Subgroups — C₅₃×#23083 #27230595

Order 13568.

1040 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#121×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1954	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1923	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1869	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#121×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₅₃	—	2	4	no	—
18	C₅₃×#431	—	2	4	no	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊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×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#390	C₄	1	1	yes	characteristic
33	C₅₃×#389	C₄	1	1	yes	characteristic
34	C₅₃×#388	C₄	1	2	yes	—
35	C₅₃×#386	C₄	1	2	yes	—
36	C₅₃×#380	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	—	2	4	no	—
50	C₂²×C₈×C₅₃	—	2	4	no	—
51	C₂²×C₈×C₅₃	—	4	2	no	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	#121×C₅₃	C₈	1	8	yes	complemented
59	C₄×D₄×C₅₃	—	2	2	no	—
60	C₄×D₄×C₅₃	—	2	2	no	—
61	C₄×D₄×C₅₃	—	4	2	no	—
62	C₄×D₄×C₅₃	—	4	2	no	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₅₃	—	2	2	no	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	—	2	2	no	—
80	C₄⋊C₈×C₅₃	—	4	2	no	—
81	C₄⋊C₈×C₅₃	D₄	1	4	yes	complemented
82	C₂²⋊C₈×C₅₃	—	2	4	no	—
83	C₂²⋊C₈×C₅₃	—	2	4	no	—
84	C₂²⋊C₈×C₅₃	—	2	2	no	—
85	C₂²⋊C₈×C₅₃	—	2	4	no	—
86	C₂²⋊C₈×C₅₃	—	2	2	no	—
87	C₂²⋊C₈×C₅₃	—	4	2	no	—
88	C₄×C₈×C₅₃	—	2	4	no	—
89	C₄×C₈×C₅₃	—	2	2	no	—
90	C₄×C₈×C₅₃	—	2	2	no	—
91	C₄×C₈×C₅₃	—	4	2	no	—
92	#95×C₅₃	C₂×C₄	1	2	yes	—
93	#95×C₅₃	C₂×C₄	1	2	yes	—
94	#95×C₅₃	C₂×C₄	1	2	yes	—
95	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×D₄×C₅₃	—	2	2	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	4	2	no	—
100	C₂×D₄×C₅₃	—	4	2	no	—
101	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
102	C₂×D₄×C₅₃	C₂×C₈	1	8	yes	complemented
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	4	1	no	—
120	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
121	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	1	no	—
145	C₂×C₈×C₅₃	—	4	8	no	—
146	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
147	C₂×C₈×C₅₃	#41	1	2	yes	—
148	C₂×C₈×C₅₃	#41	1	2	yes	—
149	C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₄⋊C₄×C₅₃	—	2	2	no	—
151	C₄⋊C₄×C₅₃	—	4	2	no	—
152	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
153	C₂²⋊C₄×C₅₃	—	2	4	no	—
154	C₂²⋊C₄×C₅₃	—	2	4	no	—
155	C₂²⋊C₄×C₅₃	—	4	2	no	—
156	C₂²⋊C₄×C₅₃	—	4	2	no	—
157	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	—	2	1	no	—
161	C₄²×C₅₃	—	2	1	no	—
162	C₄²×C₅₃	—	2	2	no	—
163	C₄²×C₅₃	—	2	2	no	—
164	C₄²×C₅₃	C₂×D₄	1	2	yes	—
165	C₄²×C₅₃	#41	1	2	yes	—
166	C₂³×C₅₃	—	2	1	no	—
167	C₂³×C₅₃	—	2	2	no	—
168	C₂³×C₅₃	—	2	1	no	—
169	C₂³×C₅₃	—	2	2	no	—
170	C₂³×C₅₃	—	2	1	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	1	no	—
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	2	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	—	2	2	no	—
178	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
179	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
180	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
181	D₄×C₅₃	—	2	8	no	—
182	D₄×C₅₃	—	4	8	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	8	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	4	4	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	1	no	—
205	C₂×C₄×C₅₃	—	4	2	no	—
206	C₂×C₄×C₅₃	—	4	1	no	—
207	C₂×C₄×C₅₃	—	4	2	no	—
208	C₂×C₄×C₅₃	—	4	1	no	—
209	C₂×C₄×C₅₃	—	4	2	no	—
210	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
211	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
212	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
213	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
214	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
215	C₂×C₄×C₅₃	#131	1	2	yes	—
216	C₂×C₄×C₅₃	#131	1	2	yes	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
223	C₈×C₅₃	—	2	2	no	—
224	C₈×C₅₃	—	2	4	no	—
225	C₈×C₅₃	—	2	2	no	—
226	C₈×C₅₃	—	4	1	no	—
227	C₈×C₅₃	—	4	2	no	—
228	C₈×C₅₃	—	4	1	no	—
229	#23083	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂²×C₅₃	—	2	2	no	—
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	2	2	no	—
233	C₂²×C₅₃	—	2	4	no	—
234	C₂²×C₅₃	—	2	2	no	—
235	C₂²×C₅₃	—	2	4	no	—
236	C₂²×C₅₃	—	4	1	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	—	4	2	no	—
240	C₂²×C₅₃	—	4	1	no	—
241	C₂²×C₅₃	—	4	2	no	—
242	C₂²×C₅₃	—	4	1	no	—
243	C₂²×C₅₃	—	4	2	no	—
