Subgroups — C₅₃×#23051 #27230563

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1924	C₂	1	2	yes	maximal
7	C₅₃×#1923	C₂	1	2	yes	maximal
8	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1771	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₅₃	—	2	4	no	—
16	C₅₃×#432	—	2	4	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	2	yes	—
25	C₂×#97×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#391	C₄	1	1	yes	characteristic
30	C₅₃×#389	C₄	1	1	yes	characteristic
31	C₅₃×#387	C₄	1	2	yes	—
32	C₅₃×#385	C₄	1	2	yes	—
33	C₅₃×#380	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#336	C₂²	1	1	yes	characteristic
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²⋊Q₈×C₅₃	—	2	4	no	—
54	C₄×Q₈×C₅₃	—	2	2	no	—
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	—	2	4	no	—
73	C₄⋊C₈×C₅₃	—	2	4	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
76	C₂²⋊C₈×C₅₃	—	2	4	no	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	#97×C₅₃	—	2	4	no	—
80	#97×C₅₃	—	2	4	no	—
81	C₄×C₈×C₅₃	—	2	4	no	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	2	2	no	—
84	#95×C₅₃	C₂×C₄	1	2	yes	—
85	#95×C₅₃	C₂×C₄	1	2	yes	—
86	#95×C₅₃	C₂×C₄	1	2	yes	—
87	#95×C₅₃	C₂×C₄	1	2	yes	—
88	#95×C₅₃	C₂×C₄	1	2	yes	—
89	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	1	no	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₅₃	#41	1	2	yes	—
132	C₂×C₈×C₅₃	#41	1	2	yes	—
133	C₄⋊C₄×C₅₃	—	2	4	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	4	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	4	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	4	no	—
140	C₂²⋊C₄×C₅₃	—	2	4	no	—
141	C₄²×C₅₃	—	2	2	no	—
142	C₄²×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	2	2	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	C₂×D₄	1	2	yes	—
150	C₄²×C₅₃	#41	1	2	yes	—
151	C₂³×C₅₃	—	2	1	no	—
152	C₂³×C₅₃	—	2	1	no	—
153	C₂³×C₅₃	—	2	1	no	—
154	C₂³×C₅₃	—	2	1	no	—
155	C₂³×C₅₃	—	2	1	no	—
156	C₂³×C₅₃	—	2	1	no	—
157	C₂³×C₅₃	#131	1	2	yes	—
158	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	Q₈×C₅₃	—	2	8	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	4	4	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	4	no	—
188	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
189	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
191	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₅₃	#131	1	2	yes	—
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
199	C₈×C₅₃	—	2	4	no	—
200	C₈×C₅₃	—	2	2	no	—
201	C₈×C₅₃	—	2	2	no	—
202	C₈×C₅₃	—	4	1	no	—
203	C₈×C₅₃	—	4	2	no	—
204	C₈×C₅₃	—	4	1	no	—
205	#23051	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	2	2	no	—
209	C₂²×C₅₃	—	4	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	—	4	1	no	—
213	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
215	C₂²×C₅₃	#519	1	1	yes	characteristic
216	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
217	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
218	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
219	C₂²×C₅₃	#568	1	1	yes	characteristic
220	C₄×C₅₃	—	2	4	no	—
221	C₄×C₅₃	—	2	2	no	—
222	C₄×C₅₃	—	4	1	no	—
223	C₄×C₅₃	—	4	1	no	—
224	C₄×C₅₃	—	4	2	no	—
225	C₄×C₅₃	—	4	1	no	—
226	C₄×C₅₃	—	4	2	no	—
227	C₄×C₅₃	—	4	2	no	—
228	C₄×C₅₃	#435	1	4	yes	—
229	C₄×C₅₃	#547	1	1	yes	characteristic
230	C₄×C₅₃	#548	1	1	yes	characteristic
231	C₄×C₅₃	#552	1	1	yes	characteristic
232	C₄×C₅₃	#556	1	1	yes	characteristic
233	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
235	C₄×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
236	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#1924	C₂×C₅₃	1	2	yes	—
239	#1923	C₂×C₅₃	1	2	yes	—
240	#1917	C₂×C₅₃	1	2	yes	complemented
241	#1771	C₂×C₅₃	1	2	yes	complemented
242	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic
244	C₂×C₅₃	—	4	2	no	—
245	C₂×C₅₃	#2362	1	1	yes	characteristic
246	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
247	C₂×C₅₃	#2933	1	1	yes	characteristic
248	C₂×C₅₃	#2966	1	1	yes	characteristic
249	C₂×C₅₃	#2971	1	1	yes	characteristic
250	C₂×C₅₃	#2984	1	1	yes	characteristic
251	C₂×C₅₃	#2985	1	1	yes	characteristic
252	C₂×C₂²⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
254	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
255	C₈×Q₈	—	2	4	no	—
256	#432	—	2	4	no	—
257	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
258	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
259	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
260	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×#97	C₂²×C₅₃	1	2	yes	—
265	C₂×#97	C₂²×C₅₃	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
