Subgroups — C₅₃×#20564 #27228076

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#435	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2290	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1750	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#436	C₄	1	8	yes	complemented
24	C₅₃×#435	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#388	C₄	1	2	yes	—
40	C₅₃×#387	C₄	1	2	yes	—
41	C₅₃×#386	C₄	1	2	yes	—
42	C₅₃×#385	C₄	1	2	yes	—
43	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#376	C₂²	1	1	yes	characteristic
47	C₅₃×#336	C₂²	1	1	yes	characteristic
48	C₅₃×#336	C₂²	1	1	yes	characteristic
49	C₅₃×#336	C₂²	1	1	yes	characteristic
50	C₅₃×#336	C₂²	1	1	yes	characteristic
51	C₅₃×#336	C₂²	1	1	yes	characteristic
52	C₅₃×#336	C₂²	1	1	yes	characteristic
53	C₅₃×#336	C₂²	1	1	yes	characteristic
54	C₅₃×#336	C₂²	1	1	yes	characteristic
55	C₅₃×#336	C₂²	1	1	yes	characteristic
56	C₅₃×#336	C₂²	1	1	yes	characteristic
57	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₅₃	—	4	1	no	—
62	C₂×M₁₆×C₅₃	—	4	1	no	—
63	C₂×M₁₆×C₅₃	—	4	1	no	—
64	C₂×M₁₆×C₅₃	—	4	1	no	—
65	C₂×M₁₆×C₅₃	—	4	1	no	—
66	C₂×M₁₆×C₅₃	—	4	1	no	—
67	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
68	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
69	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
70	C₂²×C₈×C₅₃	—	4	1	no	—
71	C₂²×C₈×C₅₃	—	4	1	no	—
72	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
82	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
83	#117×C₅₃	—	4	1	no	—
84	#117×C₅₃	—	4	1	no	—
85	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	—	4	1	no	—
94	C₂×C₄²×C₅₃	—	4	1	no	—
95	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
102	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
103	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
104	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
105	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
106	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
107	#97×C₅₃	—	4	1	no	—
108	#97×C₅₃	—	4	1	no	—
109	#97×C₅₃	—	4	1	no	—
110	#97×C₅₃	—	4	1	no	—
111	#97×C₅₃	C₂×C₄	1	4	yes	complemented
112	C₄×C₈×C₅₃	—	4	1	no	—
113	C₄×C₈×C₅₃	—	4	1	no	—
114	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
115	#95×C₅₃	C₂×C₄	1	2	yes	—
116	#95×C₅₃	C₂×C₄	1	2	yes	—
117	#95×C₅₃	C₂×C₄	1	2	yes	—
118	#95×C₅₃	C₂×C₄	1	2	yes	—
119	#95×C₅₃	C₂×C₄	1	2	yes	—
120	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄×C₅₃	—	4	2	no	—
123	C₂×D₄×C₅₃	C₄²	1	4	yes	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	1	no	—
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	2	1	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	—	2	1	no	—
131	C₂²×C₄×C₅₃	—	2	1	no	—
132	C₂²×C₄×C₅₃	—	4	2	no	—
133	C₂²×C₄×C₅₃	—	4	2	no	—
134	C₂²×C₄×C₅₃	—	4	2	no	—
135	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
139	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
148	M₁₆×C₅₃	—	4	4	no	—
149	M₁₆×C₅₃	—	4	4	no	—
150	M₁₆×C₅₃	—	4	4	no	—
151	C₂×C₈×C₅₃	—	2	1	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	2	1	no	—
154	C₂×C₈×C₅₃	—	2	1	no	—
155	C₂×C₈×C₅₃	—	2	1	no	—
156	C₂×C₈×C₅₃	—	2	1	no	—
157	C₂×C₈×C₅₃	—	2	1	no	—
158	C₂×C₈×C₅₃	—	2	1	no	—
159	C₂×C₈×C₅₃	—	2	1	no	—
160	C₂×C₈×C₅₃	—	2	1	no	—
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	4	4	no	—
168	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
169	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
170	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
171	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
172	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
173	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
176	C₄⋊C₄×C₅₃	—	4	2	no	—
177	C₄⋊C₄×C₅₃	—	4	2	no	—
178	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
179	C₂²⋊C₄×C₅₃	—	4	2	no	—
180	C₂²⋊C₄×C₅₃	—	4	2	no	—
181	C₂²⋊C₄×C₅₃	—	4	2	no	—
182	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
183	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
184	C₄²×C₅₃	—	2	1	no	—
185	C₄²×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	—	2	1	no	—
187	C₄²×C₅₃	—	2	1	no	—
188	C₄²×C₅₃	—	2	1	no	—
189	C₄²×C₅₃	—	2	1	no	—
