no—
181C2×C4×C8—21no—
182#390C4×C5311yescharacteristic
183#390C22×C5311yescharacteristic
184#389C4×C5311yescharacteristic
185#389C22×C5311yescharacteristic
186#388—21no—
187#388—21no—
188C4×C4⋊C4C4×C5311yescharacteristic
189C4×C22⋊C4—21no—
190C43C22×C5311yescharacteristic
191#336—21no—
192#336—21no—
193#336—21no—
194#336C22×C5312yes—
195#336C22×C5312yes—
196C53#758011yescharacteristic, direct product factor, Sylow
197C22×D4C2×C4×C5311yescharacteristic
198C23×C4—21no—
199C22×C8—22no—
200C22×C8—22no—
201C4×D4—22no—
202C4×D4C8×C5314yescomplemented
203#117—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×C4×C5311yescharacteristic
207C2×C4⋊C4C2×C4×C5311yescharacteristic
208C2×C4⋊C4C2×C4×C5311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42C2×C4×C5311yescharacteristic
216C2×C42C2×C4×C5311yescharacteristic
217C2×C42D4×C5311yescharacteristic
218C2×C42D4×C5311yescharacteristic
219C2×C42D4×C5311yescharacteristic
220C2×C42D4×C5311yescharacteristic
221C2×C42C23×C5311yescharacteristic
222C4×C8—24no—
223#95—21no—
224#95—21no—
225#95—21no—
226C24—21no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4C2×C8×C5312yes—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C22⋊C4×C5311yescharacteristic
238C22×C4C22⋊C4×C5311yescharacteristic
239C22×C4C22⋊C4×C5311yescharacteristic
240C22×C4C22⋊C4×C5311yescharacteristic
241C22×C4C22×C4×C5311yescharacteristic
242C22×C4C2×D4×C5311yescharacteristic
243C22×C4C2×D4×C5311yescharacteristic
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4C2×C8×C5312yes—
256C22⋊C4—24no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—41no—
265C42—41no—
266C42—41no—
267C42—42no—
268C42C2×C8×C5312yes—
269C42M16×C5312yes—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊C4×C5311yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—48no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4C22⋊C8×C5312yes—
306C2×C4C22⋊C8×C5312yes—
307C2×C4#104×C5312yes—
308C2×C4C22×C8×C5311yescharacteristic
309C2×C4C2×M16×C5311yescharacteristic
310C2×C4#135×C5312yes—
311C2×C4AGL(1,ℤ/8)×C5312yes—
312C8—44no—
313C8—82no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×C22⋊C8×C5311yescharacteristic
322C22C53×#41011yescharacteristic
323C22C53×#41311yescharacteristic
324C22C53×#41611yescharacteristic
325C22C53×#41811yescharacteristic
326C22C2×#104×C5311yescharacteristic
327C22C53×#42111yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—42no—
336C4—41no—
337C4—81no—
338C4—81no—
339C2—82no—
340C2C53×#145811yescharacteristic
341C2C53×#147611yescharacteristic
342C2C53×#147811yescharacteristic
343C2C53×#150511yescharacteristic
344C2C53×#152111yescharacteristic
345C2C53×#153211yescharacteristic
346C2C53×#155311yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C43×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1839C2semidirect
3. C53×#1744C2semidirect
4. C53×#1271C2semidirect
28. C4×D4×C53C8semidirect
140. #7580C53direct (N × H)
167. #1839C2×C53semidirect
168. #1744C2×C53semidirect
169. #1271C2×C53semidirect
196. C53#7580direct (N × H)
202. C4×D4C8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#22932
C53×#18392
C53×#17442
C53×#12712
C53×#12702
#758053