244	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
246	C₂²×C₅₃	#518	1	1	yes	characteristic
247	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
248	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
249	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
250	C₂²×C₅₃	#568	1	1	yes	characteristic
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	2	no	—
253	C₄×C₅₃	—	2	2	no	—
254	C₄×C₅₃	—	2	4	no	—
255	C₄×C₅₃	—	4	1	no	—
256	C₄×C₅₃	—	4	2	no	—
257	C₄×C₅₃	—	4	2	no	—
258	C₄×C₅₃	C₈×D₄	1	4	yes	—
259	C₄×C₅₃	#546	1	1	yes	characteristic
260	C₄×C₅₃	#550	1	1	yes	characteristic
261	C₄×C₅₃	#553	1	1	yes	characteristic
262	C₄×C₅₃	#555	1	1	yes	characteristic
263	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₂×#431	C₂×C₅₃	1	1	yes	characteristic, complemented
265	C₄×#121	C₂×C₅₃	1	1	yes	characteristic
266	#1954	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#1954	C₂×C₅₃	1	2	yes	complemented
268	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#1923	C₂×C₅₃	1	2	yes	complemented
270	#1869	C₂×C₅₃	1	2	yes	complemented
271	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₈×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
273	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
274	C₂×C₅₃	—	4	2	no	—
275	C₂×C₅₃	—	4	4	no	—
276	C₂×C₅₃	#2363	1	1	yes	characteristic
277	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
278	C₂×C₅₃	#2933	1	1	yes	characteristic
279	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
280	C₂×C₅₃	#2969	1	1	yes	characteristic
281	C₂×C₅₃	#2972	1	1	yes	characteristic
282	C₂×C₅₃	#2979	1	1	yes	characteristic
283	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
284	C₂×#121	C₄×C₅₃	1	1	yes	characteristic
285	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
287	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
288	C₈×D₄	—	2	4	no	—
289	#431	—	2	4	no	—
290	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
292	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
293	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
297	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
298	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
299	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
300	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
301	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
303	#390	C₄×C₅₃	1	1	yes	characteristic
304	#389	C₄×C₅₃	1	1	yes	characteristic
305	#388	C₄×C₅₃	1	2	yes	—
306	#386	C₄×C₅₃	1	2	yes	—
307	#380	C₄×C₅₃	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
309	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
310	#336	C₂²×C₅₃	1	2	yes	—
311	#336	C₂²×C₅₃	1	2	yes	—
312	#336	C₂²×C₅₃	1	1	yes	characteristic
313	#336	C₂²×C₅₃	1	1	yes	characteristic
314	C₅₃	#23083	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
316	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
317	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
318	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
320	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
321	C₂²×C₈	—	2	4	no	—
322	C₂²×C₈	—	2	4	no	—
323	C₂²×C₈	—	4	2	no	—
324	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
327	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
328	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
330	#121	C₈×C₅₃	1	8	yes	complemented
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	—	4	2	no	—
334	C₄×D₄	—	4	2	no	—
335	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
338	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
340	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
341	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
342	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
343	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
347	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
349	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
351	C₄⋊C₈	—	2	2	no	—
352	C₄⋊C₈	—	4	2	no	—
353	C₄⋊C₈	D₄×C₅₃	1	4	yes	complemented
354	C₂²⋊C₈	—	2	4	no	—
355	C₂²⋊C₈	—	2	4	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	C₂²⋊C₈	—	2	4	no	—
358	C₂²⋊C₈	—	2	2	no	—
359	C₂²⋊C₈	—	4	2	no	—
360	C₄×C₈	—	2	4	no	—
361	C₄×C₈	—	2	2	no	—
362	C₄×C₈	—	2	2	no	—
363	C₄×C₈	—	4	2	no	—
364	#95	C₂×C₄×C₅₃	1	2	yes	—
365	#95	C₂×C₄×C₅₃	1	2	yes	—
366	#95	C₂×C₄×C₅₃	1	2	yes	—
367	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