190	C₄²×C₅₃	—	2	1	no	—
191	C₄²×C₅₃	—	2	1	no	—
192	C₄²×C₅₃	—	4	2	no	—
193	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
194	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
195	C₂³×C₅₃	—	2	1	no	—
196	C₂³×C₅₃	—	2	1	no	—
197	C₂³×C₅₃	—	2	1	no	—
198	C₂³×C₅₃	—	2	1	no	—
199	C₂³×C₅₃	—	2	2	no	—
200	C₂³×C₅₃	—	2	2	no	—
201	C₂³×C₅₃	—	2	2	no	—
202	C₂³×C₅₃	—	2	2	no	—
203	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
204	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
205	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
206	D₄×C₅₃	—	4	4	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	2	2	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	2	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	2	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	2	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	4	8	no	—
234	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
235	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
236	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
237	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
238	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
244	C₈×C₅₃	—	2	2	no	—
245	C₈×C₅₃	—	2	2	no	—
246	C₈×C₅₃	—	2	2	no	—
247	C₈×C₅₃	—	2	2	no	—
248	C₈×C₅₃	—	2	2	no	—
249	C₈×C₅₃	—	2	2	no	—
250	C₈×C₅₃	—	2	2	no	—
251	C₈×C₅₃	—	2	2	no	—
252	#20564	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂²×C₅₃	—	2	2	no	—
254	C₂²×C₅₃	—	2	2	no	—
255	C₂²×C₅₃	—	2	2	no	—
256	C₂²×C₅₃	—	2	2	no	—
257	C₂²×C₅₃	—	2	2	no	—
258	C₂²×C₅₃	—	2	2	no	—
259	C₂²×C₅₃	—	4	2	no	—
260	C₂²×C₅₃	—	4	2	no	—
261	C₂²×C₅₃	—	4	2	no	—
262	C₂²×C₅₃	—	4	2	no	—
263	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
264	C₂²×C₅₃	#518	1	1	yes	characteristic
265	C₂²×C₅₃	#520	1	1	yes	characteristic
266	C₂²×C₅₃	#568	1	2	yes	—
267	C₂²×C₅₃	#568	1	2	yes	—
268	C₄×C₅₃	—	2	2	no	—
269	C₄×C₅₃	—	2	2	no	—
270	C₄×C₅₃	—	2	2	no	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	4	2	no	—
273	C₄×C₅₃	—	4	2	no	—
274	C₄×C₅₃	—	4	2	no	—
275	C₄×C₅₃	—	4	2	no	—
276	C₄×C₅₃	#518	1	1	yes	characteristic
277	C₄×C₅₃	#518	1	1	yes	characteristic
278	C₄×C₅₃	#520	1	1	yes	characteristic
279	C₄×C₅₃	#520	1	1	yes	characteristic
280	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
281	C₂×#435	C₂×C₅₃	1	1	yes	characteristic
282	#2290	C₂×C₅₃	1	1	yes	characteristic
283	#1771	C₂×C₅₃	1	1	yes	characteristic, complemented
284	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
285	#1768	C₂×C₅₃	1	1	yes	characteristic, complemented
286	#1753	C₂×C₅₃	1	1	yes	characteristic, complemented
287	#1753	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#1753	C₂×C₅₃	1	1	yes	characteristic, complemented
289	#1752	C₂×C₅₃	1	1	yes	characteristic, complemented
290	#1750	C₂×C₅₃	1	1	yes	characteristic, complemented
291	#1749	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
293	#1729	C₂×C₅₃	1	1	yes	characteristic, complemented
294	#1728	C₂×C₅₃	1	1	yes	characteristic, complemented
295	C₂×C₅₃	—	4	2	no	—
296	C₂×C₅₃	—	4	2	no	—
297	C₂×C₅₃	#2275	1	1	yes	characteristic
298	C₂×C₅₃	#2873	1	1	yes	characteristic
299	C₂×C₅₃	#2875	1	1	yes	characteristic
300	C₂×C₅₃	#2978	1	2	yes	—
301	C₂×C₅₃	#2995	1	2	yes	—
302	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
303	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
304	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
305	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
307	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
308	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
309	#436	C₄×C₅₃	1	8	yes	complemented
310	#435	C₄×C₅₃	1	8	yes	complemented
311	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
315	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
316	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
317	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
318	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
319	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
320	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
321	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
