Subgroups — C₅₃×#9640 #27217152

Order 13568.

582 conjugacy classes of subgroups.

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

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×D₄⋊C₈	C₂	1	2	yes	maximal
6	C₂×#124×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#391	C₄	1	1	yes	characteristic
13	C₅₃×#390	C₄	1	1	yes	characteristic
14	C₅₃×#388	—	2	2	no	—
15	C₅₃×#382	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₅₃×Q₈⋊C₈	—	2	4	no	—
20	C₅₃×D₄⋊C₈	—	2	4	no	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	#124×C₅₃	—	2	2	no	—
27	#124×C₅₃	C₈	1	4	yes	complemented
28	C₄×Q₈×C₅₃	—	2	2	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×D₄×C₅₃	—	2	2	no	—
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	—	4	4	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	D₄	1	4	yes	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	C₂⁴×C₅₃	—	2	1	no	—
52	C₂×Q₈×C₅₃	—	2	1	no	—
53	C₂×Q₈×C₅₃	—	2	2	no	—
54	C₂×Q₈×C₅₃	—	2	2	no	—
55	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	2	1	no	—
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	4	no	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	4	2	no	—
79	C₂²⋊C₄×C₅₃	—	2	8	no	—
80	C₂²⋊C₄×C₅₃	—	4	4	no	—
81	C₂²⋊C₄×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	4	2	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	C₂×C₈	1	2	yes	—
89	C₄²×C₅₃	M₁₆	1	2	yes	—
90	C₄²×C₅₃	C₂×D₄	1	2	yes	—
91	C₄²×C₅₃	#41	1	2	yes	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	—	4	1	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	4	2	no	—
97	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	Q₈×C₅₃	—	2	4	no	—
99	Q₈×C₅₃	—	4	2	no	—
100	D₄×C₅₃	—	2	4	no	—
101	D₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	—	4	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	—	8	1	no	—
117	C₂×C₄×C₅₃	—	8	1	no	—
118	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	#121	1	2	yes	—
121	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#131	1	2	yes	—
126	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₅₃	#137	1	2	yes	—
128	C₈×C₅₃	—	4	4	no	—
129	C₈×C₅₃	—	8	2	no	—
130	#9640	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	4	no	—
133	C₂²×C₅₃	—	8	1	no	—
134	C₂²×C₅₃	—	8	1	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	#390	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₅₃	#408	1	1	yes	characteristic
139	C₂²×C₅₃	#417	1	2	yes	—
140	C₂²×C₅₃	#421	1	2	yes	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	4	no	—
143	C₄×C₅₃	—	4	2	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	—	8	1	no	—
146	C₄×C₅₃	—	8	1	no	—
147	C₄×C₅₃	C₈×D₄	1	2	yes	—
148	C₄×C₅₃	#436	1	2	yes	—
149	C₄×C₅₃	#489	1	2	yes	—
150	C₄×C₅₃	#496	1	1	yes	characteristic
151	C₄×C₅₃	#497	1	1	yes	characteristic
152	C₄×#124	C₂×C₅₃	1	1	yes	characteristic
153	#1842	C₂×C₅₃	1	1	yes	characteristic, complemented
154	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	complemented
156	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	—
157	C₂×C₅₃	—	8	2	no	—
158	C₂×C₅₃	#1477	1	2	yes	—
159	C₂×C₅₃	#1495	1	2	yes	—
160	C₂×C₅₃	#1954	1	1	yes	characteristic
161	C₂×C₅₃	#1959	1	1	yes	characteristic
162	C₂×C₅₃	#1963	1	1	yes	characteristic
163	C₂×#124	C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	2	2	no	—
167	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
169	#391	C₄×C₅₃	1	1	yes	characteristic
170	#390	C₄×C₅₃	1	1	yes	characteristic
171	#388	—	2	2	no	—
172	#382	C₄×C₅₃	1	1	yes	characteristic
173	C₄×C₂²⋊C₄	—	2	2	no	—
174	C₄³	C₂²×C₅₃	1	1	yes	characteristic
175	#336	—	2	2	no	—
176	Q₈⋊C₈	—	2	4	no	—
177	D₄⋊C₈	—	2	4	no	—
178	C₅₃	#9640	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
180	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	#124	—	2	2	no	—
185	#124	C₈×C₅₃	1	4	yes	complemented
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₅₃	1	2	yes	—
200	C₂×C₄²	D₄×C₅₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	4	4	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	D₄×C₅₃	1	4	yes	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂⁴	—	2	1	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂×C₈×C₅₃	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₂×C₈×C₅₃	1	2	yes	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	C₂×C₈×C₅₃	1	2	yes	—
247	C₄²	M₁₆×C₅₃	1	2	yes	—
248	C₄²	C₂×D₄×C₅₃	1	2	yes	—
249	C₄²	#41×C₅₃	1	2	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
277	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
278	C₂×C₄	#121×C₅₃	1	2	yes	—
279	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
280	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
283	C₂×C₄	#131×C₅₃	1	2	yes	—
284	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
285	C₂×C₄	#137×C₅₃	1	2	yes	—
286	C₈	—	4	4	no	—
287	C₈	—	8	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	C₅₃×#390	1	1	yes	characteristic
294	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
295	C₂²	C₅₃×#408	1	1	yes	characteristic
296	C₂²	C₅₃×#417	1	2	yes	—
297	C₂²	C₅₃×#421	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	1	no	—
303	C₄	—	8	1	no	—
304	C₄	C₈×D₄×C₅₃	1	2	yes	—
305	C₄	C₅₃×#436	1	2	yes	—
306	C₄	C₅₃×#489	1	2	yes	—
307	C₄	C₅₃×#496	1	1	yes	characteristic
308	C₄	C₅₃×#497	1	1	yes	characteristic
309	C₂	—	8	2	no	—
310	C₂	C₅₃×#1477	1	2	yes	—
311	C₂	C₅₃×#1495	1	2	yes	—
312	C₂	C₅₃×#1954	1	1	yes	characteristic
313	C₂	C₅₃×#1959	1	1	yes	characteristic
314	C₂	C₅₃×#1963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1842	C₂	semidirect
3. C₅₃×#1744	C₂	semidirect
4. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
27. #124×C₅₃	C₈	semidirect
130. #9640	C₅₃	direct (N × H)
153. #1842	C₂×C₅₃	semidirect
154. #1744	C₂×C₅₃	semidirect
155. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
178. C₅₃	#9640	direct (N × H)
185. #124	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#124×C₅₃	2
C₅₃×#1842	2
C₅₃×#1744	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×D₄⋊C₈	2
#9640	53

Subgroups — C53×#9640 #27217152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9640 #27217152