368	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
374	C₂×D₄	C₂×C₈×C₅₃	1	8	yes	complemented
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	4	1	no	—
392	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	complemented
393	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
394	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
395	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
399	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
400	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	4	8	no	—
418	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
419	C₂×C₈	#41×C₅₃	1	2	yes	—
420	C₂×C₈	#41×C₅₃	1	2	yes	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	complemented
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₂²⋊C₄	C₂×C₈×C₅₃	1	8	yes	complemented
430	C₄²	—	2	2	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	2	no	—
436	C₄²	C₂×D₄×C₅₃	1	2	yes	—
437	C₄²	#41×C₅₃	1	2	yes	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	1	no	—
449	C₂³	—	2	2	no	—
450	C₂³	C₂²×C₈×C₅₃	1	4	yes	—
451	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
452	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
453	D₄	—	2	8	no	—
454	D₄	—	4	8	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
483	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
484	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
485	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
486	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
487	C₂×C₄	#131×C₅₃	1	2	yes	—
488	C₂×C₄	#131×C₅₃	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
493	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
494	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
495	C₈	—	2	2	no	—
496	C₈	—	2	4	no	—
497	C₈	—	2	2	no	—
498	C₈	—	4	1	no	—
499	C₈	—	4	2	no	—
500	C₈	—	4	1	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	2	no	—
515	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
516	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
517	C₂²	C₅₃×#518	1	1	yes	characteristic
518	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
519	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
520	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
521	C₂²	C₅₃×#568	1	1	yes	characteristic
522	C₄	—	2	4	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	2	no	—
529	C₄	C₈×D₄×C₅₃	1	4	yes	—
530	C₄	C₅₃×#546	1	1	yes	characteristic
531	C₄	C₅₃×#550	1	1	yes	characteristic
532	C₄	C₅₃×#553	1	1	yes	characteristic
533	C₄	C₅₃×#555	1	1	yes	characteristic
534	C₂	—	4	2	no	—
535	C₂	—	4	4	no	—
536	C₂	C₅₃×#2363	1	1	yes	characteristic
537	C₂	C₂×C₈×D₄×C₅₃	1	1	yes	characteristic
538	C₂	C₅₃×#2933	1	1	yes	characteristic
539	C₂	C₈×#41×C₅₃	1	1	yes	characteristic
540	C₂	C₅₃×#2969	1	1	yes	characteristic
541	C₂	C₅₃×#2972	1	1	yes	characteristic
542	C₂	C₅₃×#2979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂×#121×C₅₃	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
1. C₂×C₈×D₄×C₅₃	C₂	semidirect
2. C₂×C₅₃×#431	C₂	semidirect
4. C₅₃×#1954	C₂	semidirect
5. C₅₃×#1954	C₂	semidirect
6. C₅₃×#1939	C₂	semidirect
7. C₅₃×#1923	C₂	semidirect
8. C₅₃×#1869	C₂	semidirect
9. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₈×C₂²⋊C₄×C₅₃	C₂	semidirect
11. C₄×C₂²⋊C₈×C₅₃	C₂	semidirect
58. #121×C₅₃	C₈	semidirect
81. C₄⋊C₈×C₅₃	D₄	semidirect
102. C₂×D₄×C₅₃	C₂×C₈	semidirect
120. C₂²×C₄×C₅₃	C₂×C₈	semidirect
152. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
157. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
229. #23083	C₅₃	direct (N × H)
263. C₂×C₈×D₄	C₂×C₅₃	semidirect
264. C₂×#431	C₂×C₅₃	semidirect
266. #1954	C₂×C₅₃	semidirect
267. #1954	C₂×C₅₃	semidirect
268. #1939	C₂×C₅₃	semidirect
269. #1923	C₂×C₅₃	semidirect
270. #1869	C₂×C₅₃	semidirect
271. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
272. C₈×C₂²⋊C₄	C₂×C₅₃	semidirect
273. C₄×C₂²⋊C₈	C₂×C₅₃	semidirect
314. C₅₃	#23083	direct (N × H)
330. #121	C₈×C₅₃	semidirect
353. C₄⋊C₈	D₄×C₅₃	semidirect
374. C₂×D₄	C₂×C₈×C₅₃	semidirect
392. C₂²×C₄	C₂×C₈×C₅₃	semidirect
424. C₄⋊C₄	C₂×C₈×C₅₃	semidirect
429. C₂²⋊C₄	C₂×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₅₃	2
C₂×C₅₃×#431	2
C₄×#121×C₅₃	2
C₅₃×#1954	2
C₅₃×#1939	2
C₅₃×#1923	2
C₅₃×#1869	2
C₈×C₄⋊C₄×C₅₃	2
C₈×C₂²⋊C₄×C₅₃	2
C₄×C₂²⋊C₈×C₅₃	2
#23083	53

Subgroups — C53×#23083 #27230595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23083 #27230595