322	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
324	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
325	#388	C₄×C₅₃	1	2	yes	—
326	#387	C₄×C₅₃	1	2	yes	—
327	#386	C₄×C₅₃	1	2	yes	—
328	#385	C₄×C₅₃	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
330	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
331	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
332	#376	C₂²×C₅₃	1	1	yes	characteristic
333	#336	C₂²×C₅₃	1	1	yes	characteristic
334	#336	C₂²×C₅₃	1	1	yes	characteristic
335	#336	C₂²×C₅₃	1	1	yes	characteristic
336	#336	C₂²×C₅₃	1	1	yes	characteristic
337	#336	C₂²×C₅₃	1	1	yes	characteristic
338	#336	C₂²×C₅₃	1	1	yes	characteristic
339	#336	C₂²×C₅₃	1	1	yes	characteristic
340	#336	C₂²×C₅₃	1	1	yes	characteristic
341	#336	C₂²×C₅₃	1	1	yes	characteristic
342	#336	C₂²×C₅₃	1	1	yes	characteristic
343	C₅₃	#20564	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
345	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
346	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
348	C₂×M₁₆	—	4	1	no	—
349	C₂×M₁₆	—	4	1	no	—
350	C₂×M₁₆	—	4	1	no	—
351	C₂×M₁₆	—	4	1	no	—
352	C₂×M₁₆	—	4	1	no	—
353	C₂×M₁₆	—	4	1	no	—
354	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
355	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
356	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
357	C₂²×C₈	—	4	1	no	—
358	C₂²×C₈	—	4	1	no	—
359	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
360	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
361	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
368	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
369	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
370	#117	—	4	1	no	—
371	#117	—	4	1	no	—
372	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
373	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
374	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
376	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄²	—	4	1	no	—
381	C₂×C₄²	—	4	1	no	—
382	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
383	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
388	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
389	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
390	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
391	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
392	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
393	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
394	#97	—	4	1	no	—
395	#97	—	4	1	no	—
396	#97	—	4	1	no	—
397	#97	—	4	1	no	—
398	#97	C₂×C₄×C₅₃	1	4	yes	complemented
399	C₄×C₈	—	4	1	no	—
400	C₄×C₈	—	4	1	no	—
401	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
402	#95	C₂×C₄×C₅₃	1	2	yes	—
403	#95	C₂×C₄×C₅₃	1	2	yes	—
404	#95	C₂×C₄×C₅₃	1	2	yes	—
405	#95	C₂×C₄×C₅₃	1	2	yes	—
406	#95	C₂×C₄×C₅₃	1	2	yes	—
407	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
408	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
409	C₂×D₄	—	4	2	no	—
410	C₂×D₄	C₄²×C₅₃	1	4	yes	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	4	2	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
423	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
424	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
425	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
426	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
428	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
431	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
432	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
433	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
435	M₁₆	—	4	4	no	—
436	M₁₆	—	4	4	no	—
437	M₁₆	—	4	4	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	4	4	no	—
455	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
456	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
457	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
458	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
459	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
460	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
461	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