267	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
269	#391	C₄×C₅₃	1	1	yes	characteristic
270	#389	C₄×C₅₃	1	1	yes	characteristic
271	#387	C₄×C₅₃	1	2	yes	—
272	#385	C₄×C₅₃	1	2	yes	—
273	#380	C₄×C₅₃	1	1	yes	characteristic
274	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
275	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
276	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
277	#336	C₂²×C₅₃	1	1	yes	characteristic
278	#336	C₂²×C₅₃	1	2	yes	—
279	#336	C₂²×C₅₃	1	2	yes	—
280	#336	C₂²×C₅₃	1	2	yes	—
281	#336	C₂²×C₅₃	1	2	yes	—
282	#336	C₂²×C₅₃	1	1	yes	characteristic
283	C₅₃	#23051	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
285	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
286	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
291	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
292	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
294	C₂²⋊Q₈	—	2	4	no	—
295	C₄×Q₈	—	2	2	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
299	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
300	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
303	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
308	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
310	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
311	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
313	C₄⋊C₈	—	2	4	no	—
314	C₄⋊C₈	—	2	4	no	—
315	C₄⋊C₈	—	2	2	no	—
316	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
317	C₂²⋊C₈	—	2	4	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	#97	—	2	4	no	—
321	#97	—	2	4	no	—
322	C₄×C₈	—	2	4	no	—
323	C₄×C₈	—	2	2	no	—
324	C₄×C₈	—	2	2	no	—
325	#95	C₂×C₄×C₅₃	1	2	yes	—
326	#95	C₂×C₄×C₅₃	1	2	yes	—
327	#95	C₂×C₄×C₅₃	1	2	yes	—
328	#95	C₂×C₄×C₅₃	1	2	yes	—
329	#95	C₂×C₄×C₅₃	1	2	yes	—
330	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
351	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
353	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
354	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
372	C₂×C₈	#41×C₅₃	1	2	yes	—
373	C₂×C₈	#41×C₅₃	1	2	yes	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	C₂×D₄×C₅₃	1	2	yes	—
391	C₄²	#41×C₅₃	1	2	yes	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	#131×C₅₃	1	2	yes	—
399	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
400	Q₈	—	2	8	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
430	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
431	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
432	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
433	C₂×C₄	#131×C₅₃	1	2	yes	—
434	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
438	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
439	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
440	C₈	—	2	4	no	—
441	C₈	—	2	2	no	—
442	C₈	—	2	2	no	—
443	C₈	—	4	1	no	—
444	C₈	—	4	2	no	—
445	C₈	—	4	1	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
454	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
455	C₂²	C₅₃×#519	1	1	yes	characteristic
456	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
457	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
458	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
459	C₂²	C₅₃×#568	1	1	yes	characteristic
460	C₄	—	2	4	no	—
461	C₄	—	2	2	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	C₅₃×#435	1	4	yes	—
469	C₄	C₅₃×#547	1	1	yes	characteristic
470	C₄	C₅₃×#548	1	1	yes	characteristic
471	C₄	C₅₃×#552	1	1	yes	characteristic
472	C₄	C₅₃×#556	1	1	yes	characteristic
473	C₂	—	4	2	no	—
474	C₂	C₅₃×#2362	1	1	yes	characteristic
475	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
476	C₂	C₅₃×#2933	1	1	yes	characteristic
477	C₂	C₅₃×#2966	1	1	yes	characteristic
478	C₂	C₅₃×#2971	1	1	yes	characteristic
479	C₂	C₅₃×#2984	1	1	yes	characteristic
480	C₂	C₅₃×#2985	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₅₃	C₂	semidirect
4. C₅₃×#1984	C₂	semidirect
5. C₅₃×#1939	C₂	semidirect
8. C₅₃×#1917	C₂	semidirect
9. C₅₃×#1771	C₂	semidirect
10. C₅₃×#1770	C₂	semidirect
205. #23051	C₅₃	direct (N × H)
233. C₂×C₈×Q₈	C₂×C₅₃	semidirect
236. #1984	C₂×C₅₃	semidirect
237. #1939	C₂×C₅₃	semidirect
240. #1917	C₂×C₅₃	semidirect
241. #1771	C₂×C₅₃	semidirect
242. #1770	C₂×C₅₃	semidirect
283. C₅₃	#23051	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₅₃	2
C₂×C₅₃×#432	2
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#1984	2
C₅₃×#1939	2
C₅₃×#1924	2
C₅₃×#1923	2
C₅₃×#1917	2
C₅₃×#1771	2
C₅₃×#1770	2
C₄×C₂²⋊C₈×C₅₃	2
#23051	53

Subgroups — C53×#23051 #27230563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23051 #27230563