462	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
463	C₄⋊C₄	—	4	2	no	—
464	C₄⋊C₄	—	4	2	no	—
465	C₄⋊C₄	C₄²×C₅₃	1	4	yes	—
466	C₂²⋊C₄	—	4	2	no	—
467	C₂²⋊C₄	—	4	2	no	—
468	C₂²⋊C₄	—	4	2	no	—
469	C₂²⋊C₄	C₄²×C₅₃	1	4	yes	—
470	C₂²⋊C₄	C₄²×C₅₃	1	4	yes	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	1	no	—
474	C₄²	—	2	1	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	4	2	no	—
480	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
481	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
482	C₂³	—	2	1	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	2	no	—
490	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
491	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
492	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
493	D₄	—	4	4	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
522	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
523	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
524	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
525	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
526	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
527	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
528	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
529	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
530	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
531	C₈	—	2	2	no	—
532	C₈	—	2	2	no	—
533	C₈	—	2	2	no	—
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₈	—	2	2	no	—
537	C₈	—	2	2	no	—
538	C₈	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
550	C₂²	C₅₃×#518	1	1	yes	characteristic
551	C₂²	C₅₃×#520	1	1	yes	characteristic
552	C₂²	C₅₃×#568	1	2	yes	—
553	C₂²	C₅₃×#568	1	2	yes	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	2	no	—
562	C₄	C₅₃×#518	1	1	yes	characteristic
563	C₄	C₅₃×#518	1	1	yes	characteristic
564	C₄	C₅₃×#520	1	1	yes	characteristic
565	C₄	C₅₃×#520	1	1	yes	characteristic
566	C₂	—	4	2	no	—
567	C₂	—	4	2	no	—
568	C₂	C₅₃×#2275	1	1	yes	characteristic
569	C₂	C₅₃×#2873	1	1	yes	characteristic
570	C₂	C₅₃×#2875	1	1	yes	characteristic
571	C₂	C₅₃×#2978	1	2	yes	—
572	C₂	C₅₃×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1771	C₂	semidirect
5. C₅₃×#1770	C₂	semidirect
6. C₅₃×#1768	C₂	semidirect
7. C₅₃×#1753	C₂	semidirect
8. C₅₃×#1753	C₂	semidirect
9. C₅₃×#1753	C₂	semidirect
10. C₅₃×#1752	C₂	semidirect
11. C₅₃×#1750	C₂	semidirect
12. C₅₃×#1749	C₂	semidirect
13. C₅₃×#1746	C₂	semidirect
14. C₅₃×#1729	C₂	semidirect
15. C₅₃×#1728	C₂	semidirect
23. C₅₃×#436	C₄	semidirect
24. C₅₃×#435	C₄	semidirect
69. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
72. C₂²×C₈×C₅₃	C₂×C₄	semidirect
101. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
102. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
105. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
106. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
111. #97×C₅₃	C₂×C₄	semidirect
114. C₄×C₈×C₅₃	C₂×C₄	semidirect
252. #20564	C₅₃	direct (N × H)
283. #1771	C₂×C₅₃	semidirect
284. #1770	C₂×C₅₃	semidirect
285. #1768	C₂×C₅₃	semidirect
286. #1753	C₂×C₅₃	semidirect
287. #1753	C₂×C₅₃	semidirect
288. #1753	C₂×C₅₃	semidirect
289. #1752	C₂×C₅₃	semidirect
290. #1750	C₂×C₅₃	semidirect
291. #1749	C₂×C₅₃	semidirect
292. #1746	C₂×C₅₃	semidirect
293. #1729	C₂×C₅₃	semidirect
294. #1728	C₂×C₅₃	semidirect
309. #436	C₄×C₅₃	semidirect
310. #435	C₄×C₅₃	semidirect
343. C₅₃	#20564	direct (N × H)
356. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
359. C₂²×C₈	C₂×C₄×C₅₃	semidirect
388. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
389. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
392. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
393. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
398. #97	C₂×C₄×C₅₃	semidirect
401. C₄×C₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₂×C₅₃×#435	2
C₅₃×#2290	2
C₅₃×#1771	2
C₅₃×#1770	2
C₅₃×#1768	2
C₅₃×#1753	2
C₅₃×#1752	2
C₅₃×#1750	2
C₅₃×#1749	2
C₅₃×#1746	2
C₅₃×#1729	2
C₅₃×#1728	2
#20564	53

Subgroups — C53×#20564 #27228076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20564 